Finite Group Databasebeta

← #10877382

Subgroups — #10877382

Order 768.

626 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79252C211yesmaximal, characteristic, complemented
2#78491C211yesmaximal, characteristic
3C3×#1946C211yesmaximal, characteristic, complemented
4#70719C211yesmaximal, characteristic, complemented
5#70665C211yesmaximal, characteristic, complemented
6#70664C211yesmaximal, characteristic, complemented
7#70549C212yesmaximal, complemented
8#70546C212yesmaximal, complemented
9#69844C211yesmaximal, characteristic, complemented
10#68777C212yesmaximal
11#68751C212yesmaximal, complemented
12#27694—31nomaximal, Sylow
13#5745C2212yes—
14C2×#901C2211yescharacteristic
15C2×#901C2211yescharacteristic
16#5477C2211yescharacteristic
17C2×C3×#108C2211yescharacteristic
18C3×#427C2211yescharacteristic
19C3×#424C2212yes—
20C2×C3×#97C2211yescharacteristic
21#5139C2212yes—
22#5124C2212yes—
23#5110C2211yescharacteristic
24#5110C2211yescharacteristic
25C2×#819C2211yescharacteristic
26#5106C2211yescharacteristic
27#5102C2212yes—
28#5006C2212yes—
29#5005C2212yes—
30#4924C2211yescharacteristic
31#4893—22no—
32D3×#108—22no—
33#4706—22no—
34#4706—22no—
35C3×#339C2212yes—
36#4556C2212yes—
37#4533C2212yes—
38#4531C2212yes—
39#4523C2212yes—
40#2932—31no—
41#2916—31no—
42#2102—32no—
43#2098—32no—
44#1946—31no—
45#1946D311yescharacteristic, complemented
46#1884—32no—
47#1868—32no—
48#1814—31no—
49#1814—31no—
50#1812—31no—
51C2×#235C2311yescharacteristic
52C2×#235C2311yescharacteristic
53C2×C3×M16C2312yes—
54C22×C3×C8C2311yescharacteristic
55C22×C3×C8C2311yescharacteristic
56C3×#117C2312yes—
57C2×C3×C42C2311yescharacteristic
58#930—22no—
59#926C2312yes—
60C2×#208C2312yes—
61C2×#208C2311yescharacteristic
62C2×#208C2311yescharacteristic
63#907—42no—
64D3×M16—24no—
65#901—21no—
66#901—21no—
67#901—21no—
68#901—21no—
69#901—21no—
70#901—21no—
71#901—21no—
72#901—21no—
73#901—42no—
74C2×ΓL(1,𝔽25)—21no—
75C2×ΓL(1,𝔽25)D412yes—
76C2×D3×C8—21no—
77C2×D3×C8D412yes—
78#875Q811yescharacteristic
79#875Q811yescharacteristic
80#874Q811yescharacteristic
81C4×D12—22no—
82#869Q811yescharacteristic
83C3×#108—21no—
84C3×#108—21no—
85C3×#107—21no—
86C3×#106—21no—
87C3×C4⋊C8—42no—
88C3×#97—21no—
89C3×#97—21no—
90#825—42no—
91#822—24no—
92(C2×D3)⋊C8—22no—
93(C2×D3)⋊C8—22no—
94#819—21no—
95#819—21no—
96#818—21no—
97#817—21no—
98Dic3⋊C8—42no—
99#806—24no—
100#805—42no—
101C4⋊(C3⋊C8)—22no—
102#568—32no—
103C2×#131—31no—
104C2×#131—31no—
105#532—31no—
106#435—62no—
107#435—62no—
108#430—32no—
109#430—62no—
110C2×#108—31no—
111C2×#108—62no—
112C2×#108C2×D311yescharacteristic
113#427—31no—
114#427C2×D311yescharacteristic
115#424—32no—
116#424C2×D312yes—
117#423—31no—
118#421—32no—
119#421—32no—
120#410—32no—
121#408—31no—
122#408—31no—
123C2×#97C2×D311yescharacteristic
124#344—32no—
125#344—32no—
126#339—32no—
127#339C2×D312yes—
128H(3,ℤ/4)—32no—
129C22×C3×C4—21no—
130C22×C3×C4C2411yescharacteristic
131C2×#67—21no—
132C2×#67#4112yes—
133#235—21no—
134#235—21no—
135#235—21no—
136#235—21no—
137#235—21no—
138#235—21no—
139#235—21no—
140#235—21no—
141C2×D12—22no—
142C2×D12C2×Q811yescharacteristic
143C2×D12C2×Q811yescharacteristic
