Finite Group Databasebeta

← #10702175

Subgroups — #10702175

Order 768.

942 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5139C211yesmaximal, characteristic
2#83750C211yesmaximal, characteristic, complemented
3C2×#5005C211yesmaximal, characteristic
4#82070C211yesmaximal, characteristic
5C2×C3×#339C211yesmaximal, characteristic, complemented
6#71593C211yesmaximal, characteristic, complemented
7C2×#4523C211yesmaximal, characteristic, complemented
8#11056—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C22×C3×C42C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C2×#926C2211yescharacteristic
15C22×#208C2211yescharacteristic
16#5213—22no—
17#5139—22no—
18#5139—22no—
19C2×#825—22no—
20#5119—22no—
21#5118—22no—
22#5102—22no—
23#5005—22no—
24#5005—22no—
25#4939C412yescomplemented
26C4×(C2×D3)⋊C4—22no—
27#4929C412yescomplemented
28C2×#805—22no—
29C3×#339—22no—
30#4527—22no—
31#4523—22no—
32#2905—31no—
33C2×#421—31no—
34C2×#421—31no—
35#2297—31no—
36#1740—31no—
37C2×#339—31no—
38C2×#339D311yescharacteristic, complemented
39C23×C3×C4—21no—
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C2×#235—21no—
43C2×#235—21no—
44C2×#235—21no—
45C2×#235—21no—
46C2×#235—21no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235—21no—
50C2×#235—21no—
51C2×#235—21no—
52C2×#235D412yes—
53C2×#235Q812yes—
54C22×D12C2×C411yescharacteristic
55C22×C4×D3—21no—
56C22×Dic6C2×C411yescharacteristic
57C2×C3×M16—21no—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16D412yes—
63C22×C3×C8—21no—
64C3×#117—21no—
65C3×#117—21no—
66C3×#117—21no—
67C3×#117D412yes—
68C2×C3×C4⋊C4—21no—
69C2×C3×C22⋊C4—21no—
70C2×C3×C42—21no—
71C2×C3×C42—21no—
72C2×C3×C42—21no—
73C2×C3×C42—21no—
74C2×C3×C42—21no—
75C2×C3×C42—21no—
76C2×C3×C42C2×C412yes—
77C2×#217—21no—
78C2×#217D412yes—
79C2×(C2×D3)⋊C4—24no—
80#926—21no—
81#926—21no—
82#926—21no—
83#926—21no—
84#926—21no—
85#926—21no—
86#926—21no—
87#926D412yes—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3C2×C412yes—
90C2×C4×Dic3—21no—
91C2×C4×Dic3D412yes—
92C2×#208—21no—
93C2×#208—21no—
94C2×#208—21no—
95C2×#208—21no—
96C2×#208—21no—
97C2×#208D412yes—
98C22×C3⋊C8—21no—
99C3⋊#95—24no—
100#825—28no—
101Dic3⋊C8—28no—
102#805—28no—
103C22×#41—31no—
104C22×M16—31no—
105C22×M16C2×D311yescharacteristic
106C2×#117—31no—
107C2×#117—31no—
108C2×#117C2×D311yescharacteristic
109C22×C42C2×D311yescharacteristic
110#424—62no—
111#424—62no—
112#423—62no—
113#421—62no—
114#421—62no—
115#421—62no—
116#421—62no—
117C2×#104—62no—
118C2×#104—62no—
119#391—32no—
120#390—32no—
121#386—62no—
122C4×C22⋊C4—62no—
123#342—62no—
124#339—22no—
125#339—62no—
126C24×C3C2×D411yescharacteristic
127C23×D3—21no—
128C22×C3×C4—22no—
129C22×C3×C4—44no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4C22×C411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×Q811yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C22×C3×C4#4111yescharacteristic
144C22×C3×C4#4111yescharacteristic
145C22×C3×C4#4111yescharacteristic
146C22×C3×C4#4111yescharacteristic
147C2×#67—22no—
148C2×#67—22no—
149C2×#67—41no—
150C2×#67—41no—
151C2×#67—41no—
152C2×#67—41no—
153C22×Dic3—21no—
154C22×Dic3—41no—
155C22×Dic3C22⋊C414yes—
156#235—22no—
157#235—22no—
