Finite Group Databasebeta

← #34026439

Subgroups — #34026439

Order 18688.

1364 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989639C212yesmaximal, complemented
2#22626056C212yesmaximal, complemented
3#22626055C212yesmaximal, complemented
4#22626054C212yesmaximal, complemented
5#22625788C211yesmaximal, characteristic, complemented
6#22625787C211yesmaximal, characteristic, complemented
7#22625785C211yesmaximal, characteristic, complemented
8#22625658C211yesmaximal, characteristic, complemented
9#22625657C211yesmaximal, characteristic, complemented
10C4×C22⋊#10596833C211yesmaximal, characteristic, complemented
11C2×C4×#11924027C211yesmaximal, characteristic
12#17989639—28no—
13C2×#14556658C2212yes—
14C2×#14556658C2212yes—
15C2×C4⋊#10596833C2212yes—
16C2×#14556648C2212yes—
17C2×#14556648C2212yes—
18C2×C22⋊#10596833C2212yes—
19C2×C22⋊#10596833C2212yes—
20C2×C22⋊#10596833C2212yes—
21C2×C22⋊#10596833C2212yes—
22C2×#14556646C2212yes—
23C2×#14556646C2212yes—
24C2×C4×#10596833C2212yes—
25#17989453C2211yescharacteristic
26#17989453C2211yescharacteristic
27#17989453C2211yescharacteristic
28#17989453C2211yescharacteristic
29C22×#11924027C411yescharacteristic
30C4×#11924027—28no—
31C2×C4×Dic146C411yescharacteristic
32C2×#14556556C2211yescharacteristic
33C2×D4×Dic73C2211yescharacteristic
34C2×C4×C22⋊D73C411yescharacteristic
35C2×C4×C22⋊D73C411yescharacteristic
36C2×#14556551C2211yescharacteristic
37C2×#14556551C2211yescharacteristic
38C2×#14556549C411yescharacteristic
39C2×#14556547C411yescharacteristic
40C2×#14556547C411yescharacteristic
41C22×C4×Dic73C2211yescharacteristic
42C22×C4×Dic73C2211yescharacteristic
43C2×C42×D73C2211yescharacteristic
44C2×C4×D4×C73C411yescharacteristic
45C22×#10596833C2312yes—
46C22×#10596833C2312yes—
47C22×#10596833C2312yes—
48C22×#10596833C2312yes—
49#14556658—24no—
50#14556658—24no—
51C4⋊#10596833—24no—
52#14556648—24no—
53#14556648—42no—
54C22⋊#10596833—24no—
55C22⋊#10596833—24no—
56C22⋊#10596833—42no—
57C22⋊#10596833—42no—
58#14556646—24no—
59#14556646—24no—
60C4×#10596833—24no—
61C2×#11924027—24no—
62C2×#11924027—24no—
63C2×#11924027—24no—
64C22×Dic146C2×C411yescharacteristic
65C22×C22⋊D73C2×C411yescharacteristic
66C22×C22⋊D73C2×C411yescharacteristic
67C23×Dic73C2×C411yescharacteristic
68C23×Dic73C2×C411yescharacteristic
69C23×Dic73C2311yescharacteristic
70C23×Dic73C2311yescharacteristic
71C22×C4×D73C2×C411yescharacteristic
72C22×C4×D73C2×C411yescharacteristic
73C22×C4×D73C2311yescharacteristic
74C4×Dic146—24no—
75#14556556D418yes—
76D4×Dic73—24no—
77C4×C22⋊D73—24no—
78C4×C22⋊D73—24no—
79#14556551—24no—
80#14556551—24no—
81#14556549—24no—
82#14556547—24no—
83#14556547—24no—
84C2×C4⋊Dic73C2×C411yescharacteristic
85C2×#11924007C2×C411yescharacteristic
86C2×#11924007C2×C411yescharacteristic
87C2×#11924007C2×C411yescharacteristic
88C2×#11924007C2×C411yescharacteristic
89C2×C22⋊Dic73C2×C411yescharacteristic
90C2×C22⋊Dic73C2×C411yescharacteristic
