Finite Group Databasebeta

← #33007739

Subgroups — #33007739

Order 18688.

1790 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556556C211yesmaximal, characteristic
2C2×C4⋊C4×Dic73C211yesmaximal, characteristic
3C2×#17988432C211yesmaximal, characteristic
4#22608201C212yesmaximal
5#22608197C212yesmaximal
6#22608185C211yesmaximal, characteristic
7#22608184C211yesmaximal, characteristic
8#22608181C211yesmaximal, characteristic
9#95×Dic73C212yesmaximal
10#95×Dic73C212yesmaximal
11C73×#1441C211yesmaximal, characteristic
12C23×Dic146C2211yescharacteristic
13C2×#14556556—24no—
14C2×#14556556—24no—
15C2×#14556556C418yescomplemented
16C22×C4⋊Dic73C2211yescharacteristic
17C22×#11924007C2211yescharacteristic
18C22×#11924007C2211yescharacteristic
19C22×C4×Dic73C2212yes—
20C22×C4×Dic73C2211yescharacteristic
21C22×C4×Dic73C2211yescharacteristic
22C22×C4×Dic73C2212yes—
23C22×C4×Dic73C2211yescharacteristic
24C4⋊C4×Dic73—24no—
25#17988432—24no—
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2211yescharacteristic
33C2×#14556498C2212yes—
34C2×#14556498C2212yes—
35C22×C4⋊C4×C73C2211yescharacteristic
36C22×C4⋊C4×C73C2211yescharacteristic
37C2×#95×C73C2212yes—
38C2×#95×C73C2212yes—
39C2×#95×C73C2211yescharacteristic
40C22×Dic146—22no—
41C22×Dic146—22no—
42C22×Dic146—22no—
43C22×Dic146—22no—
44C22×Dic146—22no—
45C22×Dic146—22no—
46C22×Dic146C2×C414yes—
47C23×Dic73C2311yescharacteristic
48C23×Dic73C2312yes—
49C23×Dic73C2312yes—
50C23×Dic73C2311yescharacteristic
51C23×Dic73C2311yescharacteristic
52C23×Dic73C2311yescharacteristic
53#14556556—232no—
54C2×C4⋊Dic73—24no—
55C2×C4⋊Dic73—22no—
56C2×#11924007—22no—
57C2×#11924007—22no—
58C2×#11924007—22no—
59C2×#11924007—22no—
60C2×#11924007C2×C414yes—
61C2×#11924007C2×C414yes—
62C2×C4×Dic73—24no—
63C2×C4×Dic73—22no—
64C2×C4×Dic73—22no—
65C2×C4×Dic73—22no—
66C2×C4×Dic73—22no—
67C2×C4×Dic73—24no—
68C2×C4×Dic73—24no—
69C2×C4×Dic73—24no—
70C2×C4×Dic73—24no—
71C2×C4×Dic73C2×C414yes—
72C2×C4×Dic73C2×C414yes—
73C2×C4×Dic73C2×C414yes—
74C2×C4×Dic73D414yes—
75C2×C4×Dic73D414yes—
76C2×C4×Dic73Q818yescomplemented
77#14556498—24no—
78#14556498—24no—
79#14556498—24no—
80#14556498—24no—
81#14556498—24no—
82#14556498—24no—
83#14556498—22no—
84#14556498—24no—
85#14556498—24no—
86C23×C4×C73C2311yescharacteristic
87C23×C4×C73C2312yes—
88C23×C4×C73C2312yes—
89C23×C4×C73C2311yescharacteristic
90C23×C4×C73C2311yescharacteristic
91C2×C4⋊C4×C73—22no—
92C2×C4⋊C4×C73—22no—
93C2×C4⋊C4×C73—22no—
94C2×C4⋊C4×C73—24no—
95C2×C4⋊C4×C73C2×C414yes—
96#95×C73—22no—
97#95×C73—24no—
98#95×C73—24no—
99C2×Dic146—28no—
100C2×Dic146—28no—
101C2×Dic146—28no—
102C2×Dic146—28no—
103C2×Dic146—28no—
104C2×Dic146—28no—
105C2×Dic146—28no—
106C22×Dic73—21no—
107C22×Dic73—21no—
108C22×Dic73—21no—
109C22×Dic73—21no—
110C22×Dic73—21no—
111C22×Dic73—21no—
112C22×Dic73—21no—
113C22×Dic73—21no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—21no—
117C22×Dic73—21no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73—22no—
127C22×Dic73—22no—
128C22×Dic73—22no—
129C22×Dic73—22no—
130C22×Dic73C22×C412yes—
131C22×Dic73C22×C412yes—
132C22×Dic73C22×C412yes—
133C22×Dic73C22×C412yes—
134C22×Dic73C2×D412yes—
135C22×Dic73C2×D412yes—
