Finite Group Databasebeta

← #33012949

Subgroups — #33012949

Order 18688.

1558 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988486C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×Dic73C211yesmaximal, characteristic
3#22608199C211yesmaximal, characteristic, complemented
4#22608188C211yesmaximal, characteristic, complemented
5#22608184C212yesmaximal, complemented
6#22608176C211yesmaximal, characteristic
7#22608176C212yesmaximal
8#22608163C212yesmaximal, complemented
9#22608156C211yesmaximal, characteristic, complemented
10#22608153C212yesmaximal
11C2×#399×C73C211yesmaximal, characteristic
12C24×Dic73C2211yescharacteristic
13C22×C4⋊Dic73C2212yes—
14C22×C4⋊Dic73C2211yescharacteristic
15C22×C22⋊Dic73C2211yescharacteristic
16C22×C22⋊Dic73C2212yes—
17C22×C4×Dic73C2211yescharacteristic
18C22×C4×Dic73C2211yescharacteristic
19#17988486—24no—
20C22⋊C4×Dic73—24no—
21C2×#14556498C2212yes—
22C2×#14556498C2212yes—
23C2×#14556498C2211yescharacteristic
24C2×#14556498C2212yes—
25C2×#14556498C2211yescharacteristic
26C2×#14556498C2211yescharacteristic
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C22×C4⋊C4×C73C2211yescharacteristic
33C22×C4⋊C4×C73C2212yes—
34C22×C22⋊C4×C73C2211yescharacteristic
35C22×C22⋊C4×C73C2212yes—
36#399×C73C418yescomplemented
37C2×#95×C73C2211yescharacteristic
38C23×Dic73—22no—
39C23×Dic73—22no—
40C23×Dic73—42no—
41C23×Dic73—42no—
42C23×Dic73Q814yes—
43C23×Dic73C2312yes—
44C23×Dic73C2312yes—
45C23×Dic73C2311yescharacteristic
46C23×Dic73C2311yescharacteristic
47C23×Dic73C2311yescharacteristic
48C23×Dic73C2311yescharacteristic
49C2×C4⋊Dic73—24no—
50C2×C4⋊Dic73—22no—
51C2×C4⋊Dic73—42no—
52C2×C4⋊Dic73—42no—
53C2×C4⋊Dic73—42no—
54C2×C22⋊Dic73—22no—
55C2×C22⋊Dic73—22no—
56C2×C22⋊Dic73—22no—
57C2×C22⋊Dic73—24no—
58C2×C22⋊Dic73—42no—
59C2×C22⋊Dic73—42no—
60C2×C4×Dic73—22no—
61C2×C4×Dic73—42no—
62C2×C4×Dic73—42no—
63C2×C4×Dic73D414yes—
64#14556498—22no—
65#14556498—24no—
66#14556498—24no—
67#14556498—24no—
68#14556498—22no—
69#14556498—22no—
70#14556498—42no—
71#14556498—42no—
72#14556498—42no—
73#14556498—42no—
74#14556498—42no—
75C25×C73C2311yescharacteristic
76C23×C4×C73C2312yes—
77C23×C4×C73C2311yescharacteristic
78C23×C4×C73C2312yes—
79C23×C4×C73C2311yescharacteristic
80C2×C4⋊C4×C73—22no—
81C2×C4⋊C4×C73—22no—
82C2×C4⋊C4×C73—42no—
83C2×C4⋊C4×C73—42no—
84C2×C4⋊C4×C73C2×C414yescomplemented
85C2×C4⋊C4×C73C2×C418yescomplemented
86C2×C22⋊C4×C73—22no—
87C2×C22⋊C4×C73—22no—
88C2×C22⋊C4×C73—42no—
89C2×C22⋊C4×C73—42no—
90C2×C22⋊C4×C73C2×C414yes—
91C2×C22⋊C4×C73C2×C418yes—
92#95×C73C2×C414yescomplemented
93C22×Dic73—22no—
94C22×Dic73—22no—
95C22×Dic73—21no—
96C22×Dic73—21no—
