Finite Group Databasebeta

← #10877364

Subgroups — #10877364

Order 768.

626 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79252C211yesmaximal, characteristic, complemented
2#78554C211yesmaximal, characteristic
3C3×#1953C211yesmaximal, characteristic, complemented
4#70734C212yesmaximal, complemented
5#70729C212yesmaximal, complemented
6#70718C211yesmaximal, characteristic, complemented
7#70712C211yesmaximal, characteristic, complemented
8#70712C211yesmaximal, characteristic, complemented
9#69901C211yesmaximal, characteristic, complemented
10#69036C212yesmaximal
11#69016C212yesmaximal, complemented
12#27692—31nomaximal, Sylow
13#5745C2212yes—
14C2×#901C2211yescharacteristic
15C22×ΓL(1,𝔽25)C2211yescharacteristic
16#5524C2211yescharacteristic
17C3×#430C2212yes—
18C2×C3×#108C2211yescharacteristic
19C2×C3×#108C2211yescharacteristic
20C3×#405C2211yescharacteristic
21#5132C2212yes—
22#5129C2211yescharacteristic
23#5126C2211yescharacteristic
24#5126C2212yes—
25#5126C2212yes—
26#5124C2212yes—
27#5124C2212yes—
28C2×#819C2211yescharacteristic
29C2×#819C2211yescharacteristic
30#4999C2211yescharacteristic
31#4893—22no—
32D3×#108—22no—
33#4706—22no—
34#4688—22no—
35C3×#344C2212yes—
36C3⋊#338C2212yes—
37#4558C2212yes—
38#4558C2212yes—
39#4533C2212yes—
40#2930—31no—
41#2916—31no—
42#2103—32no—
43#2091—32no—
44#1953—31no—
45#1953D311yescharacteristic, complemented
46#1891—32no—
47#1858—32no—
48#1813—31no—
49#1811—31no—
50#1811—31no—
51C2×#235C2311yescharacteristic
52C22×C4×D3C2311yescharacteristic
53C2×C3×M16C2311yescharacteristic
54C2×C3×M16C2312yes—
55C2×C3×M16C2312yes—
56C22×C3×C8C2311yescharacteristic
57C3×#117C2311yescharacteristic
58C2×#208C2312yes—
59C2×#208C2312yes—
60C2×#208C2311yescharacteristic
61C22×C3⋊C8C2311yescharacteristic
62#907—42no—
63D3×M16—24no—
64#901—22no—
65#901—22no—
66#901—42no—
67C2×ΓL(1,𝔽25)—21no—
68C2×ΓL(1,𝔽25)—21no—
69C2×ΓL(1,𝔽25)—22no—
70C2×ΓL(1,𝔽25)—22no—
71C2×ΓL(1,𝔽25)D412yes—
72C2×D3×C8D412yes—
73#897—21no—
74#895—21no—
75#886Q812yes—
76#882Q812yes—
77C4×D12—22no—
78C3×#108—21no—
79C3×#108—21no—
80C3×#108—22no—
81C3×#108—42no—
82C3×#97—21no—
83C3×C4×C8—21no—
84#824—22no—
85#823—22no—
86#822—24no—
87#822—42no—
88(C2×D3)⋊C8—22no—
89(C2×D3)⋊C8—22no—
90(C2×D3)⋊C8—42no—
91(C2×D3)⋊C8—42no—
92#819—21no—
93#819—21no—
94#819—22no—
95C4⋊(C3⋊C8)—22no—
96#568—32no—
97C2×#131—31no—
98C22×M16—31no—
99#525—31no—
100#436—62no—
101#435—62no—
102#430—32no—
103#430—32no—
104#430—62no—
105#430C2×D312yes—
106C2×#108—31no—
107C2×#108—31no—
108C2×#108—62no—
109C2×#108C2×D311yescharacteristic
110C2×#108C2×D311yescharacteristic
111#424—31no—
112#413—32no—
113#408—31no—
114#407—31no—
115#407—32no—
116#407—32no—
117#405C2×D311yescharacteristic
118#344—32no—
119#344C2×D312yes—
120#341—32no—
121#341—32no—
122#338—32no—
123C23×D3C2×Q811yescharacteristic
124C22×C3×C4C2411yescharacteristic
125C2×#67C2×Q812yes—