136C22×Dic73C2×D412yes—
137C22×Dic73C2×D412yes—
138C22×Dic73C2×Q814yes—
139C22×Dic73C2×Q814yes—
140C22×Dic73#4114yes—
141C22×Dic73#4112yes—
142C22×Dic73#4112yes—
143C22×Dic73#4114yes—
144C4⋊Dic73—28no—
145#11924007—28no—
146#11924007—28no—
147C4×Dic73—28no—
148C4×Dic73—28no—
149C4×Dic73—216no—
150C4×Dic73—48no—
151C4×Dic73C22⋊C4116yes—
152C24×C73C2411yescharacteristic
153C22×C4×C73—21no—
154C22×C4×C73—21no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—21no—
158C22×C4×C73—21no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—22no—
167C22×C4×C73—22no—
168C22×C4×C73—22no—
169C22×C4×C73—22no—
170C22×C4×C73—22no—
171C22×C4×C73—22no—
172C22×C4×C73—22no—
173C22×C4×C73—22no—
174C22×C4×C73—22no—
175C22×C4×C73C22×C412yes—
176C22×C4×C73C22×C412yes—
177C22×C4×C73C22×C412yes—
178C22×C4×C73C2×D412yes—
179C22×C4×C73C2×D412yes—
180C22×C4×C73C2×Q814yes—
181C22×C4×C73#4114yes—
182C22×C4×C73#4112yes—
183C22×C4×C73#4112yes—
184C4⋊C4×C73—28no—
185C4⋊C4×C73—28no—
186C23×C73C23×C411yescharacteristic
187C23×C73C22×D411yescharacteristic
188C23×C73C22×D411yescharacteristic
189C23×C73C22×Q812yes—
190C23×C73C2×#4111yescharacteristic
191C23×C73C2×#4111yescharacteristic
192C23×C73C2×#4112yes—
193C23×C73C2×#4112yes—
194C23×C73C2×#4111yescharacteristic
195C23×C73C2×#4112yes—
196C23×C73C2×#4111yescharacteristic
197C2×C4×C73—22no—
198C2×C4×C73—22no—
199C2×C4×C73—24no—
200C2×C4×C73—22no—
201C2×C4×C73—22no—
202C2×C4×C73—22no—
203C2×C4×C73—22no—
204C2×C4×C73—22no—
205C2×C4×C73—22no—
206C2×C4×C73—24no—
207C2×C4×C73—22no—
208C2×C4×C73—24no—
209C2×C4×C73—24no—
210C2×C4×C73—24no—
211C2×C4×C73—24no—
212C2×C4×C73—24no—
213C2×C4×C73—24no—
214C2×C4×C73—24no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—42no—
218C2×C4×C73—42no—
219C2×C4×C73—42no—
220C2×C4×C73—42no—
221C2×C4×C73—42no—
222C2×C4×C73—42no—
223C2×C4×C73—42no—
224C2×C4×C73—42no—
225C2×C4×C73—42no—
226C2×C4×C73C2×C22⋊C414yes—
227C2×C4×C73#11714yes—
228C2×Dic73—22no—
229C2×Dic73—22no—
230C2×Dic73—22no—
231C2×Dic73—22no—
232C2×Dic73—22no—
233C2×Dic73—22no—
234C2×Dic73—22no—
235C2×Dic73—22no—
236C2×Dic73—22no—
237C2×Dic73—22no—
238C2×Dic73—22no—
239C2×Dic73—22no—
240C2×Dic73—22no—
241C2×Dic73—24no—
242C2×Dic73—24no—
243C2×Dic73—24no—
244C2×Dic73—24no—
245C2×Dic73—24no—
246C2×Dic73—24no—
247C2×Dic73—24no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—42no—
251C2×Dic73—42no—
252C2×Dic73—42no—
253C2×Dic73—42no—
254C2×Dic73—42no—
255C2×Dic73—42no—
256C2×Dic73—42no—
257C2×Dic73—42no—
258C2×Dic73—42no—
259C2×Dic73C2×C22⋊C414yes—
260C2×Dic73C2×C22⋊C414yes—
261C2×Dic73#11714yes—
262C2×Dic73C4×Q818yes—
263C2×Dic73C22⋊Q818yes—
264C2×Dic73#12414yes—
265C2×Dic73C4⋊Q814yes—
266Dic146—232no—
267C22×C73C22×C22⋊C411yescharacteristic
268C22×C73C2×#11711yescharacteristic
269C22×C73C2×#11711yescharacteristic
270C22×C73C2×C4×Q812yes—
271C22×C73C4×#4112yes—
272C22×C73C2×C22⋊Q812yes—
273C22×C73#52511yescharacteristic
274C22×C73#52511yescharacteristic
275C22×C73#52511yescharacteristic
276C22×C73#52511yescharacteristic
277C22×C73#52511yescharacteristic
278C22×C73#52511yescharacteristic
279C22×C73C2×#12411yescharacteristic
280C22×C73#52912yes—
281C22×C73#52912yes—
282C22×C73#52912yes—
283C22×C73#52912yes—