91C2×#11924005C2×C411yescharacteristic
92C2×#11924005C2×C411yescharacteristic
93C2×C4×Dic73—24no—
94C2×C4×Dic73—24no—
95C2×C4×Dic73C2×C411yescharacteristic
96C2×C4×Dic73C2×C411yescharacteristic
97C2×C4×Dic73C2×C411yescharacteristic
98C2×C4×Dic73C2×C411yescharacteristic
99C2×C4×Dic73C2×C411yescharacteristic
100C2×C4×Dic73C2311yescharacteristic
101C2×C4×Dic73C2311yescharacteristic
102C2×C4×Dic73C2311yescharacteristic
103C2×C4×Dic73C2311yescharacteristic
104C42×D73—24no—
105C22×D4×C73C2×C411yescharacteristic
106C23×C4×C73C2×C411yescharacteristic
107C23×C4×C73C2×C411yescharacteristic
108C4×D4×C73—24no—
109C2×C4⋊C4×C73C2×C411yescharacteristic
110C2×C22⋊C4×C73C2×C411yescharacteristic
111C2×C22⋊C4×C73C2×C411yescharacteristic
112C2×C42×C73C2×C411yescharacteristic
113C2×#10596833—22no—
114C2×#10596833—22no—
115C2×#10596833—22no—
116C2×#10596833—22no—
117C2×#10596833—22no—
118C2×#10596833—22no—
119C2×#10596833—22no—
120C2×#10596833—22no—
121C2×#10596833—22no—
122C2×#10596833—22no—
123C2×#10596833—22no—
124C2×#10596833#4114yescomplemented
125C23×D73C22×C411yescharacteristic
126#11924027—232no—
127C2×Dic146—22no—
128C2×Dic146—22no—
129C2×Dic146C22⋊C414yes—
130C2×C22⋊D73—22no—
131C2×C22⋊D73—22no—
132C2×C22⋊D73—22no—
133C2×C22⋊D73—22no—
134C2×C22⋊D73—22no—
135C2×C22⋊D73M1614yes—
136C22×Dic73—22no—
137C22×Dic73—22no—
138C22×Dic73—22no—
139C22×Dic73—22no—
140C22×Dic73—22no—
141C22×Dic73—22no—
142C22×Dic73—22no—
143C22×Dic73—22no—
144C22×Dic73—22no—
145C22×Dic73—21no—
146C22×Dic73—21no—
147C22×Dic73—21no—
148C22×Dic73—21no—
149C22×Dic73M1614yes—
150C22×Dic73C22×C411yescharacteristic
151C22×Dic73C22×C411yescharacteristic
152C22×Dic73C22×C411yescharacteristic
153C22×Dic73C22×C411yescharacteristic
154C22×Dic73C22×C411yescharacteristic
155C22×Dic73C22×C411yescharacteristic
156C22×Dic73C2411yescharacteristic
157C2×C4×D73—22no—
158C2×C4×D73—22no—
159C2×C4×D73—22no—
160C2×C4×D73—22no—
161C2×C4×D73—22no—
162C2×C4×D73—22no—
163C2×C4×D73—22no—
164C2×C4×D73—21no—
165C2×C4×D73—21no—
166C2×C4×D73—41no—
167C4⋊Dic73—22no—
168#11924007—22no—
169#11924007—22no—
170#11924007C22⋊C414yes—
171#11924007C22⋊C414yes—
172C22⋊Dic73—22no—
173C22⋊Dic73—22no—
174#11924005—22no—
175#11924005—22no—
176C4×Dic73—22no—
177C4×Dic73—22no—
178C4×Dic73—22no—
179C4×Dic73—22no—
180C4×Dic73—22no—
181C4×Dic73—22no—
182C4×Dic73C2×D414yes—
183C4×Dic73C2×D414yes—
184C4×Dic73C2×D414yes—
185C24×C73C22×C411yescharacteristic
186C24×C73C22×C411yescharacteristic
187C2×D4×C73—22no—
188C2×D4×C73—22no—
189C2×D4×C73—22no—
190C22×C4×C73—22no—
191C22×C4×C73—22no—
192C22×C4×C73—22no—
193C22×C4×C73—22no—
194C22×C4×C73—41no—
195C22×C4×C73—41no—
196C22×C4×C73C22×C411yescharacteristic
197C22×C4×C73C22×C411yescharacteristic
198C22×C4×C73C22×C411yescharacteristic
199C22×C4×C73C22×C411yescharacteristic
200C22×C4×C73C22×C411yescharacteristic
201C4⋊C4×C73C22⋊C414yes—
202C22⋊C4×C73—22no—
203C22⋊C4×C73—22no—
