Finite Group Databasebeta

← #10836673

Subgroups — #10836673

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×C3×#475C211yesmaximal, characteristic, complemented
2#83212C211yesmaximal, characteristic
3C4×#942C211yesmaximal, characteristic
4#79000C211yesmaximal, characteristic
5#71790C211yesmaximal, characteristic, complemented
6#71637C211yesmaximal, characteristic, complemented
7#70593C212yesmaximal
8#70572C212yesmaximal, complemented
9#70474C211yesmaximal, characteristic, complemented
10#70264C212yesmaximal
11#70263C212yesmaximal, complemented
12#24723—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#942C2211yescharacteristic
16C4×#235C2211yescharacteristic
17C3×#475—21no—
18C3×#475—21no—
19C3×#475—22no—
20C2×C3×#106C2212yes—
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#103C2212yes—
23C2×#836C2212yes—
24#5225C2211yescharacteristic
25#5110C2212yes—
26C2×#819C2212yes—
27#5101C2212yes—
28#5094—21no—
29#5091—21no—
30#5087—21no—
31#5084—21no—
32#5030—21no—
33C4×#216—21no—
34#5026—21no—
35C4×#215—21no—
36#4966C2212yes—
37#4966C2211yescharacteristic
38#4964C2212yes—
39#4964C2211yescharacteristic
40#4924C2211yescharacteristic
41#4924C2211yescharacteristic
42C4×#208C2211yescharacteristic
43#4840—22no—
44#4835—21no—
45#4833—21no—
46#4552C2212yes—
47#4530C2212yes—
48#4530C2212yes—
49#4530C2212yes—
50#3079—31no—
51C2×#475D311yescharacteristic, complemented
52C4×#137—31no—
53#2925—31no—
54#2079—32no—
55#2052—32no—
56#1997—31no—
57#1951—32no—
58#1950—31no—
59#1930—32no—
60#1801—31no—
61C22×C3×Q8—21no—
62C22×C3×Q8C2311yescharacteristic
63C2×#235C2311yescharacteristic
64C22×C3×C8C2312yes—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8—21no—
69C3×C4×Q8—21no—
70C3×C4×Q8—21no—
71C3×C4×Q8—21no—
72C3×C4×Q8—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C42—21no—
78C2×C3×C42C2311yescharacteristic
79C2×#216—21no—
80#942—22no—
81#942—22no—
82C2×#215—21no—
83C4×#67—21no—
84C4×#67—21no—
85#926C2311yescharacteristic
86C2×#208C2311yescharacteristic
87C2×#208C2312yes—
88C2×#208C2312yes—
89C2×#208C2311yescharacteristic
90C4×D12D411yescharacteristic
91C4×D12D411yescharacteristic
92#872—21no—
93C42×D3—21no—
94C4×Dic6D411yescharacteristic
95C4×Dic6D411yescharacteristic
96C3×#106—22no—
97C3×#106—22no—
98C3×C4⋊C8D411yescharacteristic
99C3×C4⋊C8D411yescharacteristic
100C3×C4⋊C8D412yes—
101C3×#103—22no—
102C3×#103—22no—
103#836—24no—
104#835—21no—
105#835—21no—
106(C2×D3)⋊C8—22no—
107(C2×D3)⋊C8—22no—
108#819—24no—
109#815—22no—
110C8×Dic3—22no—
111#810—21no—
112#810—22no—
113#809—21no—
114#809—22no—
115#808—21no—
116#808—22no—
117#807—21no—
118#807—22no—
119C4⋊(C3⋊C8)—21no—
120C4⋊(C3⋊C8)—21no—
121C4⋊(C3⋊C8)—21no—
122C4⋊(C3⋊C8)—21no—
123#803—21no—
124C4×C3⋊C8—21no—
125C2×#137—31no—
126C2×C4⋊Q8C2×D311yescharacteristic
127C4×#41—31no—
128C2×C4×Q8C2×D311yescharacteristic
129#477—61no—
130#476—61no—
131#475—21no—
132#475—21no—
133#475—22no—
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—32no—
145#428—31no—
146C2×#106C2×D312yes—
147#423—31no—
148#423—31no—
149C2×C4⋊C8C2×D311yescharacteristic
150#419—31no—
151#417—32no—
152#417—31no—
153C2×#103C2×D312yes—
154C2×#101—32no—
155#408—32no—
156#405—32no—
157C4×M16—31no—
158#343—32no—
