Finite Group Databasebeta

← #10818078

Subgroups — #10818078

Order 768.

844 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4896C211yesmaximal, characteristic
2#79202C212yesmaximal
3#79146C211yesmaximal, characteristic
4#78573C211yesmaximal, characteristic, complemented
5C3×#1952C211yesmaximal, characteristic, complemented
6#70702C211yesmaximal, characteristic, complemented
7#70677C212yesmaximal, complemented
8#70627C211yesmaximal, characteristic, complemented
9#69887C211yesmaximal, characteristic
10#69004C212yesmaximal
11#68982C212yesmaximal, complemented
12#23045—31nomaximal, Sylow
13C2×#909C2212yes—
14C22×Dic12C2211yescharacteristic
15C2×#901C2211yescharacteristic
16#5516C2212yes—
17#5504C2212yes—
18C2×C3×#108C2211yescharacteristic
19C3×#428C2212yes—
20C2×C3×#107C2211yescharacteristic
21C3×#405C2211yescharacteristic
22C2×#825C2211yescharacteristic
23C2×#824C2212yes—
24#5122C2212yes—
25#5120C2212yes—
26#5106C2211yescharacteristic
27C2×C8×Dic3C2211yescharacteristic
28#5002C2211yescharacteristic
29#4973C2212yes—
30C2×#810C2212yes—
31#4964C2212yes—
32#4896C418yescomplemented
33#4889—22no—
34#4881—22no—
35#4880—21no—
36D3×#107—21no—
37#4873—22no—
38#4853—22no—
39#4852—22no—
40#4716—22no—
41C4×Dic12—21no—
42C4×Dic12—21no—
43C3×#343C2212yes—
44#4558C2211yescharacteristic
45#4555C2212yes—
46#4532C2211yescharacteristic
47C2×#473—31no—
48#2950—31no—
49#2947—32no—
50#2913—31no—
51#1969—31no—
52#1952D311yescharacteristic, complemented
53#1930—32no—
54#1882—32no—
55#1857—32no—
56#1772—31no—
57#1764—31no—
58C2×#235C2311yescharacteristic
59C22×Dic6C2312yes—
60C2×C3×M16C2311yescharacteristic
61C2×C3×M16C2312yes—
62C22×C3×C8C2311yescharacteristic
63C3×#117C2311yescharacteristic
64C2×C3×C4⋊C4C2312yes—
65C4×#67—42no—
66#926C2312yes—
67C2×C4×Dic3C2311yescharacteristic
68C2×#208C2311yescharacteristic
69C22×C3⋊C8C2311yescharacteristic
70#909—24no—
71#909C2×C418yes—
72C2×Dic12—22no—
73C2×Dic12—21no—
74C2×Dic12—21no—
75C2×Dic12—22no—
76C2×Dic12C2×C414yescomplemented
77C2×#204—22no—
78#901—22no—
79#901C2×C414yes—
80C2×ΓL(1,𝔽25)—21no—
81C2×D3×C8—21no—
82#893—22no—
83#892—22no—
84D3×C4⋊C4—22no—
85#887—22no—
86#887—22no—
87#887—22no—
88#877—42no—
89C4×Dic6—22no—
90C4×Dic6—22no—
91C3×#108C2×C414yescomplemented
92C3×#107—21no—
93C3×#107—21no—
94C3×#107—22no—
95C3×#106—22no—
96C3×#97—21no—
97C3×C4×C8—21no—
98C3⋊#95—42no—
99#825—22no—
100#824C2×C418yescomplemented
101#818D411yescharacteristic
102#818D411yescharacteristic
103#817D412yes—
104#816—22no—
105#816—22no—
106#815—22no—
107C8×Dic3—22no—
108C8×Dic3—22no—
109#810—22no—
110#810—22no—
111#810D411yescharacteristic
112#810D411yescharacteristic
113#809D411yescharacteristic
114#809D411yescharacteristic
115#808—22no—
116#807—22no—
117C2×#137—32no—
118C22×Q16—31no—
119C2×#131—31no—
120#520—32no—
121#519—32no—
122#473—38no—
123#441—62no—
124#441—62no—
125#440—62no—
126C4×Q16—61no—
127C4×Q16—61no—
128C4×Q16—62no—
129C4×SD16—62no—
130C2×#108C2×D311yescharacteristic
131#428—32no—
132#428—62no—
133#428C2×D312yes—
134#427—31no—
135#427—61no—
136C2×#107—61no—
137C2×#107C2×D311yescharacteristic
138C2×#104—31no—
139#419—32no—
140#416—31no—
141C2×#103—32no—
142C2×#101—32no—
143#405—32no—
144#405C2×D311yescharacteristic
145C2×C4×C8—31no—
146#391—32no—
147#343C2×D312yes—
148#341—31no—
149#339—32no—
150#339—31no—
151C22×C3×C4—22no—
152C22×C3×C4C2411yescharacteristic
