Finite Group Databasebeta

← #10702716

Subgroups — #10702716

Order 768.

848 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5139C211yesmaximal, characteristic
2C2×#5139C211yesmaximal, characteristic
3C2×#5132C211yesmaximal, characteristic
4#83752C211yesmaximal, characteristic, complemented
5C3×#1738C211yesmaximal, characteristic, complemented
6C2×#4559C211yesmaximal, characteristic, complemented
7C2×#4556C211yesmaximal, characteristic, complemented
8#11141—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C2×#926C2211yescharacteristic
15C22×#208C2211yescharacteristic
16#5139—22no—
17#5139—22no—
18#5139—22no—
19#5139—22no—
20C2×#825—22no—
21C2×#825—22no—
22C2×#824C412yescomplemented
23#5132—22no—
24#5132—22no—
25C2×#823C412yescomplemented
26#5121—22no—
27#5121—22no—
28#5106—22no—
29C3×#341—22no—
30#4559—22no—
31#4556—22no—
32#2911—31no—
33C2×#421—31no—
34C2×#421—31no—
35C2×#413—31no—
36C2×#344—31no—
37C2×#344—31no—
38#1738D311yescharacteristic, complemented
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C2×#235—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×#235D412yes—
52C2×#235Q812yes—
53C22×D12C2×C411yescharacteristic
54C22×C4×D3—21no—
55C22×Dic6C2×C411yescharacteristic
56C2×C3×M16—21no—
57C2×C3×M16—21no—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16—21no—
63C2×C3×M16—21no—
64C2×C3×M16—21no—
65C2×C3×M16—21no—
66C2×C3×M16—21no—
67C2×C3×M16—21no—
68C2×C3×M16—21no—
69C2×C3×M16C2×C412yes—
70C2×C3×M16D412yes—
71C2×C3×M16D412yes—
72C2×C3×M16D412yes—
73C2×C3×M16D412yes—
74C22×C3×C8—21no—
75C22×C3×C8—21no—
76C22×C3×C8—21no—
77C2×#217—21no—
78C2×#217—21no—
79#926—21no—
80#926—21no—
81#926—21no—
82#926—21no—
83#926—21no—
84#926—21no—
85#926D412yes—
86#926D412yes—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4×Dic3—21no—
90C2×C4×Dic3—21no—
91C2×#208—21no—
92C2×#208—21no—
93C2×#208—21no—
94C2×#208—21no—
95C2×#208—21no—
96C2×#208C2×C412yes—
97C22×C3⋊C8—21no—
98#825—28no—
99#825—28no—
100#824—24no—
101#823—24no—
102#818—24no—
103#817—24no—
104C22×#41—31no—
105C22×M16—31no—
106C22×M16C2×D311yescharacteristic
107C22×M16C2×D311yescharacteristic
108C22×M16C2×D311yescharacteristic
109C2×#117—31no—
110C2×#117—31no—
111#428—62no—
112#428—62no—
113#427—62no—
114#421—62no—
115#421—62no—
116#421—62no—
117#421—62no—
118C2×#104—62no—
119C2×#104—62no—
120#413—62no—
121#413—62no—
122C2×#101—32no—
123C2×#100—32no—
124#344—62no—
125#344—62no—
126#341—22no—
127C24×C3#4111yescharacteristic
128C23×D3—21no—
129C22×C3×C4C22×C411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×Q811yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C2×#67—22no—
144C2×#67—22no—
145C2×#67—41no—
146C2×#67—41no—
147C2×#67—41no—
148C2×#67—41no—
149C22×Dic3—21no—
150C22×Dic3—41no—
151C22×Dic3—41no—
152#235—22no—
153#235—22no—
154#235—22no—
155#235—22no—
156#235—22no—
157#235—22no—
158#235—22no—
159#235—22no—
160#235—41no—
161#235—41no—
162#235—41no—
163#235—41no—
164#235—41no—
165#235—41no—
