Finite Group Databasebeta

← #10836669

Subgroups — #10836669

Order 768.

674 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5091C211yesmaximal, characteristic, complemented
2#83212C211yesmaximal, characteristic
3#79040C211yesmaximal, characteristic
4C4×#909C211yesmaximal, characteristic
5C3×#1997C211yesmaximal, characteristic, complemented
6#70528C211yesmaximal, characteristic, complemented
7#69875C211yesmaximal, characteristic, complemented
8#69010C212yesmaximal
9#68987C212yesmaximal
10#68985C212yesmaximal, complemented
11#68982C212yesmaximal, complemented
12#24722—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#909C2211yescharacteristic
15C4×#235C2211yescharacteristic
16C2×C4×Dic6C2211yescharacteristic
17C3×#423C2211yescharacteristic
18C3×C4×M16C2211yescharacteristic
19C2×#824C2212yes—
20#5133C2211yescharacteristic
21#5129C2212yes—
22C2×#822C2212yes—
23#5122C2212yes—
24#5121C2211yescharacteristic
25#5120C2211yescharacteristic
26#5094—21no—
27#5091—21no—
28#5091—21no—
29#5091—21no—
30#5091—22no—
31#5087—21no—
32#5084—21no—
33C2×#810C2212yes—
34#4966C2212yes—
35#4964C2212yes—
36C2×#807C2212yes—
37#4924C2211yescharacteristic
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4840—22no—
40#4835—21no—
41#4833—21no—
42#4716—21no—
43#4710—21no—
44C4×Dic12—21no—
45C4×#204—21no—
46C3×#343C2212yes—
47C3×#343C2212yes—
48#4558C2212yes—
49#4554C2212yes—
50#3079—31no—
51C2×#477—31no—
52C4×#137—31no—
53#2925—31no—
54#2078—32no—
55#2052—32no—
56#1997D311yescharacteristic, complemented
57#1952—32no—
58#1950—31no—
59#1930—32no—
60#1801—31no—
61C22×C3×Q8—21no—
62C2×#235C2311yescharacteristic
63C22×Dic6C2311yescharacteristic
64C2×C3×M16C2311yescharacteristic
65C2×C3×M16C2312yes—
66C2×C3×M16C2311yescharacteristic
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8—21no—
69C3×C4⋊Q8—21no—
70C3×C4⋊Q8—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C42C2311yescharacteristic
74C4×#67—21no—
75C4×#67—21no—
76#926C2311yescharacteristic
77C2×C4⋊Dic3C2311yescharacteristic
78C2×Dic3⋊C4—21no—
79C2×C4×Dic3—21no—
80C2×#208C2312yes—
81C22×C3⋊C8C2312yes—
82#909—22no—
83#909—22no—
84C2×Dic12—21no—
85C2×#204—21no—
86C4×D12D411yescharacteristic
87C4×D12D411yescharacteristic
88#872—21no—
89C42×D3—21no—
90C4×Dic6—21no—
91C4×Dic6—21no—
92C4×Dic6—21no—
93C4×Dic6—21no—
94C4×Dic6D411yescharacteristic
95C4×Dic6D411yescharacteristic
96C3×C4⋊C8—21no—
97C3×C4⋊C8—21no—
98C3×#97—21no—
99C3×C4×C8—21no—
100#824—24no—
101#822—24no—
102#821—21no—
103(C2×D3)⋊C8—22no—
104(C2×D3)⋊C8—22no—
105#818—21no—
106#817—21no—
107#816—21no—
108#816—21no—
109#816—21no—
110#815—22no—
111C8×Dic3—22no—
112#810—22no—
113#810—22no—
114#810—22no—
115#809—22no—
116#808—22no—
117#807—22no—
118#807—22no—
119#807—22no—
120C4⋊(C3⋊C8)—21no—
121C4⋊(C3⋊C8)—21no—
122C4⋊(C3⋊C8)D411yescharacteristic
123C4⋊(C3⋊C8)D411yescharacteristic
124C4⋊(C3⋊C8)D412yes—
125C2×#137—31no—
126C2×C4⋊Q8C2×D311yescharacteristic
127C4×#41—31no—
128C2×C4×Q8—31no—
129#477—61no—
130#477—61no—
131#477—62no—
132#477—61no—
133#476—61no—
134#475—61no—
135#474—61no—
136#441—61no—
137#440—61no—
138C4×Q16—61no—
139C4×SD16—61no—
140#433—62no—
141#432—61no—
142#431—61no—
143C2×#108—32no—
144#428—31no—
145#428—32no—
146C2×#107—32no—
147#423—31no—
148#423C2×D311yescharacteristic
149C2×C4⋊C8—31no—
150#419—31no—
151#417—31no—
152#417—32no—
153C2×#103—32no—
154C2×#101—32no—
155#408—32no—
156#405—32no—
157C4×M16C2×D311yescharacteristic
158#343—32no—
159#343C2×D312yes—
160#343C2×D312yes—