126C22×Dic3C2×Q811yescharacteristic
127#235—21no—
128#235—21no—
129#235—21no—
130#235—21no—
131C2×D12—22no—
132C2×D12#4111yescharacteristic
133C2×C4×D3—21no—
134C2×C4×D3—21no—
135C2×C4×D3—21no—
136C2×C4×D3—21no—
137C2×C4×D3—21no—
138C2×C4×D3—21no—
139C2×C4×D3C2×D412yes—
140C2×Dic6#4111yescharacteristic
141C3×M16—22no—
142C3×M16—22no—
143C3×M16—22no—
144C3×M16—21no—
145C3×M16—21no—
146C3×M16—22no—
147C2×C3×C8—21no—
148C2×C3×C8—22no—
149C2×C3×C8—22no—
150C2×C3×C8C2×D411yescharacteristic
151C2×C3×C8C2×D411yescharacteristic
152C2×C3×C8#4112yes—
153C2×C3×C8#4112yes—
154C3×C4⋊C4—21no—
155C3×C22⋊C4C2×Q812yes—
156C3×C42—21no—
157(C2×D3)⋊C4—22no—
158(C2×D3)⋊C4—22no—
159C4⋊Dic3—22no—
160#208—21no—
161#208—21no—
162#208—22no—
163#208—22no—
164#208—22no—
165#208—22no—
166C2×C3⋊C8—21no—
167C2×C3⋊C8—21no—
168C2×C3⋊C8—21no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8C2×D412yes—
172ΓL(1,𝔽25)—24no—
173ΓL(1,𝔽25)—44no—
174ΓL(1,𝔽25)—42no—
175ΓL(1,𝔽25)—42no—
176D3×C8—24no—
177D3×C8—44no—
178C2×#41—31no—
179C23×C4—31no—
180#131—62no—
181#131—62no—
182#131—122no—
183#131—122no—
184C2×M16—31no—
185C2×M16—32no—
186C2×M16—32no—
187C2×M16—32no—
188C2×M16—62no—
189C2×M16—61no—
190C2×M16—61no—
191C2×M16—62no—
192C2×M16—64no—
193C2×M16C22×D312yes—
194C2×M16C22×D312yes—
195C2×M16C22×D311yescharacteristic
196C22×C8—32no—
197C22×C8—31no—
198C22×C8C22×D311yescharacteristic
199#123—32no—
200C22⋊Q8—61no—
201#121—61no—
202#120—32no—
203C4×D4—62no—
204#117C22×D311yescharacteristic
205#108—22no—
206#108—21no—
207#108—21no—
208#108—42no—
209#108—64no—
210#108—61no—
211#108—61no—
212#108—62no—
213#108—122no—
214C4⋊C8—62no—
215#101—62no—
216#100—62no—
217C22⋊C8—62no—
218C22⋊C8—62no—
219C22⋊C8—122no—
220C22⋊C8—122no—
221#97—21no—
222C4×C8—21no—
223C23×C3C22×Q811yescharacteristic
224C22×D3—21no—
225C22×D3—21no—
226C22×D3—21no—
227C22×D3—81no—
228C22×D3C22⋊Q812yes—
229C2×C3×C4—22no—
230C2×C3×C4C22×D411yescharacteristic
231C2×C3×C4C2×#4111yescharacteristic
232C2×C3×C4C2×#4111yescharacteristic
233C2×C3×C4C2×#4111yescharacteristic
234C2×C3×C4H(5,𝔽2)11yescharacteristic
235C2×C3×C4H(5,𝔽2)11yescharacteristic
236#67—22no—
237#67—22no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3C22⋊Q812yes—
242D12—21no—
243D12—21no—
244D12—82no—
245C4×D3—22no—
246C4×D3—22no—
247C4×D3—41no—
248C4×D3—41no—
249C4×D3—41no—
250C4×D3—41no—
251Dic6—21no—
252Dic6—21no—
253C3×C8—42no—
254C3×C8—41no—
255C3×C8—42no—
256C3×C8#12312yes—
257C3×C8#12312yes—
258C3⋊C8—22no—
259C3⋊C8—41no—
260C3⋊C8—42no—
261C3⋊C8—42no—
262C24—31no—
263#41—61no—
264#41—61no—
265#41—61no—
266#41—61no—
267C2×Q8—31no—
268C2×D4—32no—
269C2×D4—31no—
270C2×D4—62no—
271C22×C4—32no—
272C22×C4—31no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4C23×D311yescharacteristic
280M16—22no—
281M16—22no—
282M16—21no—
283M16—22no—
284M16—22no—
285M16—21no—
286M16—61no—
287M16—61no—