158#235—22no—
159#235—22no—
160#235—22no—
161#235—22no—
162#235—22no—
163#235—22no—
164#235—41no—
165#235—41no—
166#235—41no—
167#235—41no—
168#235—41no—
169#235—41no—
170#235—41no—
171#235—41no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12C4⋊C412yes—
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—41no—
181C2×C4×D3—41no—
182C2×C4×D3—41no—
183C2×C4×D3—41no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6C4⋊C412yes—
190C3×M16—22no—
191C3×M16—22no—
192C3×M16—41no—
193C3×M16—41no—
194C2×C3×C8—22no—
195C2×C3×C8—41no—
196C2×C3×C8—41no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—41no—
199C3×C22⋊C4—22no—
200C3×C22⋊C4—41no—
201C3×C42—22no—
202C3×C42—22no—
203C3×C42—22no—
204C3×C42—41no—
205C3×C42—41no—
206C3×C42—41no—
207#217—22no—
208#217—22no—
209#217—22no—
210#217—41no—
211(C2×D3)⋊C4—48no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3—41no—
216C4×Dic3—22no—
217C4×Dic3—22no—
218C4×Dic3—22no—
219C4×Dic3—41no—
220#208—22no—
221#208—22no—
222#208—41no—
223#208—41no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—41no—
226C2×C3⋊C8—41no—
227C2×#41—32no—
228C2×#41—32no—
229C2×#41—61no—
230C2×#41—61no—
231C2×#41—61no—
232C2×#41—61no—
233C2×#41—61no—
234C2×#41—61no—
235C2×#41—61no—
236C2×#41—61no—
237C2×#41—61no—
238C2×#41—61no—
239C22×Q8—31no—
240C22×D4—31no—
241C22×D4—61no—
242C23×C4—21no—
243C23×C4—61no—
244C23×C4C22×D311yescharacteristic
245C2×M16—21no—
246C2×M16—21no—
247C2×M16—21no—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—32no—
251C2×M16—61no—
252C2×M16—61no—
253C2×M16—61no—
254C2×M16—61no—
255C2×M16—61no—
256C2×M16D1212yes—
257C22×C8—21no—
258C22×C8—61no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—32no—
263#117—61no—
264#117—61no—
265#117—61no—
266#117—61no—
267#117—61no—
268#117—61no—
269#117—61no—
270#117#6712yes—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—32no—
273C2×C4⋊C4—61no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—32no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—64no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—32no—
285C2×C42—61no—
286C2×C42C4×D312yes—
287C4⋊C8—68no—
288#104—68no—
289#104—68no—
290#95—64no—
291C23×C3—42no—
292C23×C3C2×C22⋊C411yescharacteristic
293C23×C3#12111yescharacteristic
294C23×C3#12111yescharacteristic
295C23×C3#12111yescharacteristic
296C23×C3#12111yescharacteristic
297C23×C3#12111yescharacteristic
298C23×C3C22⋊Q811yescharacteristic
299C22×D3—22no—
300C22×D3—44no—
301C22×D3—41no—
302C22×D3—41no—
303C2×C3×C4—24no—
304C2×C3×C4—42no—
305C2×C3×C4—44no—
306C2×C3×C4—44no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—82no—
310C2×C3×C4C2×C4⋊C411yescharacteristic
311C2×C3×C4#11711yescharacteristic
312C2×C3×C4C4×D411yescharacteristic
313C2×C3×C4C4×D411yescharacteristic
314C2×C3×C4C4×D411yescharacteristic
315C2×C3×C4C4×D411yescharacteristic
316C2×C3×C4#12011yescharacteristic
317C2×C3×C4#12011yescharacteristic
318C2×C3×C4#12011yescharacteristic
319C2×C3×C4#12111yescharacteristic
320C2×C3×C4#12111yescharacteristic
321C2×C3×C4#12111yescharacteristic
322C2×C3×C4C22⋊Q811yescharacteristic
323C2×C3×C4C22⋊Q811yescharacteristic
324C2×C3×C4C22⋊Q811yescharacteristic
325C2×C3×C4C22⋊Q811yescharacteristic