161#341—32no—
162C2×C3×Q8—41no—
163C2×C3×Q8—41no—
164C2×C3×Q8—82no—
165C22×C3×C4—22no—
166C22×C3×C4#4111yescharacteristic
167C22×C3×C4#4111yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C2×#67—21no—
170C22×Dic3—21no—
171C22×Dic3—21no—
172#235—21no—
173#235—21no—
174#235—21no—
175#235—21no—
176C2×D12C2×D411yescharacteristic
177C2×C4×D3—21no—
178C2×C4×D3—21no—
179C2×C4×D3—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6C2×D411yescharacteristic
186C2×Dic6#4112yes—
187C3×M16—21no—
188C3×M16—22no—
189C3×M16—21no—
190C3×M16—21no—
191C3×M16—22no—
192C3×M16#4112yes—
193C2×C3×C8—21no—
194C2×C3×C8—22no—
195C2×C3×C8—21no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—41no—
199C3×C4⋊C4—41no—
200C3×C42C2×D411yescharacteristic
201C3×C42C2×D411yescharacteristic
202C3×C42C2×D411yescharacteristic
203C3×C42C2×D411yescharacteristic
204#217—21no—
205(C2×D3)⋊C4—21no—
206(C2×D3)⋊C4—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3—21no—
209C4⋊Dic3C2×D411yescharacteristic
210C4⋊Dic3C2×D411yescharacteristic
211Dic3⋊C4—21no—
212Dic3⋊C4—21no—
213Dic3⋊C4—41no—
214Dic3⋊C4—41no—
215C4×Dic3—21no—
216C4×Dic3—21no—
217C4×Dic3—21no—
218C4×Dic3—41no—
219C4×Dic3—41no—
220#208—22no—
221#208—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8C2×D412yes—
226C2×C3⋊C8C2×D412yes—
227Dic12—24no—
228#204—24no—
229C2×#41—31no—
230C22×Q8—21no—
231C22×Q8—31no—
232#137—62no—
233#137—62no—
234C2×Q16—61no—
235C2×SD16—61no—
236C2×M16—32no—
237C2×M16C22×D311yescharacteristic
238C2×M16C22×D312yes—
239C2×M16C22×D311yescharacteristic
240C22×C8—32no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4×Q8—31no—
246C4×Q8—31no—
247C4×Q8—61no—
248C4×Q8—61no—
249C4×Q8—61no—
250C4×Q8—61no—
251C4×D4—31no—
252C4×D4—31no—
253C4×D4—61no—
254C4×D4—61no—
255#117—31no—
256#117—61no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4C22×D312yes—
261C2×C42—61no—
262C2×C42—61no—
263C2×C42C22×D311yescharacteristic
264#108—64no—
265#107—62no—
266#107—62no—
267#107—62no—
268#107—61no—
269#106—62no—
270#106—61no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—31no—
274C4⋊C8—31no—
275C4⋊C8—32no—
276C4⋊C8—61no—
277C4⋊C8—61no—
278#103—62no—
279#103—62no—
280#103—62no—
281#103—61no—
282#103—61no—
283#103—61no—
284#102—62no—
285#102—61no—
286#101—64no—
287C22⋊C8—62no—
288C22⋊C8—62no—
289#97—21no—
290#97—62no—
291C4×C8—21no—
292C4×C8—62no—
293C23×C3C2×#4111yescharacteristic
294C22×D3—21no—
295C3×Q8—82no—
296C3×Q8—82no—
297C2×C3×C4—22no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—82no—
301C2×C3×C4#12111yescharacteristic
302C2×C3×C4#12111yescharacteristic
303C2×C3×C4#12111yescharacteristic
304C2×C3×C4#12111yescharacteristic
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C42−1+411yescharacteristic
310C2×C3×C42−1+411yescharacteristic
311#67—41no—
312#67—41no—
313C2×Dic3—21no—
314C2×Dic3—21no—
315C2×Dic3—21no—
316C2×Dic3—22no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—81no—
321D12—21no—
322D12—21no—
323C4×D3—41no—
324C4×D3—41no—
325C4×D3—42no—
326C4×D3—81no—
327Dic6—22no—
328Dic6—21no—
329Dic6—21no—
330Dic6—22no—
331Dic6—41no—
332Dic6—41no—
333C3×C8—22no—
334C3×C8—41no—
335C3×C8—42no—
336C3⋊C8—22no—
337C3⋊C8—22no—
338C3⋊C8—42no—
339#41—61no—
340#41—61no—
341#41—61no—
342#41—61no—
343C2×Q8—31no—
344C2×Q8—32no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—61no—
348C2×Q8—61no—
349C2×Q8—61no—
350C2×Q8—61no—
351C2×Q8—61no—
352C2×Q8—82no—
353C2×D4—31no—
354C2×D4—61no—
355C22×C4—22no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4—61no—
