Finite Group Databasebeta

← #10813646

Subgroups — #10813646

Order 768.

858 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79012C212yesmaximal, complemented
2C2×#4717C211yesmaximal, characteristic
3#78494C211yesmaximal, characteristic
4C2×#4700C211yesmaximal, characteristic
5C3×#1771C211yesmaximal, characteristic, complemented
6#69844C211yesmaximal, characteristic, complemented
7#69842C211yesmaximal, characteristic, complemented
8#69794C211yesmaximal, characteristic, complemented
9#69494C212yesmaximal, complemented
10#68694C212yesmaximal, complemented
11#68694C212yesmaximal, complemented
12#22687—31nomaximal, Sylow
13C2×#904C2211yescharacteristic
14C2×#901C2212yes—
15C2×#901C2212yes—
16C4×#235C2212yes—
17C2×C3×C4⋊C8C2211yescharacteristic
18C2×C3×#97C2211yescharacteristic
19C2×C3×#97C2211yescharacteristic
20C2×C3×C4×C8C2211yescharacteristic
21C3×C4×C4⋊C4C2211yescharacteristic
22#5113C2212yes—
23C2×#819C2211yescharacteristic
24C2×#819C2211yescharacteristic
25C2×#819C2212yes—
26#5106C2211yescharacteristic
27#5102C2212yes—
28#5102C2212yes—
29C2×#805C2212yes—
30C2×#805C2212yes—
31#4924C2212yes—
32#4839—22no—
33#4839—22no—
34#4837—22no—
35D3×C4⋊C8—22no—
36#4717—24no—
37#4716C412yescomplemented
38#4711C412yescomplemented
39#4710C412yescomplemented
40#4710C412yescomplemented
41#4700—24no—
42C3×#336C2211yescharacteristic
43C3×#336C2211yescharacteristic
44#4523C2212yes—
45#4523C2212yes—
46C2×#444—31no—
47C2×#443—31no—
48#2943—31no—
49#2906—32no—
50#1946—31no—
51#1945—31no—
52#1866—32no—
53#1866—32no—
54#1849—31no—
55#1806—32no—
56#1771D311yescharacteristic, complemented
57C2×#235C2312yes—
58C22×C3×C8C2311yescharacteristic
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C2×C3×C4⋊C4—21no—
63C2×C3×C42C2311yescharacteristic
64C2×C3×C42C2311yescharacteristic
65C2×C3×C42C2311yescharacteristic
66C4×#67—22no—
67C4×#67—22no—
68#926C2312yes—
69C2×#208C2312yes—
70C2×#208C2312yes—
71C2×Dic12C2×C411yescharacteristic
72#904D414yes—
73#904Q814yes—
74C2×D24C2×C411yescharacteristic
75C2×#204C2×C412yes—
76#901—24no—
77#901—24no—
78#901—42no—
79#901—42no—
80C2×ΓL(1,𝔽25)—22no—
81C2×ΓL(1,𝔽25)—22no—
82C2×D3×C8—22no—
83C2×D3×C8—22no—
84C4×D12C2×C412yes—
85C4×D12C2×C412yes—
86#872—22no—
87C42×D3—22no—
88C4×Dic6C2×C412yes—
89C4×Dic6C2×C412yes—
90C3×C4⋊C8D412yes—
91C3×C4⋊C8D412yes—
92C3×#97—22no—
93C3×#97C2×C412yes—
94C3×#97C2×C412yes—
95C3×C4×C8—22no—
96C3×#95—21no—
97#821C2×C412yes—
98#821C2×C412yes—
99#819—22no—
100#819—22no—
101#819—42no—
102#818C2×C411yescharacteristic
103#818C2×C411yescharacteristic
104#817C2×C412yes—
105#816C2×C412yes—
106#816C2×C412yes—
107Dic3⋊C8—22no—
108Dic3⋊C8—22no—
109Dic3⋊C8—22no—
110Dic3⋊C8—22no—
111#805—24no—
112#805—24no—
113C4⋊(C3⋊C8)—22no—
114C4⋊(C3⋊C8)—22no—
115C2×#135—31no—
116C2×#131—32no—
117C2×#131—32no—
118C4×#41—32no—
119#444—64no—
120#443—64no—
121#442—32no—
122#441—32no—
123#440—32no—
124#440—32no—
125C2×#108—31no—
126C2×#108—31no—
127C2×#108—32no—
128#427—31no—
129#424—32no—
130#424—32no—
131#424—62no—
132#424—62no—
133#423—32no—
134#423—62no—
135C2×C4⋊C8—62no—
136C2×C4⋊C8C2×D311yescharacteristic
137C2×#104—32no—
138C2×#104—32no—
139#416—32no—
140C2×#97C2×D311yescharacteristic
141C2×#97C2×D311yescharacteristic
142C2×C4×C8C2×D311yescharacteristic
143C4×C4⋊C4C2×D311yescharacteristic
144#339—32no—
145#339—32no—
146#336C2×D311yescharacteristic
147#336C2×D311yescharacteristic