97C22×Dic73—21no—
98C22×Dic73—21no—
99C22×Dic73—21no—
100C22×Dic73—21no—
101C22×Dic73—22no—
102C22×Dic73—22no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—28no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73—48no—
123C22×Dic73—44no—
124C22×Dic73—44no—
125C22×Dic73—48no—
126C22×Dic73C2×D412yes—
127C22×Dic73C2×D412yes—
128C22×Dic73C2×Q812yes—
129C22×Dic73C2×Q812yes—
130C22×Dic73#4112yes—
131C22×Dic73#4112yes—
132C4⋊Dic73—48no—
133C4⋊Dic73—44no—
134C22⋊Dic73—28no—
135C22⋊Dic73—48no—
136C4×Dic73—44no—
137C4×Dic73—44no—
138C24×C73—21no—
139C24×C73—21no—
140C24×C73—21no—
141C24×C73—21no—
142C24×C73—22no—
143C24×C73—22no—
144C24×C73—22no—
145C24×C73—22no—
146C24×C73C22×C412yes—
147C24×C73C2×Q812yes—
148C24×C73#4112yes—
149C24×C73C2411yescharacteristic
150C22×C4×C73—22no—
151C22×C4×C73—22no—
152C22×C4×C73—21no—
153C22×C4×C73—22no—
154C22×C4×C73—21no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—22no—
167C22×C4×C73—22no—
168C22×C4×C73C22×C414yes—
169C22×C4×C73C22×C412yes—
170C22×C4×C73C22×C414yes—
171C22×C4×C73C22×C412yes—
172C22×C4×C73C2×D412yes—
173C22×C4×C73#4112yes—
174C4⋊C4×C73—28no—
175C4⋊C4×C73—48no—
176C22⋊C4×C73—28no—
177C22⋊C4×C73—48no—
178C23×C73—22no—
179C23×C73—22no—
180C23×C73—24no—
181C23×C73—24no—
182C23×C73—22no—
183C23×C73—22no—
184C23×C73—22no—
185C23×C73—22no—
186C23×C73—24no—
187C23×C73—24no—
188C23×C73—24no—
189C23×C73—24no—
190C23×C73—41no—
191C23×C73—41no—
192C23×C73—41no—
193C23×C73—41no—
194C23×C73—42no—
195C23×C73—42no—
196C23×C73—42no—
197C23×C73—42no—
198C23×C73—42no—
199C23×C73—42no—
200C23×C73C4×Q814yes—
201C23×C73C23×C411yescharacteristic
202C23×C73C22×D411yescharacteristic
203C23×C73C22×Q811yescharacteristic
204C23×C73C2×#4111yescharacteristic
205C23×C73C2×#4111yescharacteristic
206C23×C73C2×#4111yescharacteristic
207C23×C73C2×#4111yescharacteristic
208C23×C73H(5,𝔽2)12yes—
209C23×C73H(5,𝔽2)12yes—
210C23×C73H(5,𝔽2)12yes—
211C23×C732−1+412yes—
212C2×C4×C73—24no—
213C2×C4×C73—22no—
214C2×C4×C73—22no—
215C2×C4×C73—24no—
216C2×C4×C73—22no—
217C2×C4×C73—24no—
218C2×C4×C73—24no—
219C2×C4×C73—24no—
220C2×C4×C73—24no—
221C2×C4×C73—24no—
222C2×C4×C73—24no—
223C2×C4×C73—24no—
224C2×C4×C73—42no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—42no—
228C2×C4×C73—42no—
229C2×C4×C73—42no—
230C2×C4×C73—42no—
231C2×C4×C73—42no—
232C2×C4×C73—42no—
233C2×C4×C73—42no—
234C2×C4×C73—42no—
235C2×C4×C73C4×D414yes—
236C2×Dic73—22no—
237C2×Dic73—22no—
238C2×Dic73—24no—
239C2×Dic73—24no—
240C2×Dic73—22no—
241C2×Dic73—22no—
242C2×Dic73—22no—
243C2×Dic73—22no—
244C2×Dic73—22no—
245C2×Dic73—24no—
246C2×Dic73—24no—
247C2×Dic73—416no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—42no—
