Finite Group Databasebeta

← #10818100

Subgroups — #10818100

Order 768.

840 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4896C211yesmaximal, characteristic
2#79202C212yesmaximal
3#79116C211yesmaximal, characteristic
4#78573C211yesmaximal, characteristic, complemented
5C3×#1951C211yesmaximal, characteristic, complemented
6#70703C211yesmaximal, characteristic, complemented
7#70677C212yesmaximal, complemented
8#70575C211yesmaximal, characteristic, complemented
9#69888C211yesmaximal, characteristic
10#69004C212yesmaximal
11#68982C212yesmaximal, complemented
12#23047—31nomaximal, Sylow
13C2×#909C2212yes—
14C22×Dic12C2211yescharacteristic
15C2×#901C2211yescharacteristic
16#5516C2212yes—
17#5504C2212yes—
18C2×C3×#108C2211yescharacteristic
19C3×#428C2212yes—
20C2×C3×#106C2211yescharacteristic
21C3×#405C2211yescharacteristic
22C2×#825C2211yescharacteristic
23C2×#824C2212yes—
24#5122C2212yes—
25#5120C2212yes—
26#5106C2211yescharacteristic
27C2×#815C2211yescharacteristic
28#5002C2211yescharacteristic
29#4973C2212yes—
30C2×#810C2212yes—
31#4964C2212yes—
32#4896C418yescomplemented
33#4889—22no—
34#4873—22no—
35#4861—22no—
36#4860—21no—
37D3×#106—21no—
38#4853—22no—
39#4852—22no—
40#4716—21no—
41#4716—21no—
42C4×Dic12—22no—
43C3×#343C2212yes—
44#4558C2211yescharacteristic
45#4555C2212yes—
46#4532C2211yescharacteristic
47C2×#473—31no—
48#2950—31no—
49#2947—32no—
50#2912—31no—
51#1971—31no—
52#1951D311yescharacteristic, complemented
53#1930—32no—
54#1882—32no—
55#1857—32no—
56#1773—31no—
57#1765—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—22no—
93C3×#106—21no—
94C3×#106—21no—
95C3×#106—22no—
96C3×#97—21no—
97C3×C4×C8—21no—
98C3⋊#95—42no—
99#825—22no—
100#824C2×C418yescomplemented
101#818D412yes—
102#817D411yescharacteristic
103#817D411yescharacteristic
104#816—22no—
105#816—22no—
106#815—22no—
107#815—22no—
108C8×Dic3—22no—
109#810—22no—
110#810—22no—
111#810D412yes—
112#809D412yes—
113#808—22no—
114#807—22no—
115C2×#137—32no—
116C22×Q16—31no—
117C2×#131—31no—
118#520—32no—
119#519—32no—
120#473—38no—
121#441—61no—
122#441—61no—
123#441—62no—
124#440—62no—
125C4×Q16—62no—
126C4×Q16—62no—
127C4×SD16—62no—
128C2×#108C2×D311yescharacteristic
129#428—32no—
130#428—62no—
131#428C2×D312yes—
132#427—31no—
133#427—61no—
134C2×#106—61no—
135C2×#106C2×D311yescharacteristic
136C2×#104—31no—
137#419—32no—
138#416—31no—
139C2×#103—32no—
140C2×#101—32no—
141#405—32no—
142#405C2×D311yescharacteristic
143C2×#97—31no—
144#391—32no—
145#343C2×D312yes—
146#341—31no—
147#339—32no—
148#339—31no—
149C22×C3×C4—22no—
150C22×C3×C4C2411yescharacteristic
151C2×#67—21no—
152C22×Dic3—22no—
153C22×Dic3—21no—
154#235—21no—
155#235—21no—
156#235C22×C412yes—
157#235#4112yes—
158C2×D12C2×D411yescharacteristic
159C2×C4×D3—21no—
160C2×C4×D3—42no—
161C2×Dic6—22no—
162C2×Dic6—22no—
163C2×Dic6—22no—
164C2×Dic6—22no—
165C2×Dic6—22no—
166C2×Dic6C22×C414yes—
167C2×Dic6C2×D411yescharacteristic