284C22×C73C2×C4⋊Q811yescharacteristic
285C22×C73#53212yes—
286C22×C73#53312yes—
287C22×C73#53312yes—
288C22×C73#53312yes—
289C22×C73#53312yes—
290C4×C73—24no—
291C4×C73—24no—
292C4×C73—44no—
293C4×C73—44no—
294C4×C73—42no—
295Dic73—24no—
296Dic73—24no—
297Dic73—44no—
298Dic73—44no—
299Dic73—42no—
300Dic73#39118yes—
301#13850—731nomaximal, Sylow
302C2×C73#228011yescharacteristic
303C2×C73C2×#39111yescharacteristic
304C2×C73#229512yes—
305C2×C73#229512yes—
306C2×C73#229711yescharacteristic
307C2×C73#239612yes—
308C2×C73#239612yes—
309C2×C73#239811yescharacteristic
310C2×C73#239811yescharacteristic
311C2×C73#239911yescharacteristic
312C2×C73#239911yescharacteristic
313C22×C4×Q8—731no—
314C2×C4×C4⋊C4—731no—
315C2×#376—731no—
316#1446—732no—
317#1445—732no—
318#1441—731no—
319#1441—731no—
320#1441—731no—
321#1441D7311yescharacteristic
322C4×#95—732no—
323C4×#95—732no—
324C73#1385011yescharacteristic, complemented, Sylow
325C23×Q8—731no—
326C2×C4×Q8—738no—
327C2×C4×Q8—1464no—
328C2×C4×Q8—1464no—
329C22×C4⋊C4—731no—
330C22×C4⋊C4—731no—
331C22×C4⋊C4—731no—
332C22×C4⋊C4C2×D7311yescharacteristic
333C22×C4⋊C4C2×D7311yescharacteristic
334C22×C42—731no—
335C22×C42—731no—
336C22×C42—731no—
337C22×C42—732no—
338C22×C42—732no—
339C4×C4⋊C4—1464no—
340#376—1464no—
341C2×#95—732no—
342C2×#95—732no—
343C2×#95—732no—
344C2×#95—732no—
345C2×#95—732no—
346C2×#95—732no—
347C2×#95—731no—
348C2×#95—732no—
349C2×#95—732no—
350C2×#95C2×D7312yes—
351C2×#95C2×D7312yes—
352C2×#95C2×D7311yescharacteristic
353C22×Q8—734no—
354C22×Q8—1462no—
355C22×Q8—1462no—
356C22×Q8—1462no—
357C22×Q8—1462no—
358C22×Q8—1462no—
359C22×Q8—1462no—
360C23×C4—731no—
361C23×C4—731no—
362C23×C4—731no—
363C23×C4—731no—
364C23×C4—732no—
365C23×C4—732no—
366C23×C4C22×D7311yescharacteristic
367C23×C4C22×D7312yes—
368C23×C4C22×D7312yes—
369C23×C4C22×D7311yescharacteristic
370C23×C4C22×D7311yescharacteristic
371C4×Q8—14632no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—24no—
376C2×C4⋊C4—734no—
377C2×C4⋊C4—734no—
378C2×C4⋊C4—1462no—
379C2×C4⋊C4—1462no—
380C2×C4⋊C4—1464no—
381C2×C4⋊C4—1462no—
382C2×C4⋊C4—1462no—
383C2×C4⋊C4—1462no—
384C2×C4⋊C4C4×D7314yes—
385C2×C42—734no—
386C2×C42—734no—
387C2×C42—734no—
388C2×C42—734no—
389C2×C42—734no—
390C2×C42—738no—
391C2×C42—1464no—
392C2×C42—1464no—
393C2×C42—1464no—
394C2×C42—1462no—
395C2×C42—1462no—
396C2×C42—1462no—
397C2×C42—1462no—
398C2×C42—1464no—
399C2×C42—1464no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95—1464no—
404#95—1464no—
405#95—1464no—
406#95—1464no—
407#95—1464no—
408#95—1464no—
409#95—1462no—
410#95—1464no—
411#95—1464no—
412C24C23×D7311yescharacteristic
413C2×Q8—1468no—
414C2×Q8—1468no—
415C2×Q8—1468no—
416C2×Q8—1468no—
417C2×Q8—1468no—
418C2×Q8—1468no—
419C2×Q8—1468no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—732no—
443C22×C4—732no—
444C22×C4—732no—
445C22×C4—734no—
446C22×C4—734no—
447C22×C4—732no—
448C22×C4—732no—
449C22×C4—732no—
450C22×C4—732no—
451C22×C4—732no—
452C22×C4—732no—
453C22×C4—732no—
454C22×C4—734no—
455C22×C4—734no—
456C22×C4—1462no—
457C22×C4—1462no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4—1462no—
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—1461no—