253C2×Dic73—42no—
254C2×Dic73—42no—
255C2×Dic73—42no—
256C2×Dic73—42no—
257C2×Dic73—42no—
258C2×Dic73—42no—
259C2×Dic73—42no—
260C2×Dic73—42no—
261C2×Dic73—42no—
262C2×Dic73—42no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73—42no—
266C2×Dic73—42no—
267C2×Dic73—42no—
268C2×Dic73—42no—
269C2×Dic73—42no—
270C2×Dic73C22⋊Q814yes—
271C22×C73—28no—
272C22×C73—24no—
273C22×C73—42no—
274C22×C73—42no—
275C22×C73—44no—
276C22×C73—42no—
277C22×C73—42no—
278C22×C73—42no—
279C22×C73—42no—
280C22×C73—44no—
281C22×C73—44no—
282C22×C73C2×C4×D411yescharacteristic
283C22×C73C2×C4×Q811yescharacteristic
284C22×C73C4×#4111yescharacteristic
285C22×C73#51812yes—
286C22×C73#52012yes—
287C22×C73C2×C22⋊Q811yescharacteristic
288C22×C73#52511yescharacteristic
289C22×C73#52911yescharacteristic
290C22×C73#53311yescharacteristic
291C22×C73#53411yescharacteristic
292C22×C73#53511yescharacteristic
293C22×C73#53511yescharacteristic
294C22×C73#53611yescharacteristic
295C22×C73#53812yes—
296C22×C73#53812yes—
297C22×C73#53912yes—
298C22×C73#53912yes—
299C22×C73#53912yes—
300C22×C73#54012yes—
301C22×C73#54012yes—
302C22×C73#54212yes—
303C22×C73#54312yes—
304C22×C73#54412yes—
305C4×C73—24no—
306C4×C73—42no—
307C4×C73—44no—
308C4×C73—44no—
309Dic73—24no—
310Dic73—42no—
311Dic73—42no—
312Dic73—42no—
313Dic73—82no—
314Dic73—82no—
315#14317—731nomaximal, Sylow
316C2×C73—44no—
317C2×C73C4×C22⋊Q811yescharacteristic
318C2×C73#232211yescharacteristic
319C2×C73#232811yescharacteristic
320C2×C73#232912yes—
321C2×C73#233512yes—
322C2×C73#262611yescharacteristic
323C2×C73#262811yescharacteristic
324C2×C73#263311yescharacteristic
325C2×C73#263511yescharacteristic
326C2×C73#264912yes—
327C2×C73#266712yes—
328C2×#399D7311yescharacteristic
329C2×#389—731no—
330C2×C4×C22⋊C4—731no—
331#1445—731no—
332#1442—731no—
333#1441—732no—
334#1439—731no—
335#1439—732no—
336#1436—732no—
337#1434—731no—
338C22⋊#95—732no—
339C73#1431711yescharacteristic, complemented, Sylow
340C24×C4—731no—
341C22×C4⋊C4—732no—
342C22×C4⋊C4—731no—
343C22×C4⋊C4C2×D7311yescharacteristic
344C22×C4⋊C4C2×D7312yes—
345C22×C22⋊C4—732no—
346C22×C22⋊C4—731no—
347C22×C22⋊C4C2×D7311yescharacteristic
348C22×C22⋊C4C2×D7312yes—
349C22×C42—731no—
350C22×C42—731no—
351#399Dic7318yescomplemented
352#389—1464no—
353C4×C22⋊C4—1464no—
354C2×#95—732no—
355C2×#95—732no—
356C2×#95—732no—
357C2×#95—732no—
358C2×#95—732no—
359C2×#95—732no—
360C2×#95—732no—
361C2×#95—731no—
362C2×#95—732no—
363C2×#95—731no—
364C2×#95—731no—
365C2×#95C2×D7311yescharacteristic
366C25C22×D7311yescharacteristic
367C23×C4—2922no—
368C23×C4—2922no—
369C23×C4—734no—
370C23×C4—731no—
