Finite Group Databasebeta

← #33102692

Subgroups — #33102692

Order 18688.

1004 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11924028C211yesmaximal, characteristic
2Q8×#11924012C211yesmaximal, characteristic, complemented
3#22610912C212yesmaximal, complemented
4#22610908C211yesmaximal, characteristic, complemented
5#22610874C211yesmaximal, characteristic, complemented
6#22610874C212yesmaximal, complemented
7#22609952C211yesmaximal, characteristic
8#22609952C212yesmaximal
9#22609742C212yesmaximal
10C4×#14556519C211yesmaximal, characteristic
11C73×#2295C211yesmaximal, characteristic, complemented
12Q8×Dic146C411yescharacteristic, complemented
13#17989315C411yescharacteristic, complemented
14#17989315C412yescomplemented
15#17989297C411yescharacteristic, complemented
16#17989297C412yescomplemented
17#17989292—22no—
18#17989292—21no—
19Q8×D292C411yescharacteristic, complemented
20Q8×C22⋊D73—21no—
21#17989144C2211yescharacteristic
22#17989144C2212yes—
23#17989141—21no—
24#17989141—22no—
25C4×#11924028C2211yescharacteristic
26C4×#11924028C2212yes—
27C4×Q8×D73—21no—
28#17988718—22no—
29#17988718—21no—
30Q8×#10596831—21no—
31#17988611C2211yescharacteristic
32#17988611C2212yes—
33C2×#14556519C2211yescharacteristic
34C2×#14556519C2212yes—
35C2×#14556519C2211yescharacteristic
36C4×#11924012C2211yescharacteristic
37C4×#11924012C2212yes—
38#17988347C2212yes—
39#17988347C2212yes—
40#17988347C2212yes—
41#17988347C2211yescharacteristic
42#17988347C2211yescharacteristic
43#17988347C2212yes—
44#17988347C2212yes—
45C2×C4×Q8×C73C2211yescharacteristic
46#391×C73—21no—
47#389×C73—21no—
48#387×C73—22no—
49C4×C4⋊C4×C73C2211yescharacteristic
50C4×C4⋊C4×C73C2212yes—
51#376×C73C2212yes—
52C43×C73C2211yescharacteristic
53C2×#11924028C2311yescharacteristic
54C2×Q8×D73—21no—
55C2×#11924012C2312yes—
56C2×#11924012C2311yescharacteristic
57C2×#11924012C2311yescharacteristic
58#14556591C2×C411yescharacteristic
59#14556591C2×C412yes—
60#14556591C2×C411yescharacteristic
61#14556591C2×C412yes—
62#14556590—22no—
63#14556590—21no—
64#14556589—21no—
65#14556589—22no—
66#14556583C2×C412yes—
67#14556583C2×C412yes—
68#14556583C2×C412yes—
69#14556581—22no—
70#14556581—22no—
71#14556581—22no—
72#14556571—22no—
73#14556571—21no—
74#14556570—21no—
75#14556570—22no—
76#14556570—21no—
77#14556570—22no—
78#14556569—22no—
79#14556569—22no—
80#14556569—22no—
81#14556567—21no—
82#14556567—22no—
83Q8×Dic73—21no—
84C4×Dic146—22no—
85C4×Dic146—21no—
86C4×Dic146—21no—
87C4×Dic146—22no—
88C4×Dic146C2×C411yescharacteristic
89C4×Dic146C2×C412yes—
90C4×Dic146C2×C411yescharacteristic
91C4×Dic146C2×C412yes—
92#14556556—22no—
93#14556556—21no—
94C4×D292C2×C411yescharacteristic
95C4×D292C2×C412yes—
96C4×D292C2×C411yescharacteristic
97C4×D292C2×C412yes—
98C4×C22⋊D73—21no—
99C4×C22⋊D73—22no—
100C4×C22⋊D73—21no—
101C4×C22⋊D73—22no—
102#14556550C2312yes—
103#14556550C2311yescharacteristic
104#14556548—22no—
105#14556548—21no—
106C4⋊C4×D73—22no—
107C4⋊C4×D73—21no—
108C42×D73—21no—
109C42×D73—22no—
110C4⋊#10596831—21no—
111C4⋊#10596831—22no—