326C2×C3×C4C22⋊Q811yescharacteristic
327C2×C3×C4#12311yescharacteristic
328C2×C3×C4#12311yescharacteristic
329C2×C3×C4#12311yescharacteristic
330C2×C3×C4#12311yescharacteristic
331C2×C3×C4#12311yescharacteristic
332C2×C3×C4#12411yescharacteristic
333C2×C3×C4#12411yescharacteristic
334C2×C3×C4#12411yescharacteristic
335C2×C3×C4#12611yescharacteristic
336C2×C3×C4#12611yescharacteristic
337C2×C3×C4#12711yescharacteristic
338#67—42no—
339#67—42no—
340#67—42no—
341#67—42no—
342C2×Dic3—22no—
343C2×Dic3—24no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—44no—
347C2×Dic3—42no—
348C2×Dic3—82no—
349D12—22no—
350D12—22no—
351D12—41no—
352D12—41no—
353C4×D3—42no—
354C4×D3—42no—
355C4×D3—42no—
356C4×D3—42no—
357Dic6—22no—
358Dic6—22no—
359Dic6—41no—
360Dic6—41no—
361C3×C8—42no—
362C3⋊C8—42no—
363C24—61no—
364C24D3×D411yescharacteristic
365#41—62no—
366#41—62no—
367#41—62no—
368#41—62no—
369#41—62no—
370#41—62no—
371#41—62no—
372#41—62no—
373#41—121no—
374#41—121no—
375#41—121no—
376#41—121no—
377#41—121no—
378#41—121no—
379#41—121no—
380#41—121no—
381C2×Q8—32no—
382C2×Q8—61no—
383C2×Q8—61no—
384C2×Q8—61no—
385C2×Q8—61no—
386C2×Q8—61no—
387C2×D4—32no—
388C2×D4—62no—
389C2×D4—62no—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—61no—
393C2×D4—61no—
394C2×D4—61no—
395C2×D4—121no—
396C2×D4—121no—
397C2×D4—121no—
398C2×D4—121no—
399C22×C4—22no—
400C22×C4—34no—
401C22×C4—44no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—61no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—121no—
409C22×C4—121no—
410C22×C4—121no—
411C22×C4—121no—
412C22×C4—121no—
413C22×C4C2×C4×D311yescharacteristic
414C22×C4C2×D1211yescharacteristic
415C22×C4#23511yescharacteristic
416C22×C4#23511yescharacteristic
417C22×C4#23511yescharacteristic
418C22×C4#23511yescharacteristic
419C22×C4D3×D411yescharacteristic
420C22×C4D3×D411yescharacteristic
421C22×C4D3×D411yescharacteristic
422C22×C4D3×D411yescharacteristic
423C22×C4#23711yescharacteristic
424C22×C4D3×Q811yescharacteristic
425C22×C4#23911yescharacteristic
426C22×C4C2×#6711yescharacteristic
427M16—22no—
428M16—22no—
429M16—41no—
430M16—41no—
431M16—62no—
432M16—62no—
433M16—121no—
434M16—121no—
435C2×C8—22no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—62no—
439C2×C8—121no—
440C2×C8—121no—
441C4⋊C4—22no—
442C4⋊C4—41no—
443C4⋊C4—62no—
444C4⋊C4—62no—
445C4⋊C4—62no—
446C4⋊C4—121no—
447C22⋊C4—22no—
448C22⋊C4—41no—
449C22⋊C4—62no—
450C22⋊C4—62no—
451C22⋊C4—62no—
452C22⋊C4—128no—
453C22⋊C4—121no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—62no—
461C42—62no—
462C42—62no—
463C42—121no—
464C22×C3—22no—
465C22×C3—22no—
466C22×C3—22no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3#38011yescharacteristic
472C22×C3#38611yescharacteristic
473C22×C3#39011yescharacteristic
474C22×C3#39411yescharacteristic
475C22×C3#39411yescharacteristic
476C22×C3#39611yescharacteristic
477C22×C3#39611yescharacteristic
478C2×D3—44no—
479C2×D3—42no—
480C2×D3—82no—
481C2×D3—82no—
482C3×C4—44no—
483C3×C4—82no—
484C3×C4#38511yescharacteristic
485C3×C4#38511yescharacteristic
486C3×C4#38811yescharacteristic
487C3×C4#38811yescharacteristic
488C3×C4#39211yescharacteristic
489C3×C4#39311yescharacteristic
