Finite Group Databasebeta

← #10784415

Subgroups — #10784415

Order 768.

670 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
1C3×#3230C211yesmaximal, characteristic, complemented
2#83371C211yesmaximal, characteristic, complemented
3#83356C211yesmaximal, characteristic
4#83222C211yesmaximal, characteristic, complemented
5#83215C211yesmaximal, characteristic
6#79270C211yesmaximal, characteristic
7#71367C211yesmaximal, characteristic, complemented
8#71366C211yesmaximal, characteristic, complemented
9#71365C211yesmaximal, characteristic, complemented
10#71362C211yesmaximal, characteristic, complemented
11#71227C211yesmaximal, characteristic, complemented
12#71225C211yesmaximal, characteristic, complemented
13#71219C211yesmaximal, characteristic, complemented
14#71215C211yesmaximal, characteristic, complemented
15#70744C211yesmaximal, characteristic, complemented
16#19744—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19#5477C2211yescharacteristic
20#5470C2211yescharacteristic
21C3×#500C2211yescharacteristic
22C3×#500C2211yescharacteristic
23C3×#487C2211yescharacteristic
24C3×#487C2211yescharacteristic
25C3×#484—22no—
26C3×#477—22no—
27C3×#431C2211yescharacteristic
28#5228—22no—
29#5093C2211yescharacteristic
30#5093C2211yescharacteristic
31#5093C2211yescharacteristic
32#5092C2211yescharacteristic
33#5092C2211yescharacteristic
34#5092C2211yescharacteristic
35#5090—22no—
36#5088C2211yescharacteristic
37#5088C2211yescharacteristic
38#5088C2211yescharacteristic
39#5087—22no—
40#5085C2211yescharacteristic
41#5080C2211yescharacteristic
42#5079C2211yescharacteristic
43#5079C2211yescharacteristic
44#5078C2211yescharacteristic
45#5078C2211yescharacteristic
46#5078C2211yescharacteristic
47#5063—22no—
48#5047—22no—
49#5028—22no—
50#4996—22no—
51#4967—22no—
52#4925C2211yescharacteristic
53#4925C2211yescharacteristic
54#4847—22no—
55#4737—22no—
56#4694C2211yescharacteristic
57#4694C2211yescharacteristic
58#4682C2211yescharacteristic
59#4680C2211yescharacteristic
60#4455C2211yescharacteristic
61#4454C2211yescharacteristic
62#4454C2211yescharacteristic
63C8×C3⋊C8C2211yescharacteristic
64#3231—31no—
65#3230D311yescharacteristic, complemented
66#3228—31no—
67#3226—31no—
68#3210—31no—
69#3209—31no—
70#1723—31no—
71#1723—31no—
72#1722—31no—
73#1720—31no—
74#1719—31no—
75#1718—31no—
76#1713—31no—
77#1711—31no—
78#1454—31no—
79C22×C3×Q8—21no—
80C2×#235—21no—
81C3×C4⋊Q8—21no—
82C3×C4⋊Q8—21no—
83C3×C4⋊Q8—21no—
84C3×C4⋊Q8C2311yescharacteristic
85C3×C4⋊Q8C2311yescharacteristic
86C3×C4⋊Q8C2311yescharacteristic
87C3×C4⋊Q8C2311yescharacteristic
88C3×#125—21no—
89C3×C22⋊Q8—21no—
90C3×C22⋊Q8—21no—
91C3×C4×Q8—21no—
92C3×C4×Q8—21no—
93C3×#117—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C42C2311yescharacteristic
97C2×#216—22no—
98C2×#216—22no—
99C2×#216—22no—
100C2×#215—22no—
101#930—21no—
102#930—21no—
103#926—21no—
104#924—21no—
105#924—21no—
106#875—21no—
107#874C2311yescharacteristic
108C4×D12—21no—
109C4×D12—21no—
110#870—21no—
111#869C2311yescharacteristic
112#869C2311yescharacteristic
113#869C2311yescharacteristic
114C4×Dic6—21no—
115C4×Dic6—21no—
116C3×#107—22no—
117C3×#107—22no—
118C3×C4⋊C8D412yes—
119C3×#103—22no—
120C3×#103—22no—
121C3×C22⋊C8D412yes—
122C3×C4×C8C2311yescharacteristic
123C3×C4×C8C2311yescharacteristic
124#835—22no—
125#835—22no—
126#830D412yes—
127#830D412yes—
128#821—22no—
129#821—22no—
130#818—22no—
131#817—22no—
132#810—22no—
133#810—22no—