371C23×C4—731no—
372C23×C4—731no—
373C23×C4—731no—
374C23×C4—732no—
375C23×C4—732no—
376C23×C4—1462no—
377C23×C4—1462no—
378C23×C4C22×D7312yes—
379C23×C4C22×D7311yescharacteristic
380C23×C4C22×D7312yes—
381C23×C4C22×D7311yescharacteristic
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—42no—
385C2×C4⋊C4—42no—
386C2×C4⋊C4—2922no—
387C2×C4⋊C4—2922no—
388C2×C4⋊C4—2922no—
389C2×C4⋊C4—1464no—
390C2×C4⋊C4—1462no—
391C2×C4⋊C4C2×Dic7314yescomplemented
392C2×C4⋊C4C2×Dic7318yescomplemented
393C2×C22⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—42no—
396C2×C22⋊C4—42no—
397C2×C22⋊C4—2922no—
398C2×C22⋊C4—2922no—
399C2×C22⋊C4—1464no—
400C2×C22⋊C4—1462no—
401C2×C22⋊C4—1462no—
402C2×C22⋊C4—1462no—
403C2×C22⋊C4C2×Dic7314yes—
404C2×C22⋊C4C2×Dic7318yes—
405C2×C42—2922no—
406C2×C42—2922no—
407C2×C42—734no—
408C2×C42—1462no—
409#95—2922no—
410#95—2922no—
411#95—2922no—
412#95—2922no—
413#95—2922no—
414#95—1464no—
415#95—1464no—
416#95—1462no—
417#95—1464no—
418#95—1462no—
419#95—1462no—
420#95C2×Dic7314yescomplemented
421C24—21no—
422C24—21no—
423C24—21no—
424C24—21no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24C22×Dic7312yes—
430C24Q8×D7312yes—
431C24#1192402912yes—
432C24C23×D7311yescharacteristic
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—2924no—
452C22×C4—2924no—
453C22×C4—2928no—
454C22×C4—2928no—
455C22×C4—732no—
456C22×C4—732no—
457C22×C4—732no—
458C22×C4—732no—
459C22×C4—732no—
460C22×C4—732no—
461C22×C4—1462no—
462C22×C4—1462no—
463C22×C4—1462no—
464C22×C4—1462no—
465C22×C4—1462no—
466C22×C4—1462no—
467C22×C4—1462no—
468C22×C4—1462no—
469C22×C4—1462no—
470C22×C4—1462no—
471C22×C4—1462no—
472C22×C4—1462no—
473C22×C4—1462no—
474C22×C4—1462no—
475C22×C4—1468no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1461no—
482C22×C4—1462no—
483C22×C4—1462no—
484C22×C4—1462no—
485C22×C4—1462no—
486C22×C4—1462no—
487C22×C4—1462no—
488C22×C4—1462no—
489C22×C4—1462no—
490C22×C4C22×Dic7314yes—
491C22×C4C22×Dic7312yes—
492C22×C4C22×Dic7314yes—
493C22×C4C22×Dic7312yes—
494C22×C4D4×D7312yes—
495C22×C4#1192402712yes—
496C4⋊C4—28no—
497C4⋊C4—48no—
498C4⋊C4—2924no—
499C4⋊C4—2928no—
500C22⋊C4—28no—
501C22⋊C4—48no—
502C22⋊C4—2928no—
503C22⋊C4—1468no—
504C42—2924no—
505C42—2924no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23Q8×Dic7314yes—
529C23C23×Dic7311yescharacteristic
530C23C2×D4×D7311yescharacteristic
531C23C2×Q8×D7311yescharacteristic
532C23#41×D7311yescharacteristic
533C23#41×D7311yescharacteristic
534C23C2×#1192402711yescharacteristic
535C23C2×#1192402911yescharacteristic
536C23#1455663612yes—
537C23#1455663612yes—
538C23#1455663712yes—
