Finite Group Databasebeta

← #10877377

Subgroups — #10877377

Order 768.

626 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79260C211yesmaximal, characteristic, complemented
2#78658C211yesmaximal, characteristic
3C3×#2098C211yesmaximal, characteristic, complemented
4#70728C211yesmaximal, characteristic, complemented
5#70546C212yesmaximal, complemented
6#70538C212yesmaximal, complemented
7#70417C212yesmaximal
8#70396C212yesmaximal, complemented
9#70005C211yesmaximal, characteristic, complemented
10#68710C211yesmaximal, characteristic, complemented
11#68701C211yesmaximal, characteristic, complemented
12#27694—31nomaximal, Sylow
13#5832C2211yescharacteristic
14C2×#948C2211yescharacteristic
15#5745C2212yes—
16C2×#907C2211yescharacteristic
17C3×#430C2211yescharacteristic
18C3×#424C2212yes—
19C3×#421C2212yes—
20C3×#408C2211yescharacteristic
21#5236C2212yes—
22#5125C2211yescharacteristic
23C2×#822C2212yes—
24#5118C2211yescharacteristic
25#5110C2211yescharacteristic
26#5102C2212yes—
27C2×#815C2211yescharacteristic
28#5006C2212yes—
29#4998C2212yes—
30#4893—22no—
31D3×#108—22no—
32#4730—22no—
33#4728—22no—
34C3×H(3,ℤ/4)C2211yescharacteristic
35#4557C2211yescharacteristic
36#4556C2212yes—
37#4555C2212yes—
38#4533C2212yes—
39#4532C2212yes—
40#2932—31no—
41#2916—31no—
42#2102—31no—
43#2102—31no—
44#2098—31no—
45#2098D311yescharacteristic, complemented
46#1946—32no—
47#1884—32no—
48#1868—32no—
49#1814—32no—
50#1812—31no—
51C2×C3×#41C2311yescharacteristic
52C2×#235C2311yescharacteristic
53C2×C3×M16C2312yes—
54C2×C3×M16C2311yescharacteristic
55C22×C3×C8C2311yescharacteristic
56C3×#117C2312yes—
57#948—22no—
58#948—22no—
59#947Q811yescharacteristic
60#944Q811yescharacteristic
61#940Q811yescharacteristic
62#939—22no—
63#936Q811yescharacteristic
64C4×#67—22no—
65#926C2312yes—
66C2×C4×Dic3C2311yescharacteristic
67C2×#208—21no—
68C2×#208C2312yes—
69C2×#208C2311yescharacteristic
70C22×C3⋊C8—21no—
71C22×C3⋊C8C2311yescharacteristic
72#907—22no—
73#907—22no—
74#907—42no—
75D3×M16—24no—
76#901—42no—
77C2×ΓL(1,𝔽25)D412yes—
78C2×D3×C8D412yes—
79C3×#108—22no—
80C3×C4⋊C8—42no—
81C3×#104—42no—
82C3×C22⋊C8—22no—
83#837—42no—
84#822—22no—
85#822—22no—
86(C2×D3)⋊C8—22no—
87#819—22no—
88#815—21no—
89#815—21no—
90Dic3⋊C8—22no—
91Dic3⋊C8—42no—
92#808—22no—
93#807—22no—
94#806—24no—
95#568—32no—
96C2×#131—31no—
97C2×#131—31no—
98#532—31no—
99#435—62no—
100#435—62no—
101#430—31no—
102#430—62no—
103#430C2×D311yescharacteristic
104C2×#108—32no—
105C2×#108—62no—
106#427—32no—
107#424—32no—
108#424C2×D312yes—
109#423—31no—
110#421—32no—
111#421C2×D312yes—
112#410—32no—
113#408—31no—
114#408C2×D311yescharacteristic
115C2×#97—31no—
116#344—32no—
117#344—32no—
118#339—32no—
