Finite Group Databasebeta

← #10813669

Subgroups — #10813669

Order 768.

846 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×#4895C211yesmaximal, characteristic
2#79209C212yesmaximal, complemented
3#79136C211yesmaximal, characteristic
4#78562C211yesmaximal, characteristic
5C3×#1952C211yesmaximal, characteristic, complemented
6#70701C211yesmaximal, characteristic, complemented
7#70681C212yesmaximal, complemented
8#70607C211yesmaximal, characteristic, complemented
9#69889C211yesmaximal, characteristic, complemented
10#69003C212yesmaximal, complemented
11#69000C212yesmaximal, complemented
12#22690—31nomaximal, Sylow
13C2×#907C2212yes—
14C2×#904C2211yescharacteristic
15#5523C2212yes—
16#5504C2212yes—
17C2×C3×#108C2211yescharacteristic
18C3×#428C2212yes—
19C2×C3×#107C2211yescharacteristic
20C3×#405C2211yescharacteristic
21C2×#825C2212yes—
22#5133C2212yes—
23C2×#822C2211yescharacteristic
24#5121C2212yes—
25C2×#818C2211yescharacteristic
26#5101C2211yescharacteristic
27C2×C8×Dic3C2211yescharacteristic
28#4998C2211yescharacteristic
29#4977C2212yes—
30#4966C2212yes—
31C2×#808C2212yes—
32#4895C418yescomplemented
33#4889—22no—
34#4881—22no—
35#4880—22no—
36#4873—21no—
37#4872—21no—
38#4861—22no—
39D3×#106—22no—
40#4710—22no—
41C4×Dic12—21no—
42C4×D24—21no—
43C3×#343C2212yes—
44#4555C2211yescharacteristic
45#4554C2212yes—
46#4532C2211yescharacteristic
47C2×#443—31no—
48#2951—31no—
49#2951—31no—
50#2912—32no—
51#1952—31no—
52#1952D311yescharacteristic, complemented
53#1860—32no—
54#1860—32no—
55#1846—31no—
56#1815—32no—
57#1772—31no—
58C2×#235C2312yes—
59C2×C3×M16C2311yescharacteristic
60C2×C3×M16C2312yes—
61C22×C3×C8C2311yescharacteristic
62C3×#117C2311yescharacteristic
63C2×C3×C4⋊C4C2312yes—
64C4×#67—42no—
65#926C2311yescharacteristic
66C2×C4⋊Dic3C2312yes—
67C2×C4×Dic3C2311yescharacteristic
68C2×#208C2311yescharacteristic
69C22×C3⋊C8C2311yescharacteristic
70#907—24no—
71#907C2×C418yescomplemented
72C2×Dic12D411yescharacteristic
73#904C2×C414yes—
74#904Q814yes—
75C2×D24D411yescharacteristic
76C2×#204D412yes—
77C2×ΓL(1,𝔽25)—22no—
78C2×D3×C8—22no—
79#893—22no—
80#892—22no—
81#892—22no—
82D3×C4⋊C4—22no—
83D3×C4⋊C4—22no—
84#887—22no—
85#67⋊C4—42no—
86C4×D12—22no—
87C4×Dic6—22no—
88C3×#108C2×C414yescomplemented
89C3×#107—21no—
90C3×#107—21no—
91C3×#107—22no—
92C3×#106—22no—
93C3×#97—21no—
94C3×C4×C8—21no—
95C3⋊#95—42no—
96#825C2×C418yescomplemented
97#822—22no—
98#821—22no—
99#818—21no—
100#818—21no—
101#818—22no—
102#817—22no—
103#816—22no—
104#815—21no—
105C8×Dic3—21no—
106C8×Dic3C2×C414yescomplemented
107#810—22no—
108#809—22no—
109#808—22no—
110#808—22no—
111#808D411yescharacteristic
112#808D411yescharacteristic
113#807D411yescharacteristic
114#807D411yescharacteristic
115C2×#135—31no—
116C2×#131—32no—
117#520—32no—
118#520—32no—
119#443—38no—
120#440—62no—
121#440—62no—
122C4×Q16—61no—
123C4×Q16—61no—
124C4×D8—61no—
125C4×D8—61no—
126C2×#108—31no—
127C2×#108C2×D311yescharacteristic
128#428—32no—
129#428—62no—
130#428—62no—
131#428C2×D312yes—
132#427—31no—
133#427—62no—
134C2×#107—31no—
135C2×#107C2×D311yescharacteristic
136C2×#106—32no—
137C2×#106—62no—
138C2×#104—32no—
139#417—32no—
140#417—32no—
141#405—31no—
142#405C2×D311yescharacteristic
143C2×C4×C8—31no—
144#382—32no—
145#343—32no—
146#343C2×D312yes—
147#339—31no—