112C4⋊#10596831—22no—
113C4⋊#10596831—21no—
114#14556519—22no—
115#14556519—24no—
116#14556519Q814yescomplemented
117#14556502—22no—
118#14556502—21no—
119C4×#10596831—21no—
120C4×#10596831—22no—
121C22×Q8×C73D411yescharacteristic
122C4×Q8×C73C2×C411yescharacteristic
123C4×Q8×C73C2×C411yescharacteristic
124C4×Q8×C73C2×C412yes—
125C4×Q8×C73D411yescharacteristic
126C4×Q8×C73D412yes—
127C4×Q8×C73D411yescharacteristic
128C2×C4⋊C4×C73—22no—
129C2×C4⋊C4×C73—21no—
130C2×C4⋊C4×C73D412yes—
131C2×C4⋊C4×C73D411yescharacteristic
132C2×C42×C73—21no—
133C2×C42×C73—21no—
134C2×C42×C73C2311yescharacteristic
135C2×C42×C73C2312yes—
136C2×C42×C73C2311yescharacteristic
137C2×C42×C73C2311yescharacteristic
138C2×C42×C73C2312yes—
139#95×C73—21no—
140#95×C73—22no—
141#95×C73—22no—
142#95×C73—22no—
143#11924028—21no—
144#11924028—21no—
145#11924028C2×Q812yes—
146#11924028#4112yes—
147C2×Dic146—22no—
148C2×Dic146—21no—
149C2×Dic146—22no—
150C2×Dic146—21no—
151C2×Dic146C22×C411yescharacteristic
152Q8×D73—44no—
153C2×D292C22×C411yescharacteristic
154C2×C22⋊D73—21no—
155C2×C4×D73—21no—
156C2×C4×D73—22no—
157C2×C4×D73—21no—
158C2×C4×D73—21no—
159C2×C4×D73—22no—
160#11924012—21no—
161#11924012—21no—
162#11924012—22no—
163#11924012—22no—
164#11924012—21no—
165#11924012C2×Q812yescomplemented
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831—22no—
169C4⋊Dic73—22no—
170C4⋊Dic73—21no—
171C4⋊Dic73—22no—
172C4⋊Dic73—21no—
173C4⋊Dic73C22×C412yes—
174C4⋊Dic73C22×C411yescharacteristic
175C4⋊Dic73C22×C412yes—
176C4⋊Dic73C22×C411yescharacteristic
177#11924007—22no—
178#11924007—22no—
179#11924007—22no—
180#11924007—21no—
181#11924007—22no—
182#11924007—22no—
183#11924007—21no—
184#11924007—22no—
185#11924007—22no—
186#11924007—22no—
187C22⋊Dic73—22no—
188C22⋊Dic73—21no—
189#11924005—21no—
190#11924005—22no—
191#11924005—22no—
192#11924005—21no—
193C4×Dic73—21no—
194C4×Dic73—22no—
195C4×Dic73—22no—
196C4×Dic73—21no—
197C4×Dic73—21no—
198C4×Dic73—22no—
199C2×Q8×C73—42no—
200C2×Q8×C73C22⋊C411yescharacteristic
201C2×Q8×C73C22⋊C411yescharacteristic
202C2×Q8×C73C22⋊C412yes—
203C22×C4×C73—21no—
204C22×C4×C73—21no—
205C22×C4×C73—22no—
206C22×C4×C73C2×D411yescharacteristic
207C22×C4×C73C2×D412yes—
208C22×C4×C73C2×D411yescharacteristic
209C22×C4×C73C2×D412yes—
210C22×C4×C73C2411yescharacteristic
211C4⋊C4×C73—42no—
212C4⋊C4×C73—44no—
213C4⋊C4×C73C22⋊C412yes—
214C4⋊C4×C73C22⋊C411yescharacteristic
215C4⋊C4×C73C22⋊C412yes—
216C4⋊C4×C73C22⋊C412yes—
217C4⋊C4×C73C22⋊C412yes—
218C4⋊C4×C73C22⋊C412yes—
219C4⋊C4×C73C22⋊C411yescharacteristic
220C42×C73—21no—
221C42×C73—21no—
222C42×C73—22no—
223C42×C73—21no—
224C42×C73—22no—
225C42×C73—42no—
226C42×C73—42no—
227C42×C73C22×C411yescharacteristic
228C42×C73C22×C412yes—
229C42×C73C22×C412yes—
230C42×C73C22×C411yescharacteristic
231C42×C73C2×D411yescharacteristic
232C42×C73C2×D412yes—
233C42×C73C2×D412yes—
234C42×C73C2×D411yescharacteristic
235C42×C73C2×Q812yescomplemented
236C23×C73C22×D411yescharacteristic
237C22×D73—21no—
238Q8×C73—22no—
239Q8×C73—22no—
240Q8×C73—42no—