166#235—41no—
167#235—41no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12C4⋊C412yes—
174C2×C4×D3—22no—
175C2×C4×D3—22no—
176C2×C4×D3—41no—
177C2×C4×D3—41no—
178C2×C4×D3—41no—
179C2×C4×D3—41no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6C4⋊C412yes—
186C3×M16—22no—
187C3×M16—22no—
188C3×M16—22no—
189C3×M16—22no—
190C3×M16—22no—
191C3×M16—22no—
192C3×M16—22no—
193C3×M16—41no—
194C3×M16—41no—
195C3×M16—41no—
196C3×M16—41no—
197C3×M16C22⋊C414yes—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—41no—
204C2×C3×C8—41no—
205C2×C3×C8—41no—
206C2×C3×C8—41no—
207#217—22no—
208#217—22no—
209#217—41no—
210#217—41no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3—41no—
214C4⋊Dic3—41no—
215C4×Dic3—22no—
216C4×Dic3—22no—
217C4×Dic3—41no—
218C4×Dic3—41no—
219#208—22no—
220#208—22no—
221#208—41no—
222#208—41no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—41no—
225C2×C3⋊C8—41no—
226C2×#41—32no—
227C2×#41—32no—
228C2×#41—61no—
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—
238C22×Q8—31no—
239C22×D4—31no—
240C22×D4—61no—
241C23×C4—61no—
242C23×C4C22×D311yescharacteristic
243C2×M16—21no—
244C2×M16—21no—
245C2×M16—21no—
246C2×M16—21no—
247C2×M16—21no—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—21no—
255C2×M16—21no—
256C2×M16—32no—
257C2×M16—61no—
258C2×M16—61no—
259C2×M16—61no—
260C2×M16—61no—
261C2×M16—61no—
262C2×M16C4×D312yes—
263C2×M16D1212yes—
264C2×M16D1212yes—
265C2×M16D1212yes—
266C2×M16#6712yes—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—61no—
271#117—32no—
272#117—32no—
273#117—61no—
274#117—61no—
275#117—61no—
276#117—61no—
277#117—61no—
278#117—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C42—61no—
284C2×C42—61no—
285#107—64no—
286#106—64no—
287#104—68no—
288#104—68no—
289#101—64no—
290#100—64no—
291C23×C3—81no—
292C23×C3#11711yescharacteristic
293C23×C3#12111yescharacteristic
294C23×C3#12111yescharacteristic
295C23×C3#12111yescharacteristic
296C23×C3C22⋊Q811yescharacteristic
297C23×C3#12611yescharacteristic
298C23×C3#12611yescharacteristic
299C22×D3—22no—
300C22×D3—41no—
301C22×D3—41no—
302C22×D3—82no—
303C2×C3×C4C2×C22⋊C411yescharacteristic
304C2×C3×C4C2×C4⋊C411yescharacteristic
305C2×C3×C4C4×D411yescharacteristic
306C2×C3×C4C4×D411yescharacteristic
307C2×C3×C4C4×D411yescharacteristic
308C2×C3×C4C4×D411yescharacteristic
309C2×C3×C4#12011yescharacteristic
310C2×C3×C4#12011yescharacteristic
311C2×C3×C4#12011yescharacteristic
312C2×C3×C4#12111yescharacteristic
313C2×C3×C4#12111yescharacteristic
314C2×C3×C4#12111yescharacteristic
315C2×C3×C4#12111yescharacteristic
316C2×C3×C4#12111yescharacteristic
317C2×C3×C4C22⋊Q811yescharacteristic
318C2×C3×C4C22⋊Q811yescharacteristic
319C2×C3×C4C22⋊Q811yescharacteristic
320C2×C3×C4C22⋊Q811yescharacteristic
321C2×C3×C4C22⋊Q811yescharacteristic
322C2×C3×C4#12311yescharacteristic
323C2×C3×C4#12311yescharacteristic
324C2×C3×C4#12311yescharacteristic
325C2×C3×C4#12311yescharacteristic