159#343—32no—
160#343—32no—
161#341—32no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—41no—
168C2×C3×Q8—41no—
169C2×C3×Q8—82no—
170C2×C3×Q8#4112yes—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4#4111yescharacteristic
175C22×C3×C4#4111yescharacteristic
176C22×C3×C4C2411yescharacteristic
177C2×#67—21no—
178#235—21no—
179#235—21no—
180#235—21no—
181#235—21no—
182C2×D12C2×D411yescharacteristic
183C2×C4×D3—21no—
184C2×C4×D3—21no—
185C2×C4×D3—21no—
186C2×Dic6C2×D411yescharacteristic
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8C2×D412yes—
190C2×C3×C8C2×D412yes—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—41no—
196C3×C4⋊C4—41no—
197C3×C4⋊C4—41no—
198C3×C4⋊C4—41no—
199C3×C42—41no—
200C3×C42—41no—
201C3×C42C2×D411yescharacteristic
202C3×C42C2×D411yescharacteristic
203C3×C42C2×D411yescharacteristic
204C3×C42C2×D411yescharacteristic
205#217—21no—
206#216—24no—
207#215—24no—
208(C2×D3)⋊C4—21no—
209(C2×D3)⋊C4—21no—
210C4⋊Dic3C2×D411yescharacteristic
211C4⋊Dic3C2×D411yescharacteristic
212Dic3⋊C4—21no—
213Dic3⋊C4—21no—
214C4×Dic3—21no—
215C4×Dic3—21no—
216C4×Dic3—21no—
217#208—21no—
218#208—22no—
219#208—22no—
220#208—22no—
221#208—21no—
222#208—22no—
223#208—21no—
224#208#4112yes—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—21no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×#41—31no—
230C22×Q8—21no—
231C22×Q8C22×D311yescharacteristic
232#137—62no—
233#137—62no—
234C2×Q16—61no—
235C2×SD16—61no—
236C2×M16—31no—
237C2×M16—31no—
238C2×M16—32no—
239C2×M16—32no—
240C22×C8C22×D312yes—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8—31no—
250C4×Q8—31no—
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—21no—
259C2×C4⋊C4C22×D311yescharacteristic
260C2×C4⋊C4C22×D312yes—
261C2×C42—21no—
262C2×C42—61no—
263C2×C42C22×D311yescharacteristic
264#108—64no—
265#107—61no—
266#107—62no—
267#106—22no—
268#106—22no—
269#106—61no—
270#106—62no—
271C4⋊C8—61no—
272C4⋊C8—61no—
273C4⋊C8—61no—
274C4⋊C8—61no—
275C4⋊C8#6711yescharacteristic
276C4⋊C8#6711yescharacteristic
277C4⋊C8#6712yes—
278#103—22no—
279#103—22no—
280#103—61no—
281#103—62no—
282#103—61no—
283#103—61no—
284#102—61no—
285#102—62no—
286#101—64no—
287C22⋊C8—62no—
288C22⋊C8—62no—
289#97—62no—
290#97—61no—
291C4×C8—62no—
292C4×C8—61no—
293C23×C3C2×#4111yescharacteristic
294C22×D3—21no—
295C3×Q8—22no—
296C3×Q8—22no—
297C3×Q8—41no—
298C3×Q8—41no—
299C3×Q8—82no—
300C3×Q8—82no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4—81no—
309C2×C3×C4—82no—
310C2×C3×C4#12111yescharacteristic
311C2×C3×C4#12111yescharacteristic
312C2×C3×C4#12111yescharacteristic
313C2×C3×C4#12111yescharacteristic
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C42−1+411yescharacteristic
319C2×C3×C42−1+411yescharacteristic
320#67—41no—
321#67—41no—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3—21no—
325D12—21no—
326D12—21no—
327C4×D3—41no—
328C4×D3—41no—
329C4×D3—42no—
330C4×D3—81no—
331Dic6—21no—
332Dic6—21no—
333C3×C8—22no—
334C3×C8—22no—
335C3⋊C8—22no—
336C3⋊C8—42no—
337C3⋊C8—41no—
338C3⋊C8—42no—
339#41—61no—
340#41—61no—
341#41—61no—
342#41—61no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—31no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—82no—