168C3×M16—22no—
169C3×M16—21no—
170C3×M16—21no—
171C3×M16C22×C414yes—
172C2×C3×C8—22no—
173C2×C3×C8—21no—
174C2×C3×C8C22×C412yes—
175C2×C3×C8C2×D411yescharacteristic
176C2×C3×C8C2×D411yescharacteristic
177C2×C3×C8#4112yes—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4C2×D412yes—
181C3×C22⋊C4—21no—
182C3×C42—21no—
183#217—22no—
184(C2×D3)⋊C4—42no—
185C4⋊Dic3C2×D412yes—
186C4⋊Dic3C2×D412yes—
187Dic3⋊C4—42no—
188Dic3⋊C4—42no—
189Dic3⋊C4—42no—
190C4×Dic3—22no—
191C4×Dic3—21no—
192C4×Dic3—21no—
193C4×Dic3—42no—
194C4×Dic3—42no—
195C4×Dic3—42no—
196#208—21no—
197#208C22×C412yes—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8C2×D412yes—
202Dic12—28no—
203Dic12—24no—
204Dic12—24no—
205#204—28no—
206ΓL(1,𝔽25)—24no—
207D3×C8—24no—
208C2×#41—31no—
209C22×Q8—32no—
210#137—38no—
211#137—64no—
212C2×Q16—34no—
213C2×Q16—62no—
214C2×Q16—61no—
215C2×Q16—61no—
216C2×Q16—62no—
217C2×SD16—62no—
218#131—34no—
219#131—62no—
220C2×M16—31no—
221C2×M16—61no—
222C2×M16C22×D311yescharacteristic
223C2×M16C22×D312yes—
224C22×C8—31no—
225C22×C8—61no—
226C22×C8C22×D311yescharacteristic
227C4×Q8—62no—
228C4×Q8—62no—
229C4×Q8—62no—
230C4×Q8—62no—
231C4×Q8—62no—
232C4×D4—62no—
233C4×D4—122no—
234#117—32no—
235#117—62no—
236#117—122no—
237#117C22×D311yescharacteristic
238C2×C4⋊C4—62no—
239C2×C4⋊C4C22×D312yes—
240C2×C42—31no—
241#108C4×D314yescomplemented
242#107—22no—
243#107—32no—
244#107—62no—
245#106—21no—
246#106—21no—
247#106—22no—
248#106—31no—
249#106—31no—
250#106—62no—
251#104—62no—
252#103—32no—
253#103—62no—
254#103—62no—
255#103—62no—
256#103—62no—
257#102—32no—
258#101—38no—
259#97—21no—
260#97—62no—
261#97—62no—
262C4×C8—21no—
263C4×C8—62no—
264#95—122no—
265C23×C3H(5,𝔽2)11yescharacteristic
266C22×D3—21no—
267C2×C3×C4—22no—
268C2×C3×C4—42no—
269C2×C3×C4—82no—
270C2×C3×C4C23×C411yescharacteristic
271C2×C3×C4C22×D411yescharacteristic
272C2×C3×C4C22×D411yescharacteristic
273C2×C3×C4C2×#4111yescharacteristic
274C2×C3×C4C2×#4111yescharacteristic
275C2×C3×C42−1+411yescharacteristic
276#67—22no—
277#67—22no—
278C2×Dic3—21no—
279C2×Dic3—24no—
280C2×Dic3—22no—
281C2×Dic3—22no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—41no—
286D12—21no—
287D12C4×D412yes—
288C4×D3—22no—
289C4×D3—22no—
290C4×D3—162no—
291Dic6—21no—
292Dic6—24no—
293Dic6—24no—
294Dic6—42no—
295Dic6—42no—
296Dic6C4×D412yes—
297C3×C8—24no—
298C3×C8—41no—
299C3×C8C4×D412yes—
300C3×C8C4×D412yes—
301C3⋊C8—22no—
302C3⋊C8—22no—
303#41—32no—
304#41—32no—
305#41—61no—
306#41—61no—
307C2×Q8—34no—
308C2×Q8—31no—
309C2×Q8—62no—
310C2×Q8—62no—
311C2×Q8—62no—
312C2×Q8—62no—
313C2×Q8—62no—
314C2×D4—31no—
315C2×D4—61no—
316C22×C4—22no—
317C22×C4—61no—
318C22×C4—62no—
319C22×C4—61no—
320C22×C4—122no—
321C22×C4C23×D311yescharacteristic
322Q16—68no—
323Q16—64no—
324Q16—64no—
325SD16—68no—
326M16—22no—
327M16—21no—
328M16—21no—
329M16—32no—
330M16—61no—
331M16—64no—
332M16C2×C4×D314yes—
333C2×C8—22no—
334C2×C8—21no—