241C2×C4×C73—22no—
242C2×C4×C73—22no—
243C2×C4×C73—22no—
244C2×C4×C73—41no—
245C2×C4×C73—42no—
246C2×C4×C73—42no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—42no—
252C2×C4×C73—42no—
253C2×C4×C73—42no—
254C2×C4×C73C2×C22⋊C411yescharacteristic
255C2×C4×C73C2×C22⋊C412yes—
256C2×C4×C73C2×C22⋊C412yes—
257C2×C4×C73C2×C22⋊C411yescharacteristic
258C2×C4×C73C2×C22⋊C412yes—
259C2×C4×C73C2×C22⋊C411yescharacteristic
260C2×C4×C73C2×C22⋊C412yes—
261C2×C4×C73C2×C22⋊C411yescharacteristic
262C2×C4×C73C23×C411yescharacteristic
263C2×C4×C73C22×D411yescharacteristic
264C2×C4×C73C22×Q811yescharacteristic
265C2×C4×C73C2×#4111yescharacteristic
266C2×C4×C732−1+411yescharacteristic
267C2×C4×C732−1+411yescharacteristic
268C22⋊D73—22no—
269C22⋊D73—22no—
270C2×Dic73—21no—
271C2×Dic73—21no—
272C2×Dic73—22no—
273C2×Dic73—22no—
274C2×Dic73—21no—
275D292—21no—
276D292C4×Q812yes—
277C4×D73—22no—
278C4×D73—22no—
279C4×D73—44no—
280C4×D73—42no—
281C4×D73—81no—
282C4×D73—82no—
283Dic146—21no—
284Dic146—44no—
285Dic146—42no—
286Dic146—81no—
287Dic146—82no—
288Dic146C4×Q812yes—
289#10596831—22no—
290#10596831—41no—
291#10596831—42no—
292C22×C73—21no—
293C22×C73—21no—
294C22×C73C22×C22⋊C411yescharacteristic
295C22×C73#54811yescharacteristic
296C22×C73D4×Q811yescharacteristic
297C2×D73—22no—
298C2×D73—41no—
299C2×D73—41no—
300C4×C73—22no—
301C4×C73—21no—
302C4×C73—22no—
303C4×C73—21no—
304C4×C73—22no—
305C4×C73—21no—
306C4×C73—22no—
307C4×C73—42no—
308C4×C73—42no—
309C4×C73—82no—
310C4×C73#42112yes—
311C4×C73C2×C4×Q811yescharacteristic
312C4×C73#51911yescharacteristic
313C4×C73#54811yescharacteristic
314C4×C73D4×Q811yescharacteristic
315Dic73—22no—
316Dic73—41no—
317Dic73—42no—
318Dic73—42no—
319Dic73—41no—
320#22529—731nomaximal, Sylow
321C2×C73—21no—
322C2×C73—21no—
323C2×C73Q8×C22⋊C411yescharacteristic
324C2×C73C2×#42111yescharacteristic
325C2×C73#290111yescharacteristic
326D73—42no—
327Q8×#41—731no—
328Q8×M16—731no—
329#2295D7311yescharacteristic, complemented
330#1995—732no—
331#1994—731no—
332#1986—731no—
333#1986—732no—
334#1806—731no—
335#1806—732no—
336#1759—732no—
337C4×#104—731no—
338C73#2252911yescharacteristic, complemented, Sylow
339Q82—731no—
340#557—732no—
341#557—731no—
342#554—731no—
343#554—732no—
344#554—1462no—
345#554—1461no—
346D4×Q8—731no—
347D4×Q8—1461no—
348#533—731no—
349#533—732no—
350#533—1461no—
351#533—1462no—
352C4×#41—731no—
353C4×#41—732no—
354C2×C4×Q8—1461no—
355C2×C4×Q8C2×D7311yescharacteristic
356#446—1461no—
357#446—1462no—
358C8×Q8—1461no—
359#423—731no—
360#423—732no—
361C2×#104—731no—
362C2×#104—731no—
363C2×#104—732no—
364C4×M16—731no—
365C4×M16—732no—
366#391—21no—
367#389—21no—
368#387—22no—
369C4×C4⋊C4C2×D7311yescharacteristic
370C4×C4⋊C4C2×D7312yes—
371#376C2×D7312yes—
372C43C2×D7311yescharacteristic
373#339—732no—
374#339—732no—
375#339—732no—
376#339—731no—
377#339—732no—
378#339—732no—
379#339—731no—
380C2×#41—731no—
381C22×Q8—1461no—
382C22×Q8C22⋊D7311yescharacteristic
383C2×M16—731no—
384C2×M16—732no—
385C2×M16—731no—
386C4⋊Q8—731no—
387C4⋊Q8—732no—