153C2×#67—21no—
154C22×Dic3—21no—
155C22×Dic3—22no—
156#235—21no—
157#235—21no—
158#235C22×C412yes—
159#235#4112yes—
160C2×D12C2×D411yescharacteristic
161C2×C4×D3—21no—
162C2×C4×D3—42no—
163C2×Dic6—22no—
164C2×Dic6—22no—
165C2×Dic6—22no—
166C2×Dic6—22no—
167C2×Dic6—22no—
168C2×Dic6C22×C414yes—
169C2×Dic6C2×D411yescharacteristic
170C3×M16—22no—
171C3×M16—21no—
172C3×M16—21no—
173C3×M16C22×C414yes—
174C2×C3×C8—22no—
175C2×C3×C8—21no—
176C2×C3×C8C22×C412yes—
177C2×C3×C8C2×D411yescharacteristic
178C2×C3×C8C2×D411yescharacteristic
179C2×C3×C8#4112yes—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4C2×D411yescharacteristic
183C3×C4⋊C4C2×D411yescharacteristic
184C3×C22⋊C4—21no—
185C3×C42—21no—
186#217—22no—
187(C2×D3)⋊C4—42no—
188C4⋊Dic3C2×D412yes—
189C4⋊Dic3C2×D412yes—
190Dic3⋊C4—42no—
191Dic3⋊C4—42no—
192Dic3⋊C4—42no—
193C4×Dic3—22no—
194C4×Dic3—21no—
195C4×Dic3—21no—
196C4×Dic3—42no—
197C4×Dic3—42no—
198C4×Dic3—42no—
199#208—21no—
200#208C22×C412yes—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8C2×D411yescharacteristic
205C2×C3⋊C8C2×D411yescharacteristic
206Dic12—28no—
207Dic12—24no—
208Dic12—24no—
209#204—28no—
210ΓL(1,𝔽25)—24no—
211D3×C8—24no—
212C2×#41—31no—
213C22×Q8—32no—
214#137—38no—
215#137—64no—
216C2×Q16—34no—
217C2×Q16—62no—
218C2×Q16—61no—
219C2×Q16—61no—
220C2×Q16—62no—
221C2×SD16—62no—
222#131—34no—
223#131—62no—
224C2×M16—31no—
225C2×M16—61no—
226C2×M16C22×D311yescharacteristic
227C2×M16C22×D312yes—
228C22×C8—31no—
229C22×C8—61no—
230C22×C8C22×D311yescharacteristic
231C4×Q8—62no—
232C4×Q8—62no—
233C4×Q8—62no—
234C4×Q8—62no—
235C4×Q8—62no—
236C4×D4—62no—
237C4×D4—122no—
238#117—32no—
239#117—62no—
240#117—122no—
241#117C22×D311yescharacteristic
242C2×C4⋊C4—62no—
243C2×C4⋊C4C22×D312yes—
244C2×C42—31no—
245#108C4×D314yescomplemented
246#107—21no—
247#107—21no—
248#107—22no—
249#107—31no—
250#107—31no—
251#107—62no—
252#106—22no—
253#106—32no—
254#106—62no—
255#104—62no—
256#103—31no—
257#103—31no—
258#103—62no—
259#103—62no—
260#103—62no—
261#103—62no—
262#102—31no—
263#102—31no—
264#101—38no—
265#97—21no—
266#97—62no—
267C4×C8—21no—
268C4×C8—62no—
269C4×C8—62no—
270#95—122no—
271C23×C3H(5,𝔽2)11yescharacteristic
272C22×D3—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—42no—
276C2×C3×C4—82no—
277C2×C3×C4C23×C411yescharacteristic
278C2×C3×C4C22×D411yescharacteristic
279C2×C3×C4C22×D411yescharacteristic
280C2×C3×C4C2×#4111yescharacteristic
281C2×C3×C4C2×#4111yescharacteristic
282C2×C3×C42−1+411yescharacteristic
283#67—22no—
284#67—22no—
285C2×Dic3—21no—
286C2×Dic3—22no—
287C2×Dic3—24no—
288C2×Dic3—22no—
289C2×Dic3—22no—
290C2×Dic3—41no—
291C2×Dic3—42no—
292C2×Dic3—41no—
293C2×Dic3—42no—
294D12—21no—
295D12C4×D412yes—
296C4×D3—22no—
297C4×D3—22no—
298C4×D3—162no—
299Dic6—21no—
300Dic6—24no—
301Dic6—24no—
302Dic6—42no—
303Dic6—42no—
304Dic6C4×D412yes—
305C3×C8—24no—
306C3×C8—41no—
307C3×C8C4×D412yes—
308C3×C8C4×D412yes—
309C3⋊C8—22no—
310C3⋊C8—21no—
311C3⋊C8—21no—
312#41—32no—
313#41—32no—
314#41—61no—
315#41—61no—
316C2×Q8—34no—
317C2×Q8—31no—
318C2×Q8—62no—
319C2×Q8—62no—
320C2×Q8—62no—
321C2×Q8—62no—
322C2×Q8—62no—
323C2×D4—31no—
324C2×D4—61no—
325C22×C4—22no—
326C22×C4—61no—
327C22×C4—61no—
328C22×C4—62no—
329C22×C4—122no—
330C22×C4C23×D311yescharacteristic
331Q16—68no—
332Q16—64no—
333Q16—64no—
334SD16—68no—
335M16—22no—
336M16—21no—