352C2×Q8#23912yes—
353C2×D4—31no—
354C2×D4—61no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—61no—
359C22×C4—61no—
360C22×C4—61no—
361C22×C4#23711yescharacteristic
362C22×C4#23711yescharacteristic
363C22×C4C23×D311yescharacteristic
364Q16—64no—
365SD16—64no—
366M16—32no—
367M16—61no—
368M16—62no—
369M16—62no—
370M16—62no—
371M16—61no—
372M16—62no—
373M16—61no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—61no—
377C2×C8—61no—
378C2×C8—62no—
379C2×C8—62no—
380C2×C8C2×#6712yes—
381C2×C8C2×#6712yes—
382C4⋊C4—21no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—31no—
387C4⋊C4—31no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C22⋊C4—61no—
395C22⋊C4—61no—
396C22⋊C4—61no—
397C42—41no—
398C42—41no—
399C42—61no—
400C42—61no—
401C42—61no—
402C42D3×D411yescharacteristic
403C42D3×D411yescharacteristic
404C42C2×#6711yescharacteristic
405C42C2×#6711yescharacteristic
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—42no—
418C3×C4—82no—
419C3×C4—81no—
420C3×C4#47112yes—
421C3×C4#52511yescharacteristic
422C3×C4#53611yescharacteristic
423C3×C4#54711yescharacteristic
424C3×C4#54811yescharacteristic
425Dic3—41no—
426Dic3—41no—
427Dic3—41no—
428C23—61no—
429C23C2×#23711yescharacteristic
430Q8—22no—
431Q8—22no—
432Q8—41no—
433Q8—41no—
434Q8—61no—
435Q8—61no—
436Q8—82no—
437Q8—82no—
438D4—61no—
439D4—61no—
440D4—121no—
441D4—121no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—122no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—241no—
458C2×C4#93811yescharacteristic
459C2×C4#93811yescharacteristic
460C2×C4#93811yescharacteristic
461C2×C4#93811yescharacteristic
462C2×C4C2×D3×D411yescharacteristic
463C2×C4C2×#23911yescharacteristic
464C2×C4#100711yescharacteristic
465C2×C4D3×#4111yescharacteristic
466C2×C4#101011yescharacteristic
467C2×C4C22×#6711yescharacteristic
468C8—22no—
469C8—22no—
470C8—62no—
471C8—122no—
472C8—122no—
473C8—121no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3#241511yescharacteristic
477C2×C3C2×#47111yescharacteristic
478C2×C3#306911yescharacteristic
479D3—81no—
480C22—21no—
481C22—21no—
482C22—121no—
483C22—121no—
484C22—121no—
485C22#564111yescharacteristic
486C22#564211yescharacteristic
487C22C2×#93811yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—42no—
492C4—81no—
493C4—82no—
494C4—121no—
495C4—121no—
496C4—121no—
497C4#517712yes—
498C4#562711yescharacteristic
499C4#562811yescharacteristic
500C4#581711yescharacteristic
501C4#582411yescharacteristic
502C3#2472311yescharacteristic, complemented, Sylow
503C2—21no—
504C2—21no—
505C2—241no—
506C2#8281311yescharacteristic
507C2C2×#517711yescharacteristic
508C2#8399811yescharacteristic
subgroupG/H
centerC22C2×#938
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#475C2
socleC22×C3C2×#121
prime radicalC2×#235C23
hypercenterC2×#475D3
hypocenterC3#24723
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
1. C2×C3×#475C2semidirect
5. #71790C2semidirect
6. #71637C2semidirect
8. #70572C2semidirect
9. #70474C2semidirect
11. #70263C2semidirect
51. C2×#475D3semidirect
502. C3#24723semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4752
#832122
C4×#9422
#790002
#717902
#716372
#705932
#705722
#704742
#702642
#702632
#247233