473C22×C4—1461no—
474C22×C4—1461no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4C2×C4×D7312yes—
481C22×C4C2×C4×D7312yes—
482C22×C4C2×C4×D7312yes—
483C22×C4D4×D7312yes—
484C22×C4D4×D7312yes—
485C22×C4Q8×D7314yes—
486C22×C4#1192402714yes—
487C22×C4#1192402712yes—
488C22×C4#1192402912yes—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—1468no—
492C4⋊C4—1468no—
493C4⋊C4—1468no—
494C42—2928no—
495C42—7316no—
496C42—14616no—
497C42—1468no—
498C42—1468no—
499C23C22×C4×D7311yescharacteristic
500C23C2×D4×D7311yescharacteristic
501C23C2×D4×D7311yescharacteristic
502C23C2×Q8×D7312yes—
503C23#41×D7311yescharacteristic
504C23#41×D7312yes—
505C23#41×D7312yes—
506C23#41×D7311yescharacteristic
507C23C2×#1192402712yes—
508C23C2×#1192402711yescharacteristic
509C23C2×#1192402911yescharacteristic
510Q8—14632no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—2922no—
541C2×C4—2922no—
542C2×C4—2922no—
543C2×C4—2922no—
544C2×C4—2922no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—2922no—
548C2×C4—2922no—
549C2×C4—2922no—
550C2×C4—2922no—
551C2×C4—734no—
552C2×C4—734no—
553C2×C4—738no—
554C2×C4—738no—
555C2×C4—734no—
556C2×C4—734no—
557C2×C4—734no—
558C2×C4—1464no—
559C2×C4—1464no—
560C2×C4—1464no—
561C2×C4—1464no—
562C2×C4—1462no—
563C2×C4—1464no—
564C2×C4—1464no—
565C2×C4—1464no—
566C2×C4—1462no—
567C2×C4—1462no—
568C2×C4—1462no—
569C2×C4—1462no—
570C2×C4—1462no—
571C2×C4—1462no—
572C2×C4—1462no—
573C2×C4—1462no—
574C2×C4—1462no—
575C2×C4—1462no—
576C2×C4—1462no—
577C2×C4—1462no—
578C2×C4C22⋊C4×D7314yes—
579C2×C4#1455654914yes—
580C22C2×C22⋊C4×D7311yescharacteristic
581C22#117×D7311yescharacteristic
582C22C2×#1455654911yescharacteristic
583C22C4×Q8×D7312yes—
584C22C4×#1192402712yes—
585C22C22⋊Q8×D7312yes—
586C22#1798909311yescharacteristic
587C22#1798909511yescharacteristic
588C22#1798909711yescharacteristic
589C22#1798909711yescharacteristic
590C22#1798910011yescharacteristic
591C22#1798910011yescharacteristic
592C22#124×D7311yescharacteristic
593C22#1798911412yes—
594C22#1798911812yes—
595C22#1798912012yes—
596C22#1798912212yes—
597C22C4⋊Q8×D7311yescharacteristic
598C22#1798913212yes—
599C22#1798913912yes—
600C22#1798913912yes—
601C22#1798914212yes—
602C22#1798914312yes—
603C4—24no—
604C4—24no—
605C4—42no—
606C4—44no—
607C4—44no—
608C4—2922no—
609C4—2924no—
610C4—2924no—
611C4—738no—
612C4—1464no—
613C4—1464no—
614C2#2261231311yescharacteristic
615C2#391×D7311yescharacteristic
616C2#2261240212yes—
617C2#2261240412yes—
618C2#2261241611yescharacteristic
619C2#2261320612yes—
620C2#2261320812yes—
621C2#2261321911yescharacteristic
622C2#2261322011yescharacteristic
623C2#2261322111yescharacteristic
624C2#2261322311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1441C2
socle, prime radical, cosocleC24×C73C24
hypercenter#1441D73
hypocenterC73#13850
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
15. C2×#14556556C4semidirect
76. C2×C4×Dic73Q8semidirect
324. C73#13850semidirect

Maximal subgroups

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

subgroupindex
C22×#145565562
C2×C4⋊C4×Dic732
C2×#179884322
#226082012
#226081972
#226081852
#226081842
#226081812
#95×Dic732
C73×#14412
#1385073