204C42×C73—22no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73—21no—
214C23×C73—21no—
215C23×C73—21no—
216C23×C73—21no—
217C23×C73C2×M1612yes—
218C23×C73C2×M1612yes—
219C23×C73C23×C411yescharacteristic
220C22×D73—21no—
221C22×D73—21no—
222C22×D73—21no—
223C22×D73—21no—
224C22×D73—21no—
225C22×D73—21no—
226C22×D73C2×M1612yes—
227D4×C73—28no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—44no—
240C2×C4×C73—44no—
241C2×C4×C73C2×C22⋊C412yes—
242C2×C4×C73C2×C22⋊C412yes—
243C2×C4×C73C2×C22⋊C412yes—
244C2×C4×C73#13112yes—
245C22⋊D73—28no—
246C22⋊D73—28no—
247C2×Dic73—28no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—28no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—42no—
262C2×Dic73—42no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73C2×C22⋊C412yes—
266C2×Dic73C2×C22⋊C412yes—
267C2×Dic73C2×C22⋊C412yes—
268C2×Dic73C2×M1612yes—
269C2×Dic73C2×M1612yes—
270C2×Dic73C2×M1612yes—
271C2×Dic73C22×D412yes—
272C2×Dic73C2×#4112yes—
273C2×Dic73H(5,𝔽2)11yescharacteristic
274C2×Dic732−1+411yescharacteristic
275C4×D73—28no—
276C4×D73—44no—
277C4×D73—42no—
278C4×D73—42no—
279Dic146—28no—
280#10596833—24no—
281#10596833—42no—
282#10596833—42no—
283#10596833—42no—
284C22×C73—22no—
285C22×C73—22no—
286C22×C73—22no—
287C22×C73—22no—
288C22×C73—22no—
289C22×C73—22no—
290C22×C73—41no—
291C22×C73—41no—
292C22×C73—41no—
293C22×C73—41no—
294C22×C73—41no—
295C22×C73—41no—
296C22×C73—41no—
297C22×C73—41no—
298C22×C73C22×C22⋊C411yescharacteristic
299C22×C73C4×#4111yescharacteristic
300C22×C73#52011yescharacteristic
301C22×C73C22×M1611yescharacteristic
302C22×C73C22×M1611yescharacteristic
303C22×C73C2×#13111yescharacteristic
304C22×C73#56811yescharacteristic
305C2×D73—22no—
306C2×D73—22no—
307C2×D73—22no—
308C2×D73—41no—
309C2×D73—41no—
310C2×D73—41no—
311C2×D73—41no—
312C4×C73—22no—
313C4×C73—22no—
314C4×C73—22no—
315C4×C73—22no—
316C4×C73—41no—
317Dic73—22no—
318Dic73—22no—
319Dic73—22no—
320Dic73—41no—
321Dic73—41no—
322Dic73#40714yes—
323Dic73#54812yes—
324Dic73#55012yes—
325#22532—731nomaximal, Sylow
326C2×C73—42no—
327C2×C73—42no—
328C2×C73#234111yescharacteristic
329C2×C73C2×#40711yescharacteristic
330C2×C73#288011yescharacteristic
331C2×C73#296511yescharacteristic
332C2×C73#297011yescharacteristic
333C2×C73#298911yescharacteristic
334C2×C73#299011yescharacteristic
335D73—42no—
336C2×C4×#41—731no—
337C2×#446—732no—
338#1955—732no—
339#1955—732no—
340#1954—732no—
341#1797—731no—
342#1797—731no—
343#1797—731no—
344#1749—731no—
345#1749—731no—
346C4×C22⋊C8—731no—
347C73#2253211yescharacteristic, complemented, Sylow
348C22×#41—731no—
349C4×#41—1468no—
350C2×C4×Q8—731no—
351C2×C4×Q8—731no—
352C2×C4×D4—731no—
353C2×C4×D4—731no—
354C2×C4×D4—731no—
355C2×C4×D4—731no—
356C2×C4×D4—731no—
357C2×C4×D4#6440811yescharacteristic
358C2×#117—731no—
359C2×#117—731no—
360C2×#117—731no—
361C22×C42—731no—
362C22×C42—731no—
363C22×C42—731no—
364#446—1468no—
365C2×C4⋊C8—732no—
366C2×C4⋊C8—732no—
367C2×C4⋊C8—732no—