144C2×C4×D3—21no—
145C2×C4×D3C2×D412yes—
146C2×Dic6C2×Q811yescharacteristic
147C2×Dic6C2×Q811yescharacteristic
148C3×M16—22no—
149C3×M16—22no—
150C2×C3×C8—21no—
151C2×C3×C8—21no—
152C2×C3×C8—21no—
153C2×C3×C8—22no—
154C2×C3×C8C2×D411yescharacteristic
155C2×C3×C8C2×D411yescharacteristic
156C2×C3×C8#4111yescharacteristic
157C2×C3×C8#4111yescharacteristic
158C2×C3×C8#4111yescharacteristic
159C2×C3×C8#4111yescharacteristic
160C3×C4⋊C4—22no—
161C3×C22⋊C4—22no—
162C3×C42—21no—
163C3×C42—22no—
164C3×C42C2×Q811yescharacteristic
165C3×C42C2×Q811yescharacteristic
166#217—22no—
167(C2×D3)⋊C4—22no—
168(C2×D3)⋊C4—22no—
169C4⋊Dic3—22no—
170C4⋊Dic3—22no—
171C4×Dic3—22no—
172#208—21no—
173#208—21no—
174#208—22no—
175#208—21no—
176#208—21no—
177#208—22no—
178C2×C3⋊C8—21no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8C2×D412yes—
181ΓL(1,𝔽25)—22no—
182ΓL(1,𝔽25)—22no—
183ΓL(1,𝔽25)—44no—
184ΓL(1,𝔽25)—41no—
185ΓL(1,𝔽25)—41no—
186D3×C8—22no—
187D3×C8—22no—
188D3×C8—44no—
189D3×C8—41no—
190D3×C8—41no—
191C2×#41—31no—
192C2×#41—31no—
193#131—61no—
194#131—61no—
195#131—61no—
196#131—61no—
197#131—61no—
198#131—61no—
199#131—61no—
200#131—61no—
201#131—122no—
202#131—122no—
203C2×M16—31no—
204C2×M16—32no—
205C2×M16—31no—
206C2×M16—32no—
207C2×M16—61no—
208C2×M16—64no—
209C2×M16C22×D312yes—
210C22×C8—32no—
211C22×C8—61no—
212C22×C8C22×D311yescharacteristic
213C22×C8C22×D311yescharacteristic
214C4⋊Q8—31no—
215#127—31no—
216#124—31no—
217#124—31no—
218#121—62no—
219C4×D4—62no—
220#117—32no—
221#117C22×D312yes—
222C2×C42C22×D311yescharacteristic
223#108—21no—
224#108—21no—
225#108—64no—
226#108—61no—
227#108—61no—
228#107—21no—
229#107—61no—
230#106—21no—
231#106—61no—
232C4⋊C8—42no—
233C4⋊C8—62no—
234C4⋊C8—122no—
235#104—122no—
236#104—122no—
237#99—64no—
238C22⋊C8—62no—
239C22⋊C8—62no—
240#97—21no—
241#97—21no—
242C23×C3C2×#4111yescharacteristic
243C22×D3—21no—
244C22×D3C22⋊Q812yes—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—42no—
248C2×C3×C4—81no—
249C2×C3×C4C22×D411yescharacteristic
250C2×C3×C4C22×Q811yescharacteristic
251C2×C3×C4C2×#4111yescharacteristic
252C2×C3×C4C2×#4111yescharacteristic
253C2×C3×C4H(5,𝔽2)11yescharacteristic
254C2×C3×C4H(5,𝔽2)11yescharacteristic
255#67—22no—
256#67—22no—
257#67—41no—
258#67—41no—
259C2×Dic3—21no—
260C2×Dic3—42no—
261C2×Dic3C22⋊Q812yes—
262D12—21no—
263D12—21no—
264D12—21no—
265D12—21no—
266D12—82no—
267C4×D3—22no—
268C4×D3—22no—
269C4×D3—41no—
270C4×D3—41no—
271Dic6—21no—
272Dic6—21no—
273Dic6—21no—
274Dic6—21no—
275C3×C8—42no—
276C3×C8—41no—
277C3×C8#12311yescharacteristic
278C3×C8#12311yescharacteristic
279C3×C8#12311yescharacteristic
280C3×C8#12311yescharacteristic
281C3⋊C8—22no—
282C3⋊C8—41no—
283C3⋊C8—42no—
284#41—61no—
285#41—61no—
286#41—61no—
287#41—61no—
288#41—61no—
289#41—61no—
290#41—61no—
291#41—61no—
292C2×Q8—31no—
293C2×Q8—31no—
294C2×D4—32no—
295C2×D4—31no—
296C2×D4—31no—
297C2×D4—62no—
298C2×D4—61no—
299C22×C4—21no—
300C22×C4—32no—
301C22×C4—61no—
302C22×C4C23×D311yescharacteristic
303M16—22no—
304M16—22no—
305M16—61no—
306M16—61no—
307M16—62no—
308M16—62no—
309M16—61no—
310M16—62no—
311M16—62no—
312M16—61no—
313M16—124no—
314M16—121no—