148#339—31no—
149C22×C3×C4—22no—
150C22×C3×C4C2411yescharacteristic
151C2×#67—22no—
152C22×Dic3—21no—
153C22×Dic3—22no—
154#235—22no—
155#235—22no—
156#235C22×C414yes—
157#235C2×Q814yes—
158C2×D12C2×D412yes—
159C2×C4×D3—22no—
160C2×C4×D3—42no—
161C2×C4×D3—42no—
162C2×Dic6C2×D412yes—
163C3×M16—22no—
164C3×M16—21no—
165C3×M16—21no—
166C3×M16C22×C414yes—
167C2×C3×C8—22no—
168C2×C3×C8—21no—
169C2×C3×C8C22×C412yes—
170C2×C3×C8C2×D411yescharacteristic
171C2×C3×C8C2×D411yescharacteristic
172C2×C3×C8C2×Q812yes—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4C2×D411yescharacteristic
176C3×C4⋊C4C2×D411yescharacteristic
177C3×C22⋊C4—21no—
178C3×C42—21no—
179#217—21no—
180(C2×D3)⋊C4—42no—
181(C2×D3)⋊C4—42no—
182C4⋊Dic3—22no—
183C4⋊Dic3—22no—
184C4⋊Dic3C2×D411yescharacteristic
185C4⋊Dic3C2×D411yescharacteristic
186Dic3⋊C4—42no—
187Dic3⋊C4—42no—
188C4×Dic3—21no—
189C4×Dic3—21no—
190C4×Dic3—42no—
191C4×Dic3—42no—
192C4×Dic3C22×C412yes—
193#208—21no—
194#208—21no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8C22×C412yes—
197C2×C3⋊C8C2×D411yescharacteristic
198C2×C3⋊C8C2×D411yescharacteristic
199C2×C3⋊C8#4112yes—
200Dic12C4⋊C414yes—
201D24C4⋊C414yes—
202#204C4⋊C418yes—
203ΓL(1,𝔽25)—28no—
204D3×C8—28no—
205C2×#41—32no—
206#135—34no—
207#135—34no—
208C2×Q16—31no—
209C2×SD16—32no—
210C2×D8—31no—
211#131—38no—
212#131—64no—
213C2×M16—31no—
214C2×M16—62no—
215C2×M16C22×D311yescharacteristic
216C2×M16C22×D312yes—
217C22×C8—31no—
218C22×C8—62no—
219C22×C8C22×D311yescharacteristic
220C4×Q8—62no—
221C4×Q8—62no—
222C4×D4—62no—
223C4×D4—62no—
224C4×D4—122no—
225C4×D4—122no—
226#117—31no—
227#117—62no—
228#117—62no—
229#117C22×D311yescharacteristic
230C2×C4⋊C4—32no—
231C2×C4⋊C4—62no—
232C2×C4⋊C4—62no—
233C2×C4⋊C4C22×D312yes—
234C2×C42—31no—
235#108—62no—
236#108C4×D314yescomplemented
237#107—21no—
238#107—21no—
239#107—22no—
240#107—31no—
241#107—31no—
242#107—62no—
243#107—61no—
244#107—61no—
245#106—22no—
246#106—31no—
247#106—31no—
248#106—62no—
249#106—62no—
250#106—62no—
251#104—38no—
252#103—62no—
253#103—62no—
254#102—62no—
255#102—62no—
256#97—21no—
257#97—61no—
258C4×C8—21no—
259C4×C8—34no—
260C4×C8—61no—
261#95—122no—
262C23×C3H(5,𝔽2)11yescharacteristic
263C22×D3—22no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—42no—
267C2×C3×C4—82no—
268C2×C3×C4C23×C411yescharacteristic
269C2×C3×C4C22×D411yescharacteristic
270C2×C3×C4C22×D411yescharacteristic
271C2×C3×C4C22×Q811yescharacteristic
272C2×C3×C4C2×#4111yescharacteristic
273C2×C3×C42−1+411yescharacteristic
274#67—24no—
275#67—24no—
276C2×Dic3—22no—
277C2×Dic3—22no—
278C2×Dic3—21no—
279C2×Dic3—21no—
280C2×Dic3—41no—
281C2×Dic3—41no—
282C2×Dic3—42no—
283C2×Dic3—82no—
284D12—22no—
285D12C2×C4⋊C414yes—
286C4×D3—24no—
287C4×D3—24no—
288C4×D3—162no—
289C4×D3—162no—
290Dic6—22no—
291Dic6C2×C4⋊C414yes—
292C3×C8—24no—
293C3×C8—41no—
294C3×C8C2×C4⋊C412yes—
295C3×C8C2×C4⋊C412yes—
296C3⋊C8—41no—
297C3⋊C8C4×D412yes—
298C3⋊C8C4×D412yes—
299#41—34no—
300#41—34no—
301#41—62no—
302#41—62no—
303C2×Q8—32no—
304C2×D4—32no—
305C2×D4—62no—
306C22×C4—22no—
307C22×C4—62no—
308C22×C4—61no—
309C22×C4—62no—
310C22×C4—122no—
311C22×C4—122no—
312C22×C4C23×D311yescharacteristic
313Q16—34no—
314SD16—38no—
315D8—34no—
316M16—22no—
317M16—21no—
318M16—21no—
319M16—68no—
320M16—61no—
321M16—61no—