326C2×C3×C4#12311yescharacteristic
327C2×C3×C4#12411yescharacteristic
328C2×C3×C4#12411yescharacteristic
329C2×C3×C4#12411yescharacteristic
330C2×C3×C4#12711yescharacteristic
331#67—42no—
332#67—42no—
333#67—42no—
334#67—42no—
335C2×Dic3—22no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—82no—
341C2×Dic3—82no—
342D12—22no—
343D12—22no—
344D12—41no—
345D12—41no—
346C4×D3—42no—
347C4×D3—42no—
348C4×D3—42no—
349C4×D3—42no—
350Dic6—22no—
351Dic6—22no—
352Dic6—41no—
353Dic6—41no—
354C3×C8—24no—
355C3×C8—42no—
356C3×C8—42no—
357C3⋊C8—42no—
358C24—61no—
359C24#23911yescharacteristic
360#41—62no—
361#41—62no—
362#41—62no—
363#41—62no—
364#41—62no—
365#41—62no—
366#41—62no—
367#41—62no—
368#41—121no—
369#41—121no—
370#41—121no—
371#41—121no—
372#41—121no—
373#41—121no—
374#41—121no—
375#41—121no—
376C2×Q8—32no—
377C2×Q8—61no—
378C2×Q8—61no—
379C2×Q8—61no—
380C2×Q8—61no—
381C2×Q8—61no—
382C2×D4—32no—
383C2×D4—62no—
384C2×D4—62no—
385C2×D4—61no—
386C2×D4—61no—
387C2×D4—61no—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—121no—
391C2×D4—121no—
392C2×D4—121no—
393C2×D4—121no—
394C22×C4—61no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—121no—
398C22×C4—121no—
399C22×C4—121no—
400C22×C4—121no—
401C22×C4—121no—
402C22×C4—121no—
403C22×C4C2×C4×D311yescharacteristic
404C22×C4C2×D1211yescharacteristic
405C22×C4C2×D1211yescharacteristic
406C22×C4C2×D1211yescharacteristic
407C22×C4#23511yescharacteristic
408C22×C4#23511yescharacteristic
409C22×C4D3×D411yescharacteristic
410C22×C4D3×D411yescharacteristic
411C22×C4D3×D411yescharacteristic
412C22×C4#23711yescharacteristic
413C22×C4#23711yescharacteristic
414C22×C4#23711yescharacteristic
415C22×C4D3×Q811yescharacteristic
416C22×C4C2×#6711yescharacteristic
417M16—22no—
418M16—22no—
419M16—22no—
420M16—22no—
421M16—22no—
422M16—22no—
423M16—22no—
424M16—41no—
425M16—41no—
426M16—41no—
427M16—41no—
428M16—62no—
429M16—62no—
430M16—121no—
431M16—121no—
432M16(C2×D3)⋊C414yes—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—62no—
443C2×C8—121no—
444C2×C8—121no—
445C4⋊C4—62no—
446C4⋊C4—62no—
447C4⋊C4—121no—
448C4⋊C4—121no—
449C22⋊C4—62no—
450C22⋊C4—62no—
451C22⋊C4—121no—
452C22⋊C4—121no—
453C42—62no—
454C42—62no—
455C42—121no—
456C42—121no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3#38011yescharacteristic
465C22×C3#38811yescharacteristic
466C22×C3#38811yescharacteristic
467C22×C3#39211yescharacteristic
468C22×C3#39611yescharacteristic
469C22×C3#39611yescharacteristic
470C22×C3#39711yescharacteristic
471C2×D3—42no—
472C2×D3—82no—
473C2×D3—82no—
474C2×D3—82no—
475C3×C4#38511yescharacteristic
476C3×C4#38511yescharacteristic
477C3×C4#38611yescharacteristic
478C3×C4#39011yescharacteristic
479C3×C4#39311yescharacteristic
480C3×C4#39411yescharacteristic
481C3×C4#39411yescharacteristic
482C3×C4#39911yescharacteristic
483Dic3—42no—
484Dic3—82no—
485Dic3—82no—
486C23—62no—
487C23—81no—
488C23—121no—
489C23—121no—
490C23—242no—
491C23#89211yescharacteristic
