Finite Group Databasebeta

← #10897635

Subgroups — #10897635

Order 768.

708 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#84000C211yesmaximal, characteristic, complemented
2#78619C211yesmaximal, characteristic, complemented
3C3×#2010C211yesmaximal, characteristic
4#72117C211yesmaximal, characteristic, complemented
5#71714C212yesmaximal, complemented
6#71711C212yesmaximal, complemented
7#70401C212yesmaximal, complemented
8#70398C212yesmaximal, complemented
9#69987C211yesmaximal, characteristic, complemented
10#68701C211yesmaximal, characteristic, complemented
11#68696C211yesmaximal, characteristic, complemented
12#29384—31nomaximal, Sylow
13C3×#575C2211yescharacteristic
14C3×#525C2212yes—
15#5819C2212yes—
16C22×C4×Dic3C2211yescharacteristic
17C3×#421C2212yes—
18C3×#407C2211yescharacteristic
19#5237C2212yes—
20C2×#837C2212yes—
21C2×#837C2212yes—
22#5207C2211yescharacteristic
23#5207C2212yes—
24#5197—22no—
25#5196—22no—
26#5125C2211yescharacteristic
27#5124C2212yes—
28#5118C2211yescharacteristic
29#5118C2211yescharacteristic
30Dic3×M16C2211yescharacteristic
31Dic3×M16C2211yescharacteristic
32#4719—22no—
33#4718—22no—
34C3×H(3,ℤ/4)C2211yescharacteristic
35C3⋊#338C2211yescharacteristic
36#4533C2212yes—
37#4533C2212yes—
38#4532C2212yes—
39#4532C2212yes—
40#2956—31no—
41#2920—31no—
42#2102—31no—
43#2091—31no—
44#2050—31no—
45#2010D311yescharacteristic
46#1932—32no—
47#1896—32no—
48#1891—32no—
49#1861—32no—
50#1841—31no—
51C2×C3×#41C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×Dic3—21no—
54C3×#137—22no—
55C3×AGL(1,ℤ/8)—22no—
56C3×#135—22no—
57C2×C3×M16C2311yescharacteristic
58C2×C3×M16C2311yescharacteristic
59C3×#123—22no—
60C3×C22⋊Q8—22no—
61C3×#121—22no—
62C3×#120—22no—
63C3×C4×D4—42no—
64C3×#117C2312yes—
65#948—24no—
66#948—42no—
67C2×C4×Dic3—21no—
68C2×C4×Dic3—21no—
69C2×C4×Dic3—21no—
70C2×C4×Dic3—21no—
71C2×C4×Dic3—21no—
72C2×C4×Dic3C2311yescharacteristic
73C2×C4×Dic3C2311yescharacteristic
74C2×#208—42no—
75C2×#208C2312yes—
76C2×#208C2312yes—
77C2×#208C2311yescharacteristic
78C2×#208C2311yescharacteristic
79C3×#104—42no—
80C3×C22⋊C8—22no—
81#837—22no—
82#837—22no—
83#837—22no—
84#837—22no—
85#836—22no—
86#833—22no—
87#830—22no—
88#830—42no—
89#822—42no—
90#819—22no—
91#815—21no—
92#815D412yescomplemented
93Dic3⋊C8—22no—
94Dic3⋊C8—22no—
95C8×Dic3—21no—
96C8×Dic3D412yescomplemented
97#575C2×D311yescharacteristic
98#568—32no—
99#525C2×D312yes—
100C22×C42—31no—
101#444—62no—
102#443—62no—
103#432—62no—
104#431—62no—
105#430—31no—
106#430—32no—
107#423—31no—
108#423—31no—
109#421C2×D312yes—
110C2×#104—32no—
111C2×#104—32no—
112#413—32no—
113#407—31no—
114#407—32no—
115#407C2×D311yescharacteristic
116C4×M16—31no—
117C4×M16—31no—
118#344—32no—
119#344—32no—
120#339—32no—
121#339—32no—
122#338—31no—
123H(3,ℤ/4)C2×D311yescharacteristic
124C24×C3#4111yescharacteristic
125C3×#41—22no—
126C3×#41—22no—
127C3×#41—22no—
128C3×#41—22no—
129C2×C3×Q8#4112yes—
130C2×C3×D4—22no—
131C2×C3×D4—42no—
132C2×C3×D4#4112yes—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4#4111yescharacteristic
139C22×C3×C4C2411yescharacteristic
140C22×Dic3—21no—
141C22×Dic3—21no—
142C22×Dic3—21no—
143C22×Dic3—21no—
144C22×Dic3—21no—
145C22×Dic3—21no—
146C22×Dic3—44no—
147C3×Q16—22no—
148C3×SD16—24no—
149C3×D8—22no—
150C3×M16—22no—
151C3×M16—22no—
152C2×C3×C8—21no—
153C2×C3×C8C2×D412yes—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—42no—
156C3×C22⋊C4—22no—
157C3×C22⋊C4—42no—
158C3×C22⋊C4—42no—
159C3×C42—22no—
160C4×Dic3—21no—
161C4×Dic3—21no—
162C4×Dic3—21no—
163C4×Dic3—21no—
164C4×Dic3—21no—
165C4×Dic3—21no—
166C4×Dic3C2×D411yescharacteristic
167C4×Dic3C2×D411yescharacteristic
168C4×Dic3#4111yescharacteristic
169C4×Dic3#4111yescharacteristic
170#208—22no—
171#208—22no—
172#208—22no—
173#208—22no—
174#208—22no—
175#208—22no—
176#208—44no—
177C2×C3⋊C8—21no—
178C2×C3⋊C8—22no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—44no—
181C2×C3⋊C8C2×D412yes—
182C2×#41C22×D312yes—
183C23×C4—61no—
184C23×C4C22×D311yescharacteristic
185#137—22no—
186AGL(1,ℤ/8)—22no—
187#135—22no—
188#131—64no—
189#131—122no—
190C2×M16—31no—
191C2×M16—32no—