119#339—32no—
120H(3,ℤ/4)—31no—
121H(3,ℤ/4)C2×D311yescharacteristic
122C3×#41—22no—
123C3×#41—22no—
124C2×C3×Q8C2×Q811yescharacteristic
125C2×C3×D4—21no—
126C2×C3×D4C2×Q811yescharacteristic
127C22×C3×C4—21no—
128C22×C3×C4C2411yescharacteristic
129C2×#67—22no—
130C2×#67#4112yes—
131C22×Dic3—21no—
132#235—22no—
133#235—22no—
134C2×D12C2×Q811yescharacteristic
135C2×C4×D3C2×D412yes—
136C2×Dic6C2×Q811yescharacteristic
137C3×M16—22no—
138C3×M16—22no—
139C3×M16—22no—
140C2×C3×C8—21no—
141C2×C3×C8—22no—
142C2×C3×C8—22no—
143C2×C3×C8C2×D412yes—
144C3×C4⋊C4—22no—
145C3×C22⋊C4—22no—
146C3×C42—22no—
147#217—22no—
148#217—22no—
149(C2×D3)⋊C4—22no—
150C4⋊Dic3—22no—
151Dic3⋊C4—22no—
152C4×Dic3—21no—
153C4×Dic3—22no—
154C4×Dic3C2×Q811yescharacteristic
155C4×Dic3C2×Q811yescharacteristic
156#208—22no—
157#208—22no—
158#208—22no—
159#208—41no—
160#208—41no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—42no—
164C2×C3⋊C8C2×D411yescharacteristic
165C2×C3⋊C8C2×D411yescharacteristic
166C2×C3⋊C8#4112yes—
167C2×C3⋊C8#4112yes—
168ΓL(1,𝔽25)—24no—
169ΓL(1,𝔽25)—44no—
170D3×C8—22no—
171D3×C8—22no—
172D3×C8—44no—
173C2×#41—31no—
174C2×#41C22×D311yescharacteristic
175#131—62no—
176#131—62no—
177#131—62no—
178#131—62no—
179#131—122no—
180#131—122no—
181C2×M16—32no—
182C2×M16—31no—
183C2×M16—32no—
184C2×M16—61no—
185C2×M16—64no—
186C2×M16C22×D312yes—
187C2×M16C22×D311yescharacteristic
188C22×C8—31no—
189C22×C8—32no—
190C22×C8—61no—
191C22×C8C22×D311yescharacteristic
192C4⋊Q8—31no—
193#127—31no—
194#124—31no—
195#124—31no—
196#121—62no—
197C4×D4—62no—
198#117—32no—
199#117C22×D312yes—
200C2×C42—31no—
201#108—22no—
202#108—62no—
203#108—62no—
204#108—62no—
205#107—62no—
206#106—62no—
207C4⋊C8—42no—
208C4⋊C8—62no—
209C4⋊C8—122no—
210#104—42no—
211#104—122no—
212#99—64no—
213C22⋊C8—22no—
214C22⋊C8—62no—
215#97—61no—
216#97—61no—
217C23×C3—21no—
218C23×C3C2×#4111yescharacteristic
219C22×D3C22⋊Q812yes—
220C3×Q8—22no—
221C3×D4—22no—
222C3×D4—42no—
223C2×C3×C4—21no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4C22×D411yescharacteristic
228C2×C3×C4C22×Q811yescharacteristic
229C2×C3×C4C2×#4112yes—
230C2×C3×C4H(5,𝔽2)12yes—
231#67—24no—
232#67—82no—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—42no—
236C2×Dic3—81no—
237C2×Dic3C22⋊Q812yes—
238D12—22no—
239C4×D3—22no—
240C4×D3—22no—
241Dic6—22no—
242C3×C8—22no—
243C3×C8—42no—
244C3×C8—41no—
245C3⋊C8—41no—
246C3⋊C8—42no—
247C3⋊C8#12311yescharacteristic
248C3⋊C8#12311yescharacteristic
249C3⋊C8#12312yes—
250#41—22no—
251#41—22no—
252#41—62no—
253#41—62no—
254C2×Q8—31no—
255C2×Q8D3×Q811yescharacteristic
256C2×D4—21no—
257C2×D4—32no—
258C2×D4—31no—
259C2×D4—62no—
260C2×D4D3×Q811yescharacteristic
261C22×C4—21no—