288M16—64no—
289M16—62no—
290M16—62no—
291M16—62no—
292M16—62no—
293M16—124no—
294M16—122no—
295M16—122no—
296C2×C8—21no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—32no—
300C2×C8—64no—
301C2×C8—61no—
302C2×C8—61no—
303C2×C8—62no—
304C2×C8—61no—
305C2×C8—62no—
306C2×C8—124no—
307C2×C8#23512yes—
308C2×C8D3×D411yescharacteristic
309C2×C8D3×D411yescharacteristic
310C2×C8#23912yes—
311C4⋊C4—21no—
312C4⋊C4—62no—
313C22⋊C4—62no—
314C22⋊C4—62no—
315C22⋊C4D3×Q812yes—
316C42—21no—
317C22×C3—41no—
318C22×C3C2×C22⋊Q811yescharacteristic
319C22×C3#55411yescharacteristic
320C22×C3#55411yescharacteristic
321C2×D3—22no—
322C2×D3—42no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—42no—
326C2×D3—42no—
327C2×D3—41no—
328C3×C4—42no—
329C3×C4C2×#12311yescharacteristic
330C3×C4#53511yescharacteristic
331C3×C4#55311yescharacteristic
332C3×C4#55311yescharacteristic
333Dic3—22no—
334Dic3—41no—
335C23—32no—
336C23—61no—
337C23—61no—
338C23—61no—
339C23—241no—
340C23C2×D3×Q811yescharacteristic
341Q8—61no—
342Q8—61no—
343D4—62no—
344D4—62no—
345D4—61no—
346D4—61no—
347D4—242no—
348C2×C4—22no—
349C2×C4—32no—
350C2×C4—62no—
351C2×C4—62no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4C2×#23511yescharacteristic
360C2×C4C2×D3×D411yescharacteristic
361C2×C4#100411yescharacteristic
362C2×C4C2×#23911yescharacteristic
363C2×C4D3×#4111yescharacteristic
364C2×C4#100911yescharacteristic
365C8—42no—
366C8—42no—
367C8—41no—
368C8—62no—
369C8—122no—
370C8—121no—
371C8—122no—
372C8#89412yes—
373C8#89412yes—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3#248711yescharacteristic
378D3—42no—
379D3—81no—
380C22—41no—
381C22—62no—
382C22—122no—
383C22—121no—
384C22—121no—
385C22—122no—
386C22—122no—
387C22—121no—
388C22#557911yescharacteristic
389C22D3×C22⋊Q811yescharacteristic
390C22#573011yescharacteristic
391C4—42no—
392C4—62no—
393C4—121no—
394C4C2×#89411yescharacteristic
395C4#555811yescharacteristic
396C4#560711yescharacteristic
397C4#569511yescharacteristic
398C3#2769211yescharacteristic, complemented, Sylow
399C2—21no—
400C2—21no—
401C2—21no—
402C2—122no—
403C2—241no—
404C2#7623211yescharacteristic
subgroupG/H
centerC4#5607
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1953C2
socleC2×C3#2487
prime radical#5524C22
hypercenter#1953D3
hypocenterC3#27692
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1953

Complements

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

NH ≅ G/Ntype
1. #79252C2semidirect
3. C3×#1953C2semidirect
4. #70734C2semidirect
5. #70729C2semidirect
6. #70718C2semidirect
7. #70712C2semidirect
8. #70712C2semidirect
9. #69901C2semidirect
11. #69016C2semidirect
45. #1953D3semidirect
398. C3#27692semidirect

Maximal subgroups

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

subgroupindex
#792522
#785542
C3×#19532
#707342
#707292
#707182
#707122
#699012
#690362
#690162
#276923