315M16—121no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—22no—
319C2×C8—21no—
320C2×C8—32no—
321C2×C8—62no—
322C2×C8—62no—
323C2×C8—61no—
324C2×C8—62no—
325C2×C8—124no—
326C2×C8—121no—
327C2×C8—121no—
328C2×C8#23511yescharacteristic
329C2×C8#23511yescharacteristic
330C2×C8D3×D411yescharacteristic
331C2×C8D3×D411yescharacteristic
332C2×C8#23911yescharacteristic
333C2×C8#23911yescharacteristic
334C4⋊C4—22no—
335C4⋊C4—62no—
336C4⋊C4—62no—
337C22⋊C4—22no—
338C22⋊C4—62no—
339C22⋊C4—62no—
340C22⋊C4—62no—
341C42—21no—
342C42—22no—
343C42—62no—
344C42D3×Q811yescharacteristic
345C42D3×Q811yescharacteristic
346C22×C3—41no—
347C22×C3C2×C22⋊Q811yescharacteristic
348C22×C3#55311yescharacteristic
349C22×C3#55311yescharacteristic
350C2×D3—22no—
351C2×D3—42no—
352C2×D3—42no—
353C2×D3—41no—
354C3×C4—41no—
355C3×C4—41no—
356C3×C4—82no—
357C3×C4C2×#12311yescharacteristic
358C3×C4#53511yescharacteristic
359C3×C4#55411yescharacteristic
360C3×C4#55411yescharacteristic
361Dic3—22no—
362Dic3—41no—
363Dic3—82no—
364C23—32no—
365C23—61no—
366C23D3×#4111yescharacteristic
367Q8—61no—
368Q8—61no—
369Q8—61no—
370Q8—61no—
371D4—62no—
372D4—62no—
373D4—61no—
374D4—61no—
375D4—61no—
376D4—61no—
377D4—121no—
378D4—121no—
379D4—242no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—32no—
383C2×C4—42no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—61no—
387C2×C4—81no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—122no—
391C2×C4C2×#23511yescharacteristic
392C2×C4C2×D3×D411yescharacteristic
393C2×C4#100411yescharacteristic
394C2×C4C2×D3×Q811yescharacteristic
395C2×C4C2×#23911yescharacteristic
396C2×C4#100911yescharacteristic
397C8—41no—
398C8—42no—
399C8—62no—
400C8—121no—
401C8—122no—
402C8#89411yescharacteristic
403C8#89411yescharacteristic
404C8#89411yescharacteristic
405C8#89411yescharacteristic
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3#248711yescharacteristic
410D3—42no—
411D3—81no—
412C22—41no—
413C22—62no—
414C22—122no—
415C22—122no—
416C22—121no—
417C22#555811yescharacteristic
418C22D3×C22⋊Q811yescharacteristic
419C22#569511yescharacteristic
420C4—41no—
421C4—41no—
422C4—62no—
423C4—82no—
424C4—121no—
425C4—242no—
426C4C2×#89411yescharacteristic
427C4#557911yescharacteristic
428C4#560711yescharacteristic
429C4#573011yescharacteristic
430C3#2769411yescharacteristic, complemented, Sylow
431C2—21no—
432C2—21no—
433C2—21no—
434C2—122no—
435C2—241no—
436C2#7623211yescharacteristic
subgroupG/H
centerC4#5607
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1946C2
socleC2×C3#2487
prime radical#5477C22
hypercenter#1946D3
hypocenterC3#27694
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1946

Complements

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

NH ≅ G/Ntype
1. #79252C2semidirect
3. C3×#1946C2semidirect
4. #70719C2semidirect
5. #70665C2semidirect
6. #70664C2semidirect
7. #70549C2semidirect
8. #70546C2semidirect
9. #69844C2semidirect
11. #68751C2semidirect
45. #1946D3semidirect
430. C3#27694semidirect

Maximal subgroups

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

subgroupindex
#792522
#784912
C3×#19462
#707192
#706652
#706642
#705492
#705462
#698442
#687772
#687512
#276943