134#810—22no—
135#809—22no—
136#809—22no—
137#809—22no—
138#808—22no—
139#807—22no—
140#807—22no—
141#807—22no—
142C4⋊(C3⋊C8)D412yes—
143C4⋊(C3⋊C8)D412yes—
144C4×C3⋊C8C2311yescharacteristic
145C4×C3⋊C8C2311yescharacteristic
146C4×C3⋊C8C2311yescharacteristic
147C4×C3⋊C8C2311yescharacteristic
148#533—31no—
149#533C2×D311yescharacteristic
150#532—31no—
151C2×C4⋊Q8C2×D311yescharacteristic
152#500—31no—
153#500—31no—
154#500—31no—
155#500—31no—
156#500C2×D311yescharacteristic
157#500C2×D311yescharacteristic
158#498—31no—
159#498—31no—
160#494—31no—
161#494—31no—
162#494—31no—
163#492—31no—
164#487—31no—
165#487—31no—
166#487—31no—
167#487—31no—
168#487—31no—
169#487C2×D311yescharacteristic
170#487C2×D311yescharacteristic
171#486—31no—
172#486—31no—
173#486—31no—
174#486—31no—
175#486—31no—
176#485—62no—
177#484—22no—
178#482—62no—
179#480—62no—
180#479—62no—
181#478—62no—
182#477—22no—
183#474—62no—
184#464—62no—
185#462—62no—
186#452—62no—
187#451—62no—
188#431—31no—
189#431—31no—
190#431C2×D311yescharacteristic
191#322—31no—
192#322—31no—
193#322—31no—
194C82—31no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—41no—
200C2×C3×Q8—82no—
201C22×C3×C4—41no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C2×#67—41no—
206#235—82no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×C4×D3—41no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×C3×C8C2×D412yes—
215C2×C3×C8C2×D412yes—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—41no—
225C3×C4⋊C4—41no—
226C3×C4⋊C4—41no—
227C3×C4⋊C4—41no—
228C3×C22⋊C4—41no—
229C3×C42—41no—
230C3×C42C2×D411yescharacteristic
231C3×C42C2×D411yescharacteristic
232C3×C42C2×D411yescharacteristic
233C3×C42C2411yescharacteristic
234#217—41no—
235#216—82no—
236#216—82no—
237#216—82no—
238#215—82no—
239(C2×D3)⋊C4—41no—
240(C2×D3)⋊C4—41no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247Dic3⋊C4—41no—
248Dic3⋊C4—41no—
249C4×Dic3—41no—
250C2×C3⋊C8C2×D412yes—
251C2×C3⋊C8C2×D412yes—
252C2×C3⋊C8C2×D412yes—
253C2×C3⋊C8C2×D412yes—
254C2×#41—61no—
255C22×Q8—21no—
256C2×Q16—62no—
257C2×Q16—62no—
258C2×Q16—62no—
259C2×SD16—62no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—31no—
264C4⋊Q8—31no—
265C4⋊Q8—31no—
266C4⋊Q8C22×D311yescharacteristic
267C4⋊Q8C22×D311yescharacteristic
268C4⋊Q8C22×D311yescharacteristic
269C4⋊Q8C22×D311yescharacteristic
270#127—31no—
271#125—21no—
272#125—61no—
273#124—61no—
274C22⋊Q8—21no—
275C22⋊Q8—21no—
276C22⋊Q8—61no—
277C22⋊Q8—61no—
278#121—61no—
279#121—61no—
280C4×Q8—21no—
281C4×Q8—21no—
282C4×Q8—61no—
283C4×Q8—61no—
284C4×D4—61no—
285C4×D4—61no—
286#117—21no—
287#117—61no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C42C22×D311yescharacteristic
291#107—22no—
292#107—22no—
293#107—62no—
294#107—62no—
295#107—62no—
296#107—62no—
297#106—62no—
298#106—62no—
299C4⋊C8—32no—
300C4⋊C8—32no—
301C4⋊C8#6712yes—
302#103—22no—
303#103—22no—
304#103—62no—
305#103—62no—
306#103—62no—
307#103—62no—
308#103—62no—
309#102—62no—
310#102—62no—
311#102—62no—
312#102—62no—
313#102—62no—
314C22⋊C8—32no—
315C22⋊C8—32no—
316C22⋊C8#6712yes—
317C4×C8—31no—
318C4×C8—31no—
319C4×C8—31no—
320C4×C8—31no—
321C4×C8C22×D311yescharacteristic
322C4×C8C22×D311yescharacteristic
323C23×C3#12711yescharacteristic
324C22×D3—41no—
325C3×Q8—42no—
326C3×Q8—42no—
327C3×Q8—42no—
328C3×Q8—42no—
329C3×Q8—82no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—82no—
338C2×C3×C4#12711yescharacteristic
339C2×C3×C4#12711yescharacteristic