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4#4111yescharacteristic
151C22×C3×C4#4111yescharacteristic
152C22×C3×C4C2411yescharacteristic
153C2×#67—22no—
154#235—22no—
155#235—22no—
156#235C2×D414yes—
157#235C2×Q814yes—
158C2×D12C22×C412yes—
159C2×C4×D3—22no—
160C2×C4×D3—22no—
161C2×C4×D3—22no—
162C2×Dic6C22×C412yes—
163C2×C3×C8—21no—
164C2×C3×C8—21no—
165C2×C3×C8—21no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—21no—
169C2×C3×C8C22×C412yes—
170C2×C3×C8C22×C411yescharacteristic
171C2×C3×C8C22×C411yescharacteristic
172C2×C3×C8C2×D411yescharacteristic
173C2×C3×C8C2×D412yes—
174C2×C3×C8C2×D411yescharacteristic
175C2×C3×C8C2×D412yes—
176C2×C3×C8C2×Q812yes—
177C3×C4⋊C4—42no—
178C3×C42—21no—
179C3×C42—21no—
180C3×C42—21no—
181C3×C42—21no—
182C3×C42C22×C412yes—
183C3×C42C2×D412yes—
184#217—22no—
185(C2×D3)⋊C4—22no—
186(C2×D3)⋊C4—22no—
187C4⋊Dic3C22×C412yes—
188C4⋊Dic3C22×C412yes—
189Dic3⋊C4—22no—
190Dic3⋊C4—22no—
191C4×Dic3—22no—
192C4×Dic3—22no—
193C4×Dic3—22no—
194#208—22no—
195#208—22no—
196#208—22no—
197#208—22no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8—22no—
200Dic12C4⋊C414yescomplemented
201D24C4⋊C414yescomplemented
202#204C4⋊C418yescomplemented
203ΓL(1,𝔽25)—44no—
204ΓL(1,𝔽25)—44no—
205D3×C8—44no—
206D3×C8—44no—
207C2×#41—32no—
208#135—34no—
209#135—34no—
210C2×Q16—31no—
211C2×SD16—32no—
212C2×D8—31no—
213#131—64no—
214#131—64no—
215#131—122no—
216#131—122no—
217C2×M16—32no—
218C2×M16—32no—
219C2×M16—62no—
220C2×M16—62no—
221C22×C8—62no—
222C22×C8—62no—
223C22×C8C22×D311yescharacteristic
224C22×C8C22×D311yescharacteristic
225C22×C8C22×D311yescharacteristic
226C22×C8C22×D311yescharacteristic
227C4×Q8—32no—
228C4×Q8—32no—
229C4×D4—32no—
230C4×D4—32no—
231C4×D4—62no—
232C4×D4—62no—
233#117—32no—
234#117—62no—
235C2×C4⋊C4—21no—
236C2×C42—62no—
237C2×C42C22×D311yescharacteristic
238C2×C42C22×D311yescharacteristic
239C2×C42C22×D311yescharacteristic
240#108—62no—
241#108—62no—
242#108—122no—
243#107—31no—
244#107—31no—
245#106—32no—
246C4⋊C8—62no—
247C4⋊C8—62no—
248C4⋊C8—62no—
249C4⋊C8—62no—
250C4⋊C8—62no—
251C4⋊C8—62no—
252C4⋊C8D1212yes—
253C4⋊C8D1212yes—
254#104—64no—
255#104—64no—
256#103—32no—
257#103—32no—
258#102—32no—
259#102—32no—
260#97—22no—
261#97C4×D312yes—
262#97C4×D312yes—
263C4×C8—22no—
264#95—21no—
265C23×C3C2×#4111yescharacteristic
266C22×D3—22no—
267C2×C3×C4—22no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4C4×D412yes—
276C2×C3×C4C4×D412yes—
277C2×C3×C4C23×C411yescharacteristic
278C2×C3×C4C22×D411yescharacteristic
279C2×C3×C4C22×D411yescharacteristic
280C2×C3×C4C22×Q811yescharacteristic
281C2×C3×C4H(5,𝔽2)11yescharacteristic
282C2×C3×C42−1+411yescharacteristic
283#67—24no—
284#67—24no—
285C2×Dic3—22no—
286C2×Dic3—22no—
287C2×Dic3—22no—
288D12—22no—
289D12C2×C4⋊C414yes—
290C4×D3—24no—
291C4×D3—24no—
292C4×D3—44no—
293C4×D3—82no—
294Dic6—22no—
295Dic6C2×C4⋊C414yes—
296C3×C8—22no—
297C3×C8—21no—
298C3×C8—22no—
299C3×C8—21no—
300C3×C8C2×C4⋊C412yes—
301C3×C8C2×C4⋊C412yes—
302C3⋊C8—42no—
303C3⋊C8—42no—
304#41—34no—
305#41—34no—
306#41—62no—
307#41—62no—
308C2×Q8—32no—
309C2×D4—32no—
310C2×D4—62no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—62no—
314C22×C4—62no—
315C22×C4—62no—
316C22×C4#23511yescharacteristic
317C22×C4#23511yescharacteristic
318C22×C4C23×D311yescharacteristic
319Q16—34no—
320SD16—38no—
321D8—34no—
322M16—62no—