490C3×C4#39711yescharacteristic
491C3×C4#39911yescharacteristic
492Dic3—44no—
493Dic3—42no—
494Dic3—82no—
495C23—42no—
496C23—62no—
497C23—124no—
498C23—121no—
499C23—121no—
500C23D3×C22⋊C411yescharacteristic
501C23#88411yescharacteristic
502C23#88411yescharacteristic
503C23#88411yescharacteristic
504C23#89511yescharacteristic
505C23#89611yescharacteristic
506C23#93911yescharacteristic
507Q8—62no—
508Q8—62no—
509Q8—121no—
510Q8—121no—
511D4—62no—
512D4—62no—
513D4—122no—
514D4—122no—
515D4—122no—
516D4—122no—
517D4—121no—
518D4—121no—
519C2×C4—24no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—64no—
526C2×C4—62no—
527C2×C4—82no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—122no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4—122no—
534C2×C4—124no—
535C2×C4—122no—
536C2×C4—242no—
537C2×C4#87211yescharacteristic
538C2×C4C4×D1211yescharacteristic
539C2×C4#87511yescharacteristic
540C2×C4#87611yescharacteristic
541C2×C4#67⋊C411yescharacteristic
542C2×C4#88211yescharacteristic
543C2×C4#88211yescharacteristic
544C2×C4#88311yescharacteristic
545C2×C4#88411yescharacteristic
546C2×C4#88411yescharacteristic
547C2×C4#88511yescharacteristic
548C2×C4#88511yescharacteristic
549C2×C4D3×C4⋊C411yescharacteristic
550C2×C4#89311yescharacteristic
551C2×C4#89411yescharacteristic
552C2×C4#89411yescharacteristic
553C2×C4#89411yescharacteristic
554C2×C4#89611yescharacteristic
555C2×C4#89611yescharacteristic
556C2×C4#89611yescharacteristic
557C2×C4#89711yescharacteristic
558C2×C4#89811yescharacteristic
559C2×C4C4×#6711yescharacteristic
560C2×C4#92911yescharacteristic
561C2×C4#93011yescharacteristic
562C2×C4#93711yescharacteristic
563C2×C4#94011yescharacteristic
564C2×C4#94611yescharacteristic
565C8—42no—
566C8—122no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3#143911yescharacteristic
574C2×C3#188412yes—
575D3—82no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—124no—
584C22—122no—
585C22—242no—
586C22—242no—
587C22#466511yescharacteristic
588C22#466711yescharacteristic
589C22#467211yescharacteristic
590C22#467611yescharacteristic
591C22#495711yescharacteristic
592C22#495911yescharacteristic
593C22#499411yescharacteristic
594C4—44no—
595C4—82no—
596C4—122no—
597C4—124no—
598C4—242no—
599C4#466811yescharacteristic
600C4#466911yescharacteristic
601C4#467311yescharacteristic
602C4#467511yescharacteristic
603C4#494011yescharacteristic
604C4#496011yescharacteristic
605C4#498911yescharacteristic
606C4#499311yescharacteristic
607C3#1105611yescharacteristic, complemented, Sylow
608C2—21no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—21no—
614C2—242no—
615C2#6877712yes—
616C2#6942811yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#339C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×#339D3
hypocenterC3#11056
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
2. #83750C2semidirect
5. C2×C3×#339C2semidirect
6. #71593C2semidirect
7. C2×#4523C2semidirect
25. #4939C4semidirect
27. #4929C4semidirect
38. C2×#339D3semidirect
607. C3#11056semidirect

Maximal subgroups

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

subgroupindex
C2×#51392
#837502
C2×#50052
#820702
C2×C3×#3392
#715932
C2×#45232
#110563