368C2×C22⋊C8—732no—
369C2×C22⋊C8—732no—
370C2×C22⋊C8—732no—
371C2×C22⋊C8—732no—
372C2×C22⋊C8—732no—
373C2×C22⋊C8—732no—
374C2×#97—732no—
375C2×#97—732no—
376C2×C4×C8—732no—
377#336—731no—
378#336—731no—
379#336—731no—
380#336—731no—
381C2×#41—1464no—
382C2×#41—1464no—
383C2×#41—1464no—
384C22×Q8—731no—
385C22×D4—731no—
386C22×D4—731no—
387C22×D4C2×#6440811yescharacteristic
388C23×C4—731no—
389C23×C4—731no—
390C23×C4—731no—
391C23×C4—731no—
392C23×C4—731no—
393C23×C4—731no—
394C23×C4—731no—
395C23×C4C2×#6440811yescharacteristic
396C23×C4C2×#6440811yescharacteristic
397C22×C8—732no—
398C22×C8—732no—
399C22×C8—732no—
400C22×C8—732no—
401C4×Q8—738no—
402C4×Q8—1464no—
403C4×D4—24no—
404C4×D4—1464no—
405C4×D4—1464no—
406C4×D4—1464no—
407C4×D4—1464no—
408C4×D4—1464no—
409#117—1464no—
410#117—1464no—
411#117—1464no—
412C2×C4⋊C4—731no—
413C2×C4⋊C4—731no—
414C2×C4⋊C4—731no—
415C2×C4⋊C4—731no—
416C2×C4⋊C4—731no—
417C2×C4⋊C4C2×#6440811yescharacteristic
418C2×C22⋊C4—731no—
419C2×C22⋊C4—731no—
420C2×C22⋊C4—731no—
421C2×C22⋊C4—731no—
422C2×C22⋊C4C2×#6440811yescharacteristic
423C2×C22⋊C4C2×#6440811yescharacteristic
424C2×C42—731no—
425C2×C42—731no—
426C2×C42—731no—
427C2×C42—731no—
428C2×C42—731no—
429C2×C42—731no—
430C2×C42—731no—
431C2×C42—731no—
432C2×C42—731no—
433C2×C42—1464no—
434C2×C42—1464no—
435C2×C42—1464no—
436C2×C42C2×#6440811yescharacteristic
437C4⋊C8—1464no—
438C4⋊C8—1464no—
439C4⋊C8—1464no—
440C22⋊C8—2922no—
441C22⋊C8—2922no—
442C22⋊C8—2922no—
443C22⋊C8—1464no—
444C22⋊C8—1464no—
445C22⋊C8—1464no—
446#97—1464no—
447#97—1464no—
448C4×C8—1464no—
449C24—731no—
450C24C22×#6440811yescharacteristic
451C24C22×#6440811yescharacteristic
452#41—14632no—
453C2×Q8—734no—
454C2×Q8—1462no—
455C2×Q8—1462no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—734no—
460C2×D4—1462no—
461C2×D4—1462no—
462C2×D4—1462no—
463C2×D4—1462no—
464C2×D4—1462no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—2921no—
472C22×C4—731no—
473C22×C4—731no—
474C22×C4—734no—
475C22×C4—731no—
476C22×C4—731no—
477C22×C4—731no—
478C22×C4—731no—
479C22×C4—731no—
480C22×C4—1461no—
481C22×C4—1462no—
482C22×C4—1462no—
483C22×C4—1462no—
484C22×C4—1461no—
485C22×C4—1462no—
486C22×C4—1462no—
487C22×C4—1461no—
488C22×C4—1461no—
489C22×C4—1462no—
490C22×C4—1462no—
491C22×C4—1462no—
492C22×C4—1462no—
493C22×C4—1461no—
494C22×C4—1461no—
495C22×C4—1462no—
496C22×C4—1462no—
497C22×C4—1462no—
498C22×C4—1462no—
499C22×C4—1462no—
500C22×C4—1462no—
501C22×C4—1462no—
502C22×C4C22×#6440811yescharacteristic
503C22×C4C22×#6440811yescharacteristic
504C22×C4C22×#6440811yescharacteristic
505C22×C4C22×#6440811yescharacteristic
506C22×C4C22×#6440811yescharacteristic
507C2×C8—734no—
508C2×C8—1462no—
509C2×C8—1462no—
510C2×C8—1462no—
511C2×C8—1462no—
512C2×C8—1462no—
513C2×C8—1462no—
514C2×C8—1462no—
515C2×C8—1462no—
516C2×C8—1462no—
517C2×C8—1462no—
518C2×C8—1462no—
519C4⋊C4—734no—
520C4⋊C4—734no—
521C4⋊C4—1462no—
