Finite Group Databasebeta

← #10813664

Subgroups — #10813664

Order 768.

846 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5144C211yesmaximal, characteristic
2#83784C211yesmaximal, characteristic, complemented
3#79116C211yesmaximal, characteristic, complemented
4#78868C212yesmaximal
5C3×#1815C211yesmaximal, characteristic, complemented
6#70720C211yesmaximal, characteristic, complemented
7#70702C211yesmaximal, characteristic, complemented
8#70555C211yesmaximal, characteristic, complemented
9#70298C212yesmaximal, complemented
10#69003C212yesmaximal, complemented
11#68985C212yesmaximal, complemented
12#22690—31nomaximal, Sylow
13C2×C3×#131C2211yescharacteristic
14#5827C2212yes—
15C2×#950C2211yescharacteristic
16C2×#901C2211yescharacteristic
17#5504C2212yes—
18C2×C3×#108C2211yescharacteristic
19C2×C3×#106C2211yescharacteristic
20C3×#417C2212yes—
21C2×#837C2211yescharacteristic
22#5144C418yescomplemented
23C2×#825C2211yescharacteristic
24#5133C2212yes—
25C2×#822C2211yescharacteristic
26#5122C2212yes—
27#5121—22no—
28#5121C2212yes—
29#5106—21no—
30#5106C2211yescharacteristic
31C2×#817—21no—
32C2×#817C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34#4977C2212yes—
35#4964C2212yes—
36C2×#807C2212yes—
37#4861—22no—
38#4860—21no—
39D3×#106—21no—
40#4817—22no—
41#4790—22no—
42#4757—22no—
43#4756—22no—
44C3×#343C2212yes—
45#4555C2212yes—
46#4554C2212yes—
47C2×#443—31no—
48#2951—32no—
49#2912—31no—
50#2912—31no—
51#1952—32no—
52#1860—32no—
53#1860—32no—
54#1846—31no—
55#1815—31no—
56#1815D311yescharacteristic, complemented
57#1772—31no—
58C2×C3×#41C2311yescharacteristic
59C2×#235C2311yescharacteristic
60C3×#131—22no—
61C3×#131C2×C414yescomplemented
62C2×C3×M16—21no—
63C2×C3×M16C2312yes—
64C2×C3×M16C2311yescharacteristic
65C22×C3×C8—21no—
66C22×C3×C8C2311yescharacteristic
67C2×C3×C4⋊C4C2312yes—
68#950C2×C414yes—
69#950Q814yes—
70Q8×Dic3—22no—
71C2×#216D411yescharacteristic
72C2×#215D411yescharacteristic
73D4×Dic3—22no—
74D4×Dic3—42no—
75C2×#214D411yescharacteristic
76C2×#213D411yescharacteristic
77C4×#67—42no—
78#926—22no—
79#926C2312yes—
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3C2312yes—
82C2×C4×Dic3C2311yescharacteristic
83C2×#208C2311yescharacteristic
84C22×C3⋊C8C2311yescharacteristic
85#901—22no—
86#901C2×C414yescomplemented
87C2×ΓL(1,𝔽25)—21no—
88C2×D3×C8—21no—
89#893—22no—
90#892—22no—
91D3×C4⋊C4—22no—
92#887—22no—
93C3×#108—22no—
94C3×#106—21no—
95C3×#106—21no—
96C3×#103—22no—
97C3×#102—22no—
98#837C2×C414yescomplemented
99C3⋊#95—42no—
100#825C2×C414yescomplemented
101#822C2×C414yescomplemented
102#821—22no—
103#818—22no—
104#818D412yes—
105#817—21no—
106#817—21no—
107#817—22no—
108#817D411yescharacteristic
109#817D411yescharacteristic
110#816—22no—
111#815—22no—
112C8×Dic3—22no—
113C8×Dic3C2×C414yescomplemented
114#808—22no—
115#807—22no—
116#807—22no—
117#807—22no—
118C2×#135—31no—
119C2×#131—31no—
120C2×#131C2×D311yescharacteristic
121#520—32no—
122#520—32no—
123#443—38no—
124#440—62no—
125#440—62no—
126C4×Q16—62no—
127C4×D8—62no—
128C2×#108—31no—
129C2×#108C2×D311yescharacteristic
130#428—32no—
131#428—32no—
132#428—62no—
133#428—62no—
134#427—31no—
135#427—61no—
136#427—61no—
137C2×#107—32no—
138C2×#106—31no—
139C2×#106—61no—
140C2×#106—61no—
141C2×#106C2×D311yescharacteristic
142C2×#104—31no—
143C2×#104—31no—
144#417—32no—
145#417C2×D312yes—
146#405—32no—
147C2×C4×C8—31no—
148#382—32no—
149#343—32no—
150#343C2×D312yes—
151#339—32no—
152C3×#41—21no—
153C3×#41—21no—
154C3×#41C22×C412yes—
155C3×#41C2×Q812yes—