492C23#89511yescharacteristic
493C23#89511yescharacteristic
494C23#89511yescharacteristic
495C23#89811yescharacteristic
496C23#89811yescharacteristic
497C23#94611yescharacteristic
498Q8—62no—
499Q8—62no—
500Q8—121no—
501Q8—121no—
502D4—62no—
503D4—62no—
504D4—122no—
505D4—122no—
506D4—122no—
507D4—122no—
508D4—121no—
509D4—121no—
510C2×C4—62no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—242no—
520C2×C4—242no—
521C2×C4C4×D1211yescharacteristic
522C2×C4C4×D1211yescharacteristic
523C2×C4#87411yescharacteristic
524C2×C4#87511yescharacteristic
525C2×C4#67⋊C411yescharacteristic
526C2×C4#67⋊C411yescharacteristic
527C2×C4#88211yescharacteristic
528C2×C4#88211yescharacteristic
529C2×C4#88211yescharacteristic
530C2×C4#88311yescharacteristic
531C2×C4#88311yescharacteristic
532C2×C4#88511yescharacteristic
533C2×C4#88511yescharacteristic
534C2×C4#88611yescharacteristic
535C2×C4#88611yescharacteristic
536C2×C4#88611yescharacteristic
537C2×C4D3×C4⋊C411yescharacteristic
538C2×C4#89611yescharacteristic
539C2×C4#89611yescharacteristic
540C2×C4#89711yescharacteristic
541C2×C4#89711yescharacteristic
542C2×C4#89711yescharacteristic
543C2×C4C2×(C2×D3)⋊C411yescharacteristic
544C2×C4#93011yescharacteristic
545C2×C4#93011yescharacteristic
546C2×C4#93811yescharacteristic
547C2×C4#93911yescharacteristic
548C2×C4#93911yescharacteristic
549C8—24no—
550C8—42no—
551C8—42no—
552C8—122no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3#143911yescharacteristic
561D3—82no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—122no—
570C22—242no—
571C22—242no—
572C22—242no—
573C22#467011yescharacteristic
574C22#467011yescharacteristic
575C22#467111yescharacteristic
576C22#467411yescharacteristic
577C22#498711yescharacteristic
578C22#499411yescharacteristic
579C22#499411yescharacteristic
580C4—122no—
581C4—242no—
582C4—242no—
583C4#466811yescharacteristic
584C4#466811yescharacteristic
585C4#467311yescharacteristic
586C4#467711yescharacteristic
587C4#493911yescharacteristic
588C4#495811yescharacteristic
589C4#496111yescharacteristic
590C4#496111yescharacteristic
591C3#1114111yescharacteristic, complemented, Sylow
592C2—21no—
593C2—21no—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—242no—
600C2#6942711yescharacteristic
subgroupG/H
centerC4#4961
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1738C2
socleC2×C3#1439
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenter#1738D3
hypocenterC3#11141
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C23×C4 ◁ #1738

Complements

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

NH ≅ G/Ntype
4. #83752C2semidirect
5. C3×#1738C2semidirect
6. C2×#4559C2semidirect
7. C2×#4556C2semidirect
22. C2×#824C4semidirect
25. C2×#823C4semidirect
38. #1738D3semidirect
591. C3#11141semidirect

Maximal subgroups

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

subgroupindex
C2×#51392
C2×#51322
#837522
C3×#17382
C2×#45592
C2×#45562
#111413