323M16—62no—
324M16—62no—
325M16—62no—
326M16—124no—
327M16—124no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—62no—
335C2×C8—62no—
336C2×C8—124no—
337C2×C8—124no—
338C2×C8C2×C4×D312yes—
339C2×C8C2×C4×D311yescharacteristic
340C2×C8C2×C4×D311yescharacteristic
341C2×C8C2×D1211yescharacteristic
342C2×C8C2×D1212yes—
343C2×C8C2×D1211yescharacteristic
344C2×C8D3×D412yes—
345C2×C8D3×Q812yes—
346C4⋊C4—32no—
347C4⋊C4—32no—
348C4⋊C4—42no—
349C4⋊C4—62no—
350C4⋊C4—62no—
351C22⋊C4—62no—
352C22⋊C4—62no—
353C22⋊C4—62no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—62no—
359C42—62no—
360C42—62no—
361C42C2×C4×D312yes—
362C42C2×D1212yes—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3C2×C4×D411yescharacteristic
366C22×C3#54711yescharacteristic
367C22×C3#55411yescharacteristic
368C2×D3—24no—
369C2×D3—42no—
370C2×D3—42no—
371C3×C4—22no—
372C3×C4—22no—
373C3×C4—42no—
374C3×C4—81no—
375C3×C4C22×C4⋊C411yescharacteristic
376C3×C4#52011yescharacteristic
377C3×C4D4211yescharacteristic
378C3×C4D4×Q811yescharacteristic
379Dic3—24no—
380Dic3—42no—
381Dic3—42no—
382C23—62no—
383C23C2×#23511yescharacteristic
384Q8—34no—
385Q8—62no—
386D4—34no—
387D4—64no—
388D4—64no—
389D4—62no—
390C2×C4—22no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—62no—
399C2×C4—64no—
400C2×C4—64no—
401C2×C4—62no—
402C2×C4—62no—
403C2×C4—124no—
404C2×C4—242no—
405C2×C4C4×D1212yes—
406C2×C4C4×D1212yes—
407C2×C4C22×C4×D311yescharacteristic
408C2×C4C22×D1211yescharacteristic
409C2×C4C2×D3×D411yescharacteristic
410C2×C4C2×D3×Q811yescharacteristic
411C2×C4#100911yescharacteristic
412C2×C4#101011yescharacteristic
413C8—22no—
414C8—21no—
415C8—22no—
416C8—21no—
417C8—122no—
418C8—122no—
419C8D3×C4⋊C412yes—
420C8D3×C4⋊C412yes—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3D4×C4⋊C411yescharacteristic
424C2×C3#295511yescharacteristic
425C2×C3#295611yescharacteristic
426D3—44no—
427C22—21no—
428C22—21no—
429C22—64no—
430C22—122no—
431C22—122no—
432C22C2×C4×D1211yescharacteristic
433C22#555111yescharacteristic
434C22#557911yescharacteristic
435C4—22no—
436C4—22no—
437C4—42no—
438C4—64no—
439C4—81no—
440C4—122no—
441C4—122no—
442C4C2×D3×C4⋊C411yescharacteristic
443C4#552311yescharacteristic
444C4D4×D1211yescharacteristic
445C4Q8×D1211yescharacteristic
446C3#2268711yescharacteristic, complemented, Sylow
447C2—21no—
448C2—21no—
449C2—124no—
450C2C4⋊C4×D1211yescharacteristic
451C2#7854111yescharacteristic
452C2#7857911yescharacteristic
subgroupG/H
centerC2×C4C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1771C2
socleC22×C3C2×C4×D4
prime radicalC2×#904C22
cosocleC22×C3×C4C24
hypercenter#1771D3
hypocenterC3#22687
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ #1771

Complements

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

NH ≅ G/Ntype
1. #79012C2semidirect
5. C3×#1771C2semidirect
6. #69844C2semidirect
7. #69842C2semidirect
8. #69794C2semidirect
9. #69494C2semidirect
10. #68694C2semidirect
11. #68694C2semidirect
37. #4716C4semidirect
38. #4711C4semidirect
39. #4710C4semidirect
40. #4710C4semidirect
56. #1771D3semidirect
200. Dic12C4⋊C4semidirect
201. D24C4⋊C4semidirect
202. #204C4⋊C4semidirect
446. C3#22687semidirect

Maximal subgroups

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

subgroupindex
#790122
C2×#47172
#784942
C2×#47002
C3×#17712
#698442
#698422
#697942
#694942
#686942
#226873