156C2×C3×Q8C2×D411yescharacteristic
157C2×C3×D4—21no—
158C2×C3×D4C2×D411yescharacteristic
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4C2411yescharacteristic
162C2×#67—21no—
163C22×Dic3—21no—
164C22×Dic3—22no—
165C22×Dic3—42no—
166#235—21no—
167#235—21no—
168#235C22×C412yes—
169#235C2×Q812yes—
170C2×D12C2×D411yescharacteristic
171C2×C4×D3—21no—
172C2×C4×D3—42no—
173C2×Dic6C2×D411yescharacteristic
174C3×M16—21no—
175C3×M16—24no—
176C3×M16—22no—
177C3×M16—22no—
178C3×M16C22×C412yes—
179C2×C3×C8—21no—
180C2×C3×C8—24no—
181C2×C3×C8—22no—
182C2×C3×C8C22×C412yes—
183C2×C3×C8C2×D411yescharacteristic
184C2×C3×C8C2×D411yescharacteristic
185C2×C3×C8#4112yes—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188#217—22no—
189#217—42no—
190#216C4⋊C414yes—
191#215C4⋊C414yes—
192#214C4⋊C414yes—
193#213C4⋊C414yes—
194(C2×D3)⋊C4—42no—
195C4⋊Dic3—22no—
196C4⋊Dic3—22no—
197C4⋊Dic3—42no—
198C4⋊Dic3C2×D412yes—
199C4⋊Dic3C2×D412yes—
200Dic3⋊C4—42no—
201C4×Dic3—22no—
202C4×Dic3—21no—
203C4×Dic3—42no—
204C4×Dic3—42no—
205C4×Dic3C22×C412yes—
206#208—21no—
207#208C22×C412yes—
208C2×C3⋊C8—21no—
209C2×C3⋊C8C22×C412yes—
210C2×C3⋊C8C2×D411yescharacteristic
211C2×C3⋊C8C2×D411yescharacteristic
212C2×C3⋊C8C2×Q812yes—
213ΓL(1,𝔽25)—24no—
214D3×C8—24no—
215C2×#41—31no—
216C2×#41C22×D311yescharacteristic
217#135—34no—
218#135—34no—
219C2×Q16—31no—
220C2×SD16—31no—
221C2×SD16—31no—
222C2×D8—31no—
223#131—22no—
224#131—34no—
225#131—62no—
226#131C4×D314yescomplemented
227C2×M16—21no—
228C2×M16—31no—
229C2×M16—61no—
230C2×M16C22×D312yes—
231C2×M16C22×D311yescharacteristic
232C22×C8—21no—
233C22×C8—31no—
234C22×C8—61no—
235C22×C8C22×D311yescharacteristic
236C4×Q8—62no—
237C4×Q8—62no—
238C4×D4—62no—
239C4×D4—62no—
240C4×D4—122no—
241C4×D4—122no—
242#117—32no—
243#117—62no—
244#117—62no—
245C2×C4⋊C4—32no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4C22×D312yes—
249C2×C42—31no—
250#108—22no—
251#108—34no—
252#107—32no—
253#107—62no—
254#107—62no—
255#107—62no—
256#107—62no—
257#106—21no—
258#106—21no—
259#106—31no—
260#106—31no—
261#106—62no—
262#106—62no—
263#106—61no—
264#106—61no—
265#104—34no—
266#104—34no—
267#103—22no—
268#103—62no—
269#102—22no—
270#102—62no—
271#97—62no—
272C4×C8—34no—
273C4×C8—62no—
274#95—122no—
275C23×C3—21no—
276C23×C3H(5,𝔽2)11yescharacteristic
277C22×D3—21no—
278C3×Q8—21no—
279C3×Q8C2×C4⋊C412yes—
280C3×D4—22no—
281C3×D4—22no—
282C3×D4—21no—
283C3×D4C2×C4⋊C412yes—
284C2×C3×C4—21no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—42no—
288C2×C3×C4—82no—
289C2×C3×C4C23×C411yescharacteristic
290C2×C3×C4C22×D411yescharacteristic
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4C22×Q811yescharacteristic
293C2×C3×C4C2×#4111yescharacteristic
294C2×C3×C42−1+411yescharacteristic
295#67—22no—
296#67—22no—
297C2×Dic3—21no—
298C2×Dic3—22no—
299C2×Dic3—22no—
300C2×Dic3—22no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—82no—
304C2×Dic3—162no—
305D12—21no—
306D12C2×C4⋊C412yes—
307C4×D3—22no—
308C4×D3—22no—
309C4×D3—162no—
310Dic6—21no—
311Dic6C2×C4⋊C412yes—
312C3×C8—22no—
313C3×C8—42no—
314C3×C8C4×D412yes—
315C3×C8C4×D412yes—
316C3⋊C8—22no—
317C3⋊C8C2×C4⋊C412yes—
318C3⋊C8C2×C4⋊C412yes—
319#41—21no—
320#41—21no—
321#41—32no—
322#41—32no—
323#41—61no—
324#41—61no—
325#41C2×C4×D312yes—
326#41D3×Q812yes—
327C2×Q8—31no—
328C2×Q8D3×D411yescharacteristic
329C2×D4—21no—
330C2×D4—31no—
331C2×D4—61no—
332C2×D4D3×D411yescharacteristic
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—61no—
336C22×C4—61no—
337C22×C4—62no—
338C22×C4—122no—
339C22×C4—122no—