388C4⋊Q8—731no—
389C4⋊Q8—732no—
390C4⋊Q8—1461no—
391C4⋊Q8—1462no—
392C4⋊Q8—1462no—
393C4⋊Q8—1461no—
394#125—732no—
395#125—732no—
396#125—732no—
397#125—1462no—
398#125—1462no—
399#125—1462no—
400C22⋊Q8—1461no—
401C22⋊Q8—1462no—
402C22⋊Q8—1461no—
403C22⋊Q8—1462no—
404C22⋊Q8—1461no—
405C22⋊Q8—1462no—
406C22⋊Q8—1462no—
407C22⋊Q8—1461no—
408C22⋊Q8—1462no—
409C22⋊Q8—1462no—
410C22⋊Q8—1462no—
411C4×Q8—731no—
412C4×Q8—732no—
413C4×Q8—731no—
414C4×Q8—732no—
415C4×Q8—1461no—
416C4×Q8—1462no—
417C4×Q8—1461no—
418C4×Q8—1461no—
419C4×Q8—1462no—
420C4×Q8—1462no—
421C4×Q8—1461no—
422C4×Q8C4×D7311yescharacteristic
423C4×Q8C4×D7311yescharacteristic
424C4×Q8C4×D7312yes—
425C4×Q8D29211yescharacteristic
426C4×Q8D29211yescharacteristic
427C4×Q8D29212yes—
428C4×D4—731no—
429C4×D4—732no—
430C4×D4—731no—
431C4×D4—732no—
432C4×D4—1461no—
433C4×D4—1462no—
434C4×D4—1461no—
435C4×D4—1462no—
436#117—731no—
437#117—732no—
438#117—1462no—
439#117—1461no—
440C2×C4⋊C4—22no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—1462no—
443C2×C4⋊C4—1461no—
444C2×C4⋊C4C22⋊D7312yes—
445C2×C4⋊C4C22⋊D7311yescharacteristic
446C2×C42—21no—
447C2×C42—21no—
448C2×C42—1461no—
449C2×C42—1462no—
450C2×C42C22×D7311yescharacteristic
451C2×C42C22×D7312yes—
452C2×C42C22×D7311yescharacteristic
453C2×C42C22×D7311yescharacteristic
454C2×C42C22×D7312yes—
455C4⋊C8—1461no—
456C4⋊C8—1462no—
457C4⋊C8—1462no—
458C4⋊C8—1461no—
459#104—734no—
460#104—1462no—
461#104—1464no—
462#97—1462no—
463#97—1461no—
464C4×C8—1461no—
465C4×C8—1462no—
466#95—22no—
467#95—22no—
468#95—21no—
469#95—22no—
470#41—732no—
471#41—732no—
472#41—1461no—
473#41—1461no—
474C2×Q8—42no—
475C2×Q8—2924no—
476C2×Q8—731no—
477C2×Q8—1462no—
478C2×Q8—1461no—
479C2×Q8—1462no—
480C2×Q8—1461no—
481C2×Q8#1192400511yescharacteristic
482C2×Q8#1192400511yescharacteristic
483C2×Q8#1192400512yes—
484C2×D4—731no—
485C2×D4—1461no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—1462no—
490C22×C4—1461no—
491C22×C4—1461no—
492C22×C4—1462no—
493C22×C4—1461no—
494C22×C4C2×C22⋊D7311yescharacteristic
495C22×C4C2×C22⋊D7312yes—
496C22×C4C2×C22⋊D7311yescharacteristic
497C22×C4C2×C22⋊D7312yes—
498C22×C4C23×D7311yescharacteristic
499M16—732no—
500M16—1461no—
501M16—1462no—
502M16—1462no—
503M16—1461no—
504M16—1461no—
505C2×C8—1461no—
506C2×C8—1462no—
507C2×C8—1461no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—731no—
511C4⋊C4—732no—
512C4⋊C4—731no—
513C4⋊C4—732no—
514C4⋊C4—1462no—
515C4⋊C4—1462no—
516C4⋊C4—1462no—
517C4⋊C4—1461no—
518C4⋊C4—1462no—
519C4⋊C4—1462no—
520C4⋊C4—1461no—
521C4⋊C4—1462no—
522C4⋊C4—1462no—
523C4⋊C4—1462no—
524C4⋊C4—1462no—
525C4⋊C4—1461no—
526C4⋊C4—1462no—
527C4⋊C4—1461no—
528C4⋊C4#1192400512yes—
529C4⋊C4#1192400511yescharacteristic
530C4⋊C4#1192400512yes—
531C4⋊C4#1192400512yes—
532C4⋊C4#1192400512yes—
533C4⋊C4#1192400512yes—
534C4⋊C4#1192400511yescharacteristic
535C22⋊C4—1461no—
536C22⋊C4—1462no—
537C22⋊C4—1461no—
538C22⋊C4—1462no—
539C22⋊C4—1462no—
540C22⋊C4—1461no—
541C42—21no—
542C42—21no—
543C42—22no—
544C42—21no—
545C42—22no—
546C42—42no—