539C23#1455663812yes—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—2922no—
568C2×C4—2922no—
569C2×C4—2922no—
570C2×C4—2922no—
571C2×C4—29216no—
572C2×C4—2922no—
573C2×C4—2922no—
574C2×C4—2922no—
575C2×C4—2922no—
576C2×C4—2922no—
577C2×C4—2922no—
578C2×C4—2922no—
579C2×C4—2922no—
580C2×C4—2922no—
581C2×C4—2922no—
582C2×C4—2922no—
583C2×C4—2922no—
584C2×C4—2922no—
585C2×C4—2922no—
586C2×C4—734no—
587C2×C4—1462no—
588C2×C4—1462no—
589C2×C4—1464no—
590C2×C4—1464no—
591C2×C4—1462no—
592C2×C4—1462no—
593C2×C4—1462no—
594C2×C4—1462no—
595C2×C4—1462no—
596C2×C4—1464no—
597C2×C4—1464no—
598C2×C4D4×Dic7314yes—
599C22—28no—
600C22—24no—
601C22—42no—
602C22—44no—
603C22—42no—
604C22—44no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—44no—
609C22—42no—
610C22C2×D4×Dic7311yescharacteristic
611C22C2×Q8×Dic7311yescharacteristic
612C22#41×Dic7311yescharacteristic
613C22#1798906212yes—
614C22#1798907012yes—
615C22C22⋊Q8×D7311yescharacteristic
616C22#1798910011yescharacteristic
617C22#1798912311yescharacteristic
618C22#1798913911yescharacteristic
619C22#1798914811yescharacteristic
620C22#1798915011yescharacteristic
621C22#1798915011yescharacteristic
622C22#1798915911yescharacteristic
623C22#1798917312yes—
624C22#1798917712yes—
625C22#1798918012yes—
626C22#1798918112yes—
627C22#1798918612yes—
628C22#1798918912yes—
629C22#1798919112yes—
630C22#1798920512yes—
631C22#1798921212yes—
632C22#1798921612yes—
633C4—24no—
634C4—42no—
635C4—44no—
636C4—44no—
637C4—2922no—
638C4—2922no—
639C4—2922no—
640C4—5842no—
641C4—5842no—
642C4—1464no—
643C2—44no—
644C2C22⋊Q8×Dic7311yescharacteristic
645C2#2261260311yescharacteristic
646C2#2261265711yescharacteristic
647C2#2261266612yes—
648C2#2261272512yes—
649C2#2261573711yescharacteristic
650C2#2261575711yescharacteristic
651C2#2261580511yescharacteristic
652C2#2261582111yescharacteristic
653C2#2261593712yes—
654C2#2261612412yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC2×#399×C73C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC2×#399D73
hypocenterC73#14317
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C2×#17988486C2semidirect
3. #22608199C2semidirect
4. #22608188C2semidirect
5. #22608184C2semidirect
8. #22608163C2semidirect
9. #22608156C2semidirect
36. #399×C73C4semidirect
84. C2×C4⋊C4×C73C2×C4semidirect
85. C2×C4⋊C4×C73C2×C4semidirect
92. #95×C73C2×C4semidirect
339. C73#14317semidirect
351. #399Dic73semidirect
391. C2×C4⋊C4C2×Dic73semidirect
392. C2×C4⋊C4C2×Dic73semidirect
420. #95C2×Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179884862
C2×C22⋊C4×Dic732
#226081992
#226081882
#226081842
#226081762
#226081632
#226081562
#226081532
C2×#399×C732
#1431773