192C2×M16—32no—
193C2×M16—31no—
194C2×M16—122no—
195C2×M16C22×D311yescharacteristic
196C2×M16C22×D311yescharacteristic
197#123—22no—
198C22⋊Q8—22no—
199#121—22no—
200#120—22no—
201C4×D4—42no—
202#117C22×D312yes—
203C2×C42—31no—
204C2×C42—31no—
205C2×C42—61no—
206C2×C42—61no—
207C2×C42—61no—
208C2×C42—61no—
209C2×C42—61no—
210#108—62no—
211#108—122no—
212C4⋊C8—62no—
213C4⋊C8—62no—
214#104—42no—
215#104—62no—
216#104—62no—
217#104—62no—
218#104—62no—
219#101—62no—
220#100—62no—
221C22⋊C8—22no—
222C22⋊C8—62no—
223C22⋊C8—122no—
224#97—32no—
225#97—61no—
226C4×C8—32no—
227C4×C8—61no—
228C23×C3—21no—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—42no—
232C23×C3C2×#4111yescharacteristic
233C3×Q8—22no—
234C3×Q8—22no—
235C3×D4—22no—
236C3×D4—22no—
237C3×D4—42no—
238C3×D4—42no—
239C3×D4—84no—
240C2×C3×C4—21no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4C22×D411yescharacteristic
250C2×C3×C4C2×#4112yes—
251C2×C3×C4C2×#4111yescharacteristic
252C2×C3×C4H(5,𝔽2)12yes—
253C2×Dic3—24no—
254C2×Dic3—21no—
255C2×Dic3—21no—
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C3×C8—22no—
264C3×C8—41no—
265C3⋊C8—22no—
266C3⋊C8—41no—
267C3⋊C8—42no—
268C3⋊C8—42no—
269C24#23711yescharacteristic
270#41—22no—
271#41—22no—
272#41—22no—
273#41—22no—
274C2×Q8#23712yes—
275C2×D4—22no—
276C2×D4—42no—
277C2×D4#23712yes—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4—124no—
290C22×C4#23711yescharacteristic
291C22×C4C23×D311yescharacteristic
292Q16—22no—
293SD16—24no—
294D8—22no—
295M16—22no—
296M16—22no—
297M16—62no—
298M16—62no—
299M16—62no—
300M16—62no—
301M16—62no—
302M16—62no—
303M16—124no—
304C2×C8—21no—
305C2×C8—32no—
306C2×C8—61no—
307C2×C8—62no—
308C2×C8—62no—
309C2×C8—124no—
310C2×C8D3×D412yes—
311C4⋊C4—22no—
312C4⋊C4—42no—
313C22⋊C4—22no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C42—22no—
317C42—31no—
318C42—31no—
319C42—31no—
320C42—31no—
321C42—61no—
322C42—61no—
323C42—61no—
324C42—61no—
325C42—61no—
326C42—61no—
327C22×C3—22no—
328C22×C3—24no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—42no—
332C22×C3—41no—
333C22×C3—42no—
334C22×C3—82no—
335C22×C3#52511yescharacteristic
336C22×C3#55112yes—
337C3×C4—41no—
338C3×C4—42no—
339C3×C4—82no—
340C3×C4#52511yescharacteristic
341C3×C4#53511yescharacteristic
342C3×C4#55312yes—
343Dic3—22no—
344Dic3—22no—
345Dic3—41no—
346Dic3—41no—
347C23—22no—
348C23—21no—
349C23—22no—
350C23—42no—
351C23C2×#23711yescharacteristic
352Q8—22no—
353Q8—22no—
354D4—22no—
355D4—22no—
356D4—42no—
357D4—42no—
358D4—84no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—64no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4C2×D3×D411yescharacteristic
379C2×C4C2×#23712yes—
380C2×C4#100412yes—
381C2×C4D3×#4111yescharacteristic
382C8—22no—
383C8—41no—
384C8—62no—
385C8—122no—
386C8—121no—
387C8—122no—
388C2×C3—21no—
389C2×C3—22no—
390C2×C3—42no—
391C2×C3—82no—
392C2×C3#259811yescharacteristic
393C22—22no—
394C22—24no—
395C22—41no—
396C22—41no—
397C22—42no—
398C22—41no—
399C22—42no—
400C22—82no—
401C22#558712yes—
402C22#560611yescharacteristic
403C4—41no—
404C4—42no—
405C4—62no—
406C4—62no—
407C4—82no—
408C4—121no—
409C4—121no—
410C4#558611yescharacteristic
411C4#560112yes—
412C4#571511yescharacteristic
413C3#2938411yescharacteristic, complemented, Sylow
414C2—21no—
415C2—22no—
416C2—42no—
417C2—82no—
418C2#7669211yescharacteristic
subgroupG/H
centerC4#5715
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2010C2
socleC2×C3#2598
hypercenter#2010D3
hypocenterC3#29384
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2010

Complements

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

NH ≅ G/Ntype
1. #84000C2semidirect
2. #78619C2semidirect
4. #72117C2semidirect
5. #71714C2semidirect
6. #71711C2semidirect
7. #70401C2semidirect
8. #70398C2semidirect
9. #69987C2semidirect
10. #68701C2semidirect
11. #68696C2semidirect
92. #815D4semidirect
96. C8×Dic3D4semidirect
413. C3#29384semidirect

Maximal subgroups

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

subgroupindex
#840002
#786192
C3×#20102
#721172
#717142
#717112
#704012
#703982
#699872
#687012
#686962
#293843