340C2×C3×C4#12711yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344#67—82no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348D12—42no—
349D12—42no—
350D12—42no—
351C4×D3—82no—
352Dic6—42no—
353Dic6—42no—
354Dic6—42no—
355C3×C8—22no—
356C3×C8—22no—
357C3⋊C8—22no—
358C3⋊C8—22no—
359C3⋊C8—22no—
360C3⋊C8—22no—
361#41—242no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—41no—
367C2×Q8—61no—
368C2×Q8—61no—
369C2×Q8—61no—
370C2×Q8—82no—
371C2×D4—61no—
372C2×D4—61no—
373C2×D4—61no—
374C2×D4—121no—
375C22×C4—41no—
376C22×C4—121no—
377C22×C4D3×D411yescharacteristic
378C22×C4D3×D411yescharacteristic
379C22×C4C2×#6711yescharacteristic
380Q16—242no—
381Q16—242no—
382Q16—242no—
383SD16—242no—
384C2×C8—32no—
385C2×C8—32no—
386C2×C8—32no—
387C2×C8—32no—
388C2×C8C2×#6712yes—
389C2×C8C2×#6712yes—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—61no—
403C4⋊C4—61no—
404C4⋊C4—61no—
405C4⋊C4—61no—
406C4⋊C4—61no—
407C4⋊C4—61no—
408C4⋊C4—121no—
409C4⋊C4—121no—
410C22⋊C4—41no—
411C22⋊C4—121no—
412C22⋊C4—121no—
413C22⋊C4—121no—
414C42—41no—
415C42—121no—
416C42D3×D411yescharacteristic
417C42D3×D411yescharacteristic
418C42C2×#6711yescharacteristic
419C42C23×D311yescharacteristic
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3C2×#12711yescharacteristic
424C2×D3—81no—
425C2×D3—81no—
426C2×D3—81no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—81no—
431C3×C4—81no—
432C3×C4—81no—
433C3×C4—81no—
434C3×C4#57612yes—
435C3×C4#57812yes—
436C3×C4#57812yes—
437Dic3—81no—
438Dic3—81no—
439Dic3—81no—
440C23—121no—
441C23#94011yescharacteristic
442Q8—42no—
443Q8—42no—
444Q8—42no—
445Q8—42no—
446Q8—82no—
447Q8—122no—
448Q8—122no—
449Q8—122no—
450D4—122no—
451D4—122no—
452D4—122no—
453D4—242no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—82no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—242no—
466C2×C4#94011yescharacteristic
467C2×C4#94011yescharacteristic
468C2×C4#94011yescharacteristic
469C2×C4C2×D3×D411yescharacteristic
470C2×C4C2×D3×D411yescharacteristic
471C2×C4C22×#6711yescharacteristic
472C8—22no—
473C8—22no—
474C8—62no—
475C8—62no—
476C8—62no—
477C8—62no—
478C2×C3—41no—
479C2×C3#315511yescharacteristic
480C2×C3#315611yescharacteristic
481C2×C3#315611yescharacteristic
482D3—161no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—241no—
487C22—241no—
488C22—241no—
489C22C2×#94011yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—81no—
497C4—241no—
498C4—241no—
499C4—241no—
500C4#576612yes—
501C4#577412yes—
502C4#583812yes—
503C3#1974411yescharacteristic, complemented, Sylow
504C2—41no—
505C2—481no—
506C2#8340311yescharacteristic
507C2#8388111yescharacteristic
508C2#8394011yescharacteristic
subgroupG/H
centerC22C2×#940
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3230C2
socleC22×C3C2×#127
prime radical#5477C22
hypercenter#3230D3
hypocenterC3#19744
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
1. C3×#3230C2semidirect
2. #83371C2semidirect
4. #83222C2semidirect
7. #71367C2semidirect
8. #71366C2semidirect
9. #71365C2semidirect
10. #71362C2semidirect
11. #71227C2semidirect
12. #71225C2semidirect
13. #71219C2semidirect
14. #71215C2semidirect
15. #70744C2semidirect
65. #3230D3semidirect
503. C3#19744semidirect

Maximal subgroups

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

subgroupindex
C3×#32302
#833712
#833562
#832222
#832152
#792702
#713672
#713662
#713652
#713622
#712272
#712252
#712192
#712152
#707442
#197443