340C22×C4C23×D311yescharacteristic
341Q16—34no—
342SD16—34no—
343SD16—34no—
344D8—34no—
345M16—21no—
346M16—24no—
347M16—22no—
348M16—22no—
349M16—32no—
350M16—61no—
351M16—64no—
352M16C2×C4×D312yes—
353C2×C8—21no—
354C2×C8—24no—
355C2×C8—22no—
356C2×C8—32no—
357C2×C8—31no—
358C2×C8—31no—
359C2×C8—32no—
360C2×C8—61no—
361C2×C8—64no—
362C2×C8C2×C4×D312yes—
363C2×C8D3×D411yescharacteristic
364C2×C8D3×D411yescharacteristic
365C2×C8#23712yes—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—32no—
369C4⋊C4—32no—
370C4⋊C4—62no—
371C4⋊C4—62no—
372C4⋊C4—122no—
373C4⋊C4—122no—
374C22⋊C4—62no—
375C22⋊C4—122no—
376C22⋊C4—122no—
377C42—32no—
378C42—62no—
379C42—61no—
380C42—122no—
381C42—122no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—42no—
385C22×C3#52011yescharacteristic
386C22×C3D4211yescharacteristic
387C22×C3#55411yescharacteristic
388C2×D3—22no—
389C2×D3—42no—
390C3×C4—22no—
391C3×C4—82no—
392C3×C4C22×C4⋊C411yescharacteristic
393C3×C4C2×C4×D411yescharacteristic
394C3×C4#54711yescharacteristic
395C3×C4D4×Q811yescharacteristic
396Dic3—22no—
397Dic3—42no—
398Dic3—42no—
399Dic3—42no—
400Dic3—82no—
401C23—21no—
402C23—61no—
403C23#100911yescharacteristic
404Q8—21no—
405Q8—32no—
406Q8—61no—
407Q8D3×C4⋊C412yes—
408D4—22no—
409D4—22no—
410D4—21no—
411D4—32no—
412D4—62no—
413D4—62no—
414D4—61no—
415D4D3×C4⋊C412yes—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—42no—
420C2×C4—61no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—62no—
424C2×C4—62no—
425C2×C4—62no—
426C2×C4—82no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—242no—
430C2×C4—482no—
431C2×C4—482no—
432C2×C4C22×C4×D311yescharacteristic
433C2×C4C2×D3×D411yescharacteristic
434C2×C4C2×D3×D411yescharacteristic
435C2×C4C2×#23711yescharacteristic
436C2×C4C2×D3×Q811yescharacteristic
437C2×C4#101011yescharacteristic
438C8—22no—
439C8—32no—
440C8—32no—
441C8—42no—
442C8—62no—
443C8#67⋊C412yes—
444C8#67⋊C412yes—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—42no—
448C2×C3D4×C4⋊C411yescharacteristic
449C2×C3#329012yes—
450D3—42no—
451C22—22no—
452C22—22no—
453C22—42no—
454C22—62no—
455C22—122no—
456C22#552311yescharacteristic
457C22#560911yescharacteristic
458C22#563511yescharacteristic
459C4—22no—
460C4—62no—
461C4—82no—
462C4—122no—
463C4—122no—
464C4—122no—
465C4—242no—
466C4C2×#67⋊C411yescharacteristic
467C4C2×D3×C4⋊C411yescharacteristic
468C4#561111yescharacteristic
469C4#563211yescharacteristic
470C3#2269011yescharacteristic, complemented, Sylow
471C2—21no—
472C2—21no—
473C2—42no—
474C2—122no—
475C2#7401011yescharacteristic
476C2#8093412yes—
subgroupG/H
centerC22#5609
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1815C2
socleC22×C3D42
prime radicalC2×#950C22
cosocleC22×C3×C4C24
hypercenter#1815D3
hypocenterC3#22690
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1815

Complements

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

NH ≅ G/Ntype
2. #83784C2semidirect
3. #79116C2semidirect
5. C3×#1815C2semidirect
6. #70720C2semidirect
7. #70702C2semidirect
8. #70555C2semidirect
9. #70298C2semidirect
10. #69003C2semidirect
11. #68985C2semidirect
22. #5144C4semidirect
56. #1815D3semidirect
61. C3×#131C2×C4semidirect
86. #901C2×C4semidirect
98. #837C2×C4semidirect
100. #825C2×C4semidirect
101. #822C2×C4semidirect
113. C8×Dic3C2×C4semidirect
226. #131C4×D3semidirect
470. C3#22690semidirect

Maximal subgroups

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

subgroupindex
C2×#51442
#837842
#791162
#788682
C3×#18152
#707202
#707022
#705552
#702982
#690032
#689852
#226903