360C22×C4—61no—
361C22×C4#23511yescharacteristic
362C22×C4#23511yescharacteristic
363C22×C4C23×D311yescharacteristic
364Q16—64no—
365SD16—64no—
366M16—21no—
367M16—21no—
368M16—22no—
369M16—22no—
370M16—21no—
371M16—62no—
372M16—62no—
373M16#23912yes—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—22no—
377C2×C8—32no—
378C2×C8—32no—
379C2×C8—62no—
380C2×C8—62no—
381C2×C8—62no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—31no—
385C4⋊C4—31no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—61no—
389C4⋊C4—61no—
390C4⋊C4—61no—
391C4⋊C4—61no—
392C4⋊C4—121no—
393C4⋊C4—121no—
394C22⋊C4—61no—
395C22⋊C4—61no—
396C22⋊C4—61no—
397C42—61no—
398C42—61no—
399C42—61no—
400C42—121no—
401C42—121no—
402C42D3×D411yescharacteristic
403C42D3×D411yescharacteristic
404C42D3×D411yescharacteristic
405C42D3×D411yescharacteristic
406C22×C3—21no—
407C22×C3—21no—
408C22×C3C2×#12111yescharacteristic
409C22×C3#55211yescharacteristic
410C22×C3#55611yescharacteristic
411C2×D3—41no—
412C2×D3—41no—
413C2×D3—41no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—82no—
418C3×C4#47312yes—
419C3×C4#52511yescharacteristic
420C3×C4#53611yescharacteristic
421C3×C4#54711yescharacteristic
422C3×C4#54811yescharacteristic
423Dic3—42no—
424Dic3—41no—
425Dic3—41no—
426Dic3—41no—
427Dic3—81no—
428C23—61no—
429C23C2×#23511yescharacteristic
430Q8—62no—
431Q8—62no—
432Q8—61no—
433Q8—61no—
434Q8—82no—
435Q8—82no—
436Q8—121no—
437Q8—121no—
438D4—61no—
439D4—61no—
440D4—121no—
441D4—121no—
442C2×C4—22no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—62no—
449C2×C4—82no—
450C2×C4—121no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—121no—
454C2×C4—122no—
455C2×C4—121no—
456C2×C4—241no—
457C2×C4—241no—
458C2×C4#88411yescharacteristic
459C2×C4#88411yescharacteristic
460C2×C4#88411yescharacteristic
461C2×C4#88411yescharacteristic
462C2×C4C2×D3×D411yescharacteristic
463C2×C4C2×D3×D411yescharacteristic
464C2×C4C2×#23911yescharacteristic
465C2×C4#100711yescharacteristic
466C2×C4D3×#4111yescharacteristic
467C2×C4#101011yescharacteristic
468C8—22no—
469C8—42no—
470C8—41no—
471C8—62no—
472C8—62no—
473C8—122no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3#241511yescharacteristic
477C2×C3C2×#47311yescharacteristic
478C2×C3#306811yescharacteristic
479D3—81no—
480C22—21no—
481C22—21no—
482C22—121no—
483C22—121no—
484C22—121no—
485C22C2×#88411yescharacteristic
486C22#558111yescharacteristic
487C22#558411yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—82no—
492C4—121no—
493C4—122no—
494C4—121no—
495C4—121no—
496C4—241no—
497C4#475512yes—
498C4#562711yescharacteristic
499C4#562811yescharacteristic
500C4#567711yescharacteristic
501C4#572611yescharacteristic
502C3#2472211yescharacteristic, complemented, Sylow
503C2—21no—
504C2—21no—
505C2—241no—
506C2#7662011yescharacteristic
507C2#7870311yescharacteristic
508C2C2×#475511yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1997C2
socleC22×C3C2×#121
prime radicalC2×#235C23
hypercenter#1997D3
hypocenterC3#24722
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
1. C2×#5091C2semidirect
5. C3×#1997C2semidirect
6. #70528C2semidirect
7. #69875C2semidirect
10. #68985C2semidirect
11. #68982C2semidirect
56. #1997D3semidirect
502. C3#24722semidirect

Maximal subgroups

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

subgroupindex
C2×#50912
#832122
#790402
C4×#9092
C3×#19972
#705282
#698752
#690102
#689872
#689852
#689822
#247223