337M16—21no—
338M16—32no—
339M16—61no—
340M16—64no—
341M16C2×C4×D314yes—
342C2×C8—22no—
343C2×C8—21no—
344C2×C8—31no—
345C2×C8—31no—
346C2×C8—61no—
347C2×C8—61no—
348C2×C8—61no—
349C2×C8—64no—
350C2×C8C2×C4×D312yes—
351C2×C8D3×D411yescharacteristic
352C2×C8D3×D411yescharacteristic
353C2×C8#23912yes—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—32no—
357C4⋊C4—32no—
358C4⋊C4—122no—
359C4⋊C4—122no—
360C4⋊C4—122no—
361C4⋊C4D3×D411yescharacteristic
362C4⋊C4D3×D411yescharacteristic
363C22⋊C4—21no—
364C22⋊C4—62no—
365C22⋊C4—122no—
366C42—21no—
367C42—62no—
368C42—61no—
369C42—61no—
370C42—122no—
371C42—122no—
372C42—122no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3#52011yescharacteristic
376C22×C3D4211yescharacteristic
377C22×C3#55311yescharacteristic
378C2×D3—22no—
379C2×D3—41no—
380C2×D3—41no—
381C3×C4—41no—
382C3×C4—41no—
383C3×C4—82no—
384C3×C4C2×C4×D411yescharacteristic
385C3×C4C2×C4×D411yescharacteristic
386C3×C4#54711yescharacteristic
387C3×C4#54811yescharacteristic
388Dic3—22no—
389Dic3—42no—
390Dic3—44no—
391Dic3—42no—
392Dic3—42no—
393C23—61no—
394C23#100911yescharacteristic
395Q8—32no—
396Q8—61no—
397Q8—64no—
398Q8—64no—
399Q8—122no—
400Q8—122no—
401D4—32no—
402D4—62no—
403D4—62no—
404D4—61no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—42no—
408C2×C4—61no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—64no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—82no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4—482no—
421C2×C4C22×C4×D311yescharacteristic
422C2×C4C2×D3×D411yescharacteristic
423C2×C4C2×D3×D411yescharacteristic
424C2×C4C2×#23911yescharacteristic
425C2×C4#100711yescharacteristic
426C2×C4D3×#4111yescharacteristic
427C8—24no—
428C8—41no—
429C8—61no—
430C8—61no—
431C8—62no—
432C8#89312yes—
433C8#89312yes—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3#236111yescharacteristic
437C2×C3#329011yescharacteristic
438C2×C3#329311yescharacteristic
439D3—42no—
440C22—21no—
441C22—21no—
442C22—62no—
443C22—121no—
444C22—121no—
445C22#557411yescharacteristic
446C22#560911yescharacteristic
447C22#569511yescharacteristic
448C4—41no—
449C4—41no—
450C4—62no—
451C4—82no—
452C4—122no—
453C4—122no—
454C4—122no—
455C4—124no—
456C4C2×#89311yescharacteristic
457C4C4×D3×D411yescharacteristic
458C4#563311yescharacteristic
459C4#572611yescharacteristic
460C3#2304511yescharacteristic, complemented, Sylow
461C2—21no—
462C2—21no—
463C2—122no—
464C2#7620611yescharacteristic
465C2#8094011yescharacteristic
466C2#8094711yescharacteristic
subgroupG/H
centerC22#5609
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1952C2
socleC22×C3D42
prime radicalC2×#235C23
cosocleC22×C3×C4C24
hypercenter#1952D3
hypocenterC3#23045
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1952

Complements

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

NH ≅ G/Ntype
4. #78573C2semidirect
5. C3×#1952C2semidirect
6. #70702C2semidirect
7. #70677C2semidirect
8. #70627C2semidirect
11. #68982C2semidirect
32. #4896C4semidirect
52. #1952D3semidirect
76. C2×Dic12C2×C4semidirect
91. C3×#108C2×C4semidirect
100. #824C2×C4semidirect
245. #108C4×D3semidirect
460. C3#23045semidirect

Maximal subgroups

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

subgroupindex
C2×#48962
#792022
#791462
#785732
C3×#19522
#707022
#706772
#706272
#698872
#690042
#689822
#230453