262C22×C4—32no—
263C22×C4—61no—
264C22×C4C23×D311yescharacteristic
265M16—22no—
266M16—22no—
267M16—22no—
268M16—64no—
269M16—62no—
270M16—62no—
271M16—62no—
272M16—121no—
273M16—121no—
274M16—124no—
275C2×C8—21no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—31no—
279C2×C8—31no—
280C2×C8—32no—
281C2×C8—32no—
282C2×C8—62no—
283C2×C8—62no—
284C2×C8—61no—
285C2×C8—62no—
286C2×C8—122no—
287C2×C8—124no—
288C2×C8D3×D412yes—
289C4⋊C4—22no—
290C4⋊C4—62no—
291C4⋊C4—62no—
292C22⋊C4—22no—
293C22⋊C4—62no—
294C22⋊C4—62no—
295C22⋊C4—62no—
296C42—22no—
297C42—31no—
298C42—31no—
299C42—61no—
300C42—62no—
301C22×C3—41no—
302C22×C3—42no—
303C22×C3—41no—
304C22×C3C2×C22⋊Q811yescharacteristic
305C22×C3#55312yes—
306C2×D3—22no—
307C2×D3—42no—
308C3×C4—41no—
309C3×C4—82no—
310C3×C4C2×#12311yescharacteristic
311C3×C4#53511yescharacteristic
312C3×C4#55412yes—
313Dic3—22no—
314Dic3—41no—
315Dic3—41no—
316Dic3—82no—
317C23—21no—
318C23—32no—
319C23D3×#4111yescharacteristic
320Q8—22no—
321Q8—62no—
322D4—22no—
323D4—42no—
324D4—64no—
325D4—62no—
326D4—242no—
327C2×C4—21no—
328C2×C4—32no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—62no—
333C2×C4—62no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—122no—
337C2×C4—241no—
338C2×C4C2×D3×D411yescharacteristic
339C2×C4#100412yes—
340C2×C4C2×D3×Q811yescharacteristic
341C2×C4D3×#4112yes—
342C8—22no—
343C8—31no—
344C8—31no—
345C8—32no—
346C8—41no—
347C8—42no—
348C8—121no—
349C8—122no—
350C2×C3—21no—
351C2×C3—22no—
352C2×C3—81no—
353C2×C3#248711yescharacteristic
354D3—42no—
355C22—41no—
356C22—42no—
357C22—41no—
358C22—62no—
359C22—122no—
360C22D3×C22⋊Q811yescharacteristic
361C22#566112yes—
362C4—41no—
363C4—62no—
364C4—82no—
365C4—121no—
366C4—121no—
367C4—242no—
368C4#563612yes—
369C4D3×#12311yescharacteristic
370C4#571511yescharacteristic
371C3#2769411yescharacteristic, complemented, Sylow
372C2—22no—
373C2—21no—
374C2—81no—
375C2—122no—
376C2#7802911yescharacteristic
subgroupG/H
centerC4#5715
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2098C2
socleC2×C3#2487
prime radical#5832C22
hypercenter#2098D3
hypocenterC3#27694
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2098

Complements

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

NH ≅ G/Ntype
1. #79260C2semidirect
3. C3×#2098C2semidirect
4. #70728C2semidirect
5. #70546C2semidirect
6. #70538C2semidirect
8. #70396C2semidirect
9. #70005C2semidirect
10. #68710C2semidirect
11. #68701C2semidirect
45. #2098D3semidirect
371. C3#27694semidirect

Maximal subgroups

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

subgroupindex
#792602
#786582
C3×#20982
#707282
#705462
#705382
#704172
#703962
#700052
#687102
#687012
#276943