547C42—42no—
548C42—1461no—
549C42—1462no—
550C42—1462no—
551C42—1461no—
552C42—1461no—
553C42—1462no—
554C42C2×C4×D7311yescharacteristic
555C42C2×C4×D7312yes—
556C42C2×C4×D7312yes—
557C42C2×C4×D7311yescharacteristic
558C42C2×D29211yescharacteristic
559C42C2×D29212yes—
560C42C2×D29212yes—
561C42C2×D29211yescharacteristic
562C42Q8×D7312yescomplemented
563C23—1461no—
564C23C22×C22⋊D7311yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—42no—
568Q8—2924no—
569Q8—2922no—
570Q8—5841no—
571Q8—5842no—
572Q8—732no—
573Q8—1461no—
574D4—732no—
575D4—1462no—
576D4—1462no—
577D4—1461no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—2924no—
592C2×C4—2922no—
593C2×C4—5841no—
594C2×C4—5842no—
595C2×C4—1462no—
596C2×C4—1461no—
597C2×C4—1462no—
598C2×C4—1461no—
599C2×C4—1461no—
600C2×C4—1462no—
601C2×C4—1462no—
602C2×C4C2×#1192400511yescharacteristic
603C2×C4C2×#1192400512yes—
604C2×C4C2×#1192400512yes—
605C2×C4C2×#1192400511yescharacteristic
606C2×C4C2×#1192400512yes—
607C2×C4C2×#1192400512yes—
608C2×C4C2×#1192400511yescharacteristic
609C2×C4C2×#1192400511yescharacteristic
610C2×C4C22×C4×D7311yescharacteristic
611C2×C4C22×D29211yescharacteristic
612C2×C4C2×Q8×D7311yescharacteristic
613C2×C4#41×D7311yescharacteristic
614C2×C4#1455663811yescharacteristic
615C2×C4#1455663911yescharacteristic
616C8—2921no—
617C8—2922no—
618C8—1462no—
619C22—21no—
620C22—21no—
621C22—2921no—
622C22—2921no—
623C22—1462no—
624C22C22×#1192400511yescharacteristic
625C22#1798924411yescharacteristic
626C22Q8×C22⋊D7311yescharacteristic
627C4—21no—
628C4—22no—
629C4—22no—
630C4—21no—
631C4—22no—
632C4—21no—
633C4—22no—
634C4—42no—
635C4—42no—
636C4—82no—
637C4—2922no—
638C4—2921no—
639C4—2921no—
640C4—2922no—
641C4—1462no—
642C4#1798860512yes—
643C4C4×Q8×D7311yescharacteristic
644C4#1798906311yescharacteristic
645C4#1798924911yescharacteristic
646C4Q8×D29211yescharacteristic
647C2—21no—
648C2—21no—
649C2—2922no—
650C2Q8×#1192400511yescharacteristic
651C2C2×#1798860511yescharacteristic
652C2#2261847911yescharacteristic
subgroupG/H
centerC2×C4C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2295C2
socleC22×C73C22×C22⋊C4
prime radicalC2×#11924028C23
cosocleC22×C4×C73C24
hypercenter#2295D73
hypocenterC73#22529
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8 ◁ #2295

Complements

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

NH ≅ G/Ntype
2. Q8×#11924012C2semidirect
3. #22610912C2semidirect
4. #22610908C2semidirect
5. #22610874C2semidirect
6. #22610874C2semidirect
11. C73×#2295C2semidirect
12. Q8×Dic146C4semidirect
13. #17989315C4semidirect
14. #17989315C4semidirect
15. #17989297C4semidirect
16. #17989297C4semidirect
19. Q8×D292C4semidirect
116. #14556519Q8semidirect
165. #11924012C2×Q8semidirect
235. C42×C73C2×Q8semidirect
329. #2295D73semidirect
338. C73#22529semidirect
562. C42Q8×D73semidirect

Maximal subgroups

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

subgroupindex
Q8×#119240282
Q8×#119240122
#226109122
#226109082
#226108742
#226099522
#226097422
C4×#145565192
C73×#22952
#2252973