522C4⋊C4—1462no—
523C4⋊C4—1462no—
524C4⋊C4C22⋊#6440814yes—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—1462no—
528C22⋊C4—1462no—
529C22⋊C4—1462no—
530C22⋊C4—1462no—
531C42—22no—
532C42—734no—
533C42—734no—
534C42—734no—
535C42—1462no—
536C42—1462no—
537C42—1462no—
538C42—1462no—
539C42—1462no—
540C42—1462no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—732no—
554C23—1461no—
555C23—1461no—
556C23—1461no—
557C23—1461no—
558C23—1461no—
559C23—1461no—
560C23D73⋊M1612yes—
561C23D73⋊M1612yes—
562C23C23×#6440811yescharacteristic
563Q8—1468no—
564D4—28no—
565D4—1468no—
566D4—1468no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—2924no—
581C2×C4—2922no—
582C2×C4—2922no—
583C2×C4—2922no—
584C2×C4—2922no—
585C2×C4—2922no—
586C2×C4—2922no—
587C2×C4—732no—
588C2×C4—732no—
589C2×C4—732no—
590C2×C4—731no—
591C2×C4—731no—
592C2×C4—732no—
593C2×C4—732no—
594C2×C4—732no—
595C2×C4—732no—
596C2×C4—732no—
597C2×C4—1468no—
598C2×C4—1461no—
599C2×C4—1461no—
600C2×C4—1461no—
601C2×C4—1461no—
602C2×C4—1461no—
603C2×C4—1461no—
604C2×C4—1468no—
605C2×C4—1468no—
606C2×C4—1461no—
607C2×C4—1461no—
608C2×C4—1461no—
609C2×C4—1461no—
610C2×C4—1461no—
611C2×C4—1461no—
612C2×C4C2×C22⋊#6440812yes—
613C2×C4C2×C22⋊#6440812yes—
614C2×C4C2×C22⋊#6440812yes—
615C2×C4#1455667812yes—
616C8—2922no—
617C8—2922no—
618C8—2922no—
619C8—1464no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—2921no—
635C22—2921no—
636C22—2921no—
637C22—2921no—
638C22—1462no—
639C22—1462no—
640C22—1462no—
641C22C22×C22⋊#6440811yescharacteristic
642C22#41×#6440811yescharacteristic
643C22#1798965611yescharacteristic
644C22C2×D73⋊M1611yescharacteristic
645C22C2×D73⋊M1611yescharacteristic
646C22C2×#1455667811yescharacteristic
647C22#1798966511yescharacteristic
648C4—22no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—41no—
653C4—2921no—
654C4—2921no—
655C4—732no—
656C4—732no—
657C4—734no—
658C4—1462no—
659C4—1462no—
660C4—1462no—
661C2—42no—
662C2—42no—
663C2—2922no—
664C2#2262643611yescharacteristic
665C2D73⋊#40711yescharacteristic
666C2#2262651911yescharacteristic
667C2#2262674111yescharacteristic
668C2#2262677111yescharacteristic
669C2#2262685111yescharacteristic
670C2#2262685611yescharacteristic
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
FittingC2×C4×D4×C73C4
prime radicalC22×#11924027C4
cosocleC22×Dic73C24
hypercenterC2×C4×D4#64408
hypocenterC73#22532
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#17989639C2semidirect
2. #22626056C2semidirect
3. #22626055C2semidirect
4. #22626054C2semidirect
5. #22625788C2semidirect
6. #22625787C2semidirect
7. #22625785C2semidirect
8. #22625658C2semidirect
9. #22625657C2semidirect
10. C4×C22⋊#10596833C2semidirect
124. C2×#10596833#41semidirect
347. C73#22532semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179896392
#226260562
#226260552
#226260542
#226257882
#226257872
#226257852
#226256582
#226256572
C4×C22⋊#105968332
C2×C4×#119240272
#2253273