322M16C2×C4×D314yes—
323C2×C8—22no—
324C2×C8—21no—
325C2×C8—32no—
326C2×C8—31no—
327C2×C8—31no—
328C2×C8—32no—
329C2×C8—68no—
330C2×C8—61no—
331C2×C8C2×C4×D312yes—
332C2×C8D3×D411yescharacteristic
333C2×C8D3×D411yescharacteristic
334C2×C8D3×Q812yes—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—31no—
338C4⋊C4—31no—
339C4⋊C4—62no—
340C4⋊C4—62no—
341C4⋊C4—122no—
342C4⋊C4—122no—
343C4⋊C4D3×D411yescharacteristic
344C4⋊C4D3×D411yescharacteristic
345C22⋊C4—21no—
346C22⋊C4—61no—
347C22⋊C4—122no—
348C22⋊C4—122no—
349C42—21no—
350C42—32no—
351C42—61no—
352C42—61no—
353C42—122no—
354C42—122no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3#52011yescharacteristic
358C22×C3D4211yescharacteristic
359C22×C3#55411yescharacteristic
360C2×D3—24no—
361C2×D3—42no—
362C2×D3—42no—
363C3×C4—41no—
364C3×C4—41no—
365C3×C4—82no—
366C3×C4C22×C4⋊C411yescharacteristic
367C3×C4C2×C4×D411yescharacteristic
368C3×C4#54711yescharacteristic
369C3×C4D4×Q811yescharacteristic
370Dic3—24no—
371Dic3—42no—
372Dic3—41no—
373Dic3—41no—
374Dic3—82no—
375C23—62no—
376C23#100911yescharacteristic
377Q8—34no—
378Q8—62no—
379D4—34no—
380D4—64no—
381D4—64no—
382D4—62no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—42no—
386C2×C4—62no—
387C2×C4—64no—
388C2×C4—64no—
389C2×C4—62no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—82no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—122no—
396C2×C4—242no—
397C2×C4—482no—
398C2×C4—482no—
399C2×C4C22×C4×D311yescharacteristic
400C2×C4C2×D3×D411yescharacteristic
401C2×C4C2×D3×D411yescharacteristic
402C2×C4C2×D3×Q811yescharacteristic
403C2×C4#100711yescharacteristic
404C2×C4D3×#4111yescharacteristic
405C8—24no—
406C8—32no—
407C8—32no—
408C8—41no—
409C8—121no—
410C8D3×C4⋊C412yes—
411C8D3×C4⋊C412yes—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3D4×C4⋊C411yescharacteristic
415C2×C3#329011yescharacteristic
416C2×C3#329011yescharacteristic
417D3—44no—
418C22—21no—
419C22—21no—
420C22—64no—
421C22—122no—
422C22—122no—
423C22#557411yescharacteristic
424C22#560911yescharacteristic
425C22#569011yescharacteristic
426C4—41no—
427C4—41no—
428C4—64no—
429C4—82no—
430C4—122no—
431C4—121no—
432C4—121no—
433C4—242no—
434C4C2×D3×C4⋊C411yescharacteristic
435C4C4×D3×D411yescharacteristic
436C4#563111yescharacteristic
437C4#572711yescharacteristic
438C3#2269011yescharacteristic, complemented, Sylow
439C2—21no—
440C2—21no—
441C2—124no—
442C2#7620211yescharacteristic
443C2#8094011yescharacteristic
444C2#8094611yescharacteristic
subgroupG/H
centerC22#5609
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1952C2
socleC22×C3D42
prime radicalC2×#904C22
cosocleC22×C3×C4C24
hypercenter#1952D3
hypocenterC3#22690
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
2. #79209C2semidirect
5. C3×#1952C2semidirect
6. #70701C2semidirect
7. #70681C2semidirect
8. #70607C2semidirect
9. #69889C2semidirect
10. #69003C2semidirect
11. #69000C2semidirect
32. #4895C4semidirect
52. #1952D3semidirect
71. #907C2×C4semidirect
88. C3×#108C2×C4semidirect
96. #825C2×C4semidirect
106. C8×Dic3C2×C4semidirect
236. #108C4×D3semidirect
438. C3#22690semidirect

Maximal subgroups

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

subgroupindex
C2×#48952
#792092
#791362
#785622
C3×#19522
#707012
#706812
#706072
#698892
#690032
#690002
#226903