335C2×C8—32no—
336C2×C8—61no—
337C2×C8—61no—
338C2×C8—61no—
339C2×C8—64no—
340C2×C8C2×C4×D312yes—
341C2×C8D3×D411yescharacteristic
342C2×C8D3×D411yescharacteristic
343C2×C8#23912yes—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—32no—
347C4⋊C4—32no—
348C4⋊C4—122no—
349C4⋊C4—122no—
350C4⋊C4—122no—
351C4⋊C4D3×D412yes—
352C22⋊C4—21no—
353C22⋊C4—62no—
354C22⋊C4—122no—
355C42—21no—
356C42—62no—
357C42—61no—
358C42—61no—
359C42—122no—
360C42—122no—
361C42—122no—
362C22×C3—22no—
363C22×C3#52011yescharacteristic
364C22×C3D4211yescharacteristic
365C22×C3#55311yescharacteristic
366C2×D3—22no—
367C2×D3—42no—
368C3×C4—42no—
369C3×C4—82no—
370C3×C4C2×C4×D411yescharacteristic
371C3×C4C2×C4×D411yescharacteristic
372C3×C4#54711yescharacteristic
373C3×C4#54811yescharacteristic
374Dic3—22no—
375Dic3—44no—
376Dic3—42no—
377Dic3—42no—
378Dic3—81no—
379C23—61no—
380C23#100911yescharacteristic
381Q8—32no—
382Q8—61no—
383Q8—64no—
384Q8—64no—
385Q8—122no—
386Q8—122no—
387D4—32no—
388D4—62no—
389D4—62no—
390D4—61no—
391C2×C4—22no—
392C2×C4—42no—
393C2×C4—61no—
394C2×C4—62no—
395C2×C4—62no—
396C2×C4—64no—
397C2×C4—62no—
398C2×C4—62no—
399C2×C4—82no—
400C2×C4—122no—
401C2×C4—122no—
402C2×C4—122no—
403C2×C4—121no—
404C2×C4—482no—
405C2×C4C22×C4×D311yescharacteristic
406C2×C4C2×D3×D411yescharacteristic
407C2×C4C2×D3×D411yescharacteristic
408C2×C4C2×#23911yescharacteristic
409C2×C4#100711yescharacteristic
410C2×C4D3×#4111yescharacteristic
411C8—24no—
412C8—41no—
413C8—62no—
414C8—62no—
415C8#89312yes—
416C8#89312yes—
417C2×C3—21no—
418C2×C3—21no—
419C2×C3#236111yescharacteristic
420C2×C3#328912yes—
421D3—42no—
422C22—22no—
423C22—62no—
424C22—122no—
425C22#557411yescharacteristic
426C22#560911yescharacteristic
427C22#569511yescharacteristic
428C4—42no—
429C4—62no—
430C4—82no—
431C4—122no—
432C4—122no—
433C4—124no—
434C4—241no—
435C4C2×#89311yescharacteristic
436C4C4×D3×D411yescharacteristic
437C4#563311yescharacteristic
438C4#572611yescharacteristic
439C3#2304711yescharacteristic, complemented, Sylow
440C2—21no—
441C2—21no—
442C2—122no—
443C2#7620611yescharacteristic
444C2#8093312yes—
subgroupG/H
centerC22#5609
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1951C2
socleC22×C3D42
prime radicalC2×#235C23
cosocleC22×C3×C4C24
hypercenter#1951D3
hypocenterC3#23047
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1951

Complements

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

NH ≅ G/Ntype
4. #78573C2semidirect
5. C3×#1951C2semidirect
6. #70703C2semidirect
7. #70677C2semidirect
8. #70575C2semidirect
11. #68982C2semidirect
32. #4896C4semidirect
52. #1951D3semidirect
76. C2×Dic12C2×C4semidirect
91. C3×#108C2×C4semidirect
100. #824C2×C4semidirect
241. #108C4×D3semidirect
439. C3#23047semidirect

Maximal subgroups

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

subgroupindex
C2×#48962
#792022
#791162
#785732
C3×#19512
#707032
#706772
#705752
#698882
#690042
#689822
#230473