Finite Group Databasebeta

← #10897646

Subgroups — #10897646

Order 768.

662 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#84000C211yesmaximal, characteristic, complemented
2#78620C211yesmaximal, characteristic, complemented
3C3×#2010C211yesmaximal, characteristic
4#72117C211yesmaximal, characteristic, complemented
5#71714C212yesmaximal, complemented
6#71712C212yesmaximal, complemented
7#70401C212yesmaximal, complemented
8#70398C212yesmaximal, complemented
9#69987C211yesmaximal, characteristic, complemented
10#68701C211yesmaximal, characteristic, complemented
11#68696C211yesmaximal, characteristic, complemented
12#29385—31nomaximal, Sylow
13C3×#575C2211yescharacteristic
14C3×#525C2212yes—
15#5819C2212yes—
16C2×#926C2211yescharacteristic
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#4720—22no—
33#4720—22no—
34C3×H(3,ℤ/4)C2211yescharacteristic
35C3⋊#338C2211yescharacteristic
36#4533C2212yes—
37#4533C2212yes—
38#4532C2212yes—
39#4532C2212yes—
40#2956—31no—
41#2921—31no—
42#2102—31no—
43#2091—31no—
44#2050—31no—
45#2010D311yescharacteristic
46#1932—32no—
47#1897—32no—
48#1891—32no—
49#1861—32no—
50#1841—31no—
51C2×C3×#41C2312yes—
52C23×C3×C4C2311yescharacteristic
53C3×#137—22no—
54C3×AGL(1,ℤ/8)—22no—
55C3×#135—22no—
56C2×C3×M16C2311yescharacteristic
57C2×C3×M16C2311yescharacteristic
58C3×#123—22no—
59C3×C22⋊Q8—22no—
60C3×#121—22no—
61C3×#120—22no—
62C3×C4×D4—42no—
63C3×#117C2312yes—
64C2×#217—21no—
65#948—24no—
66#948—42no—
67#926—21no—
68#926—21no—
69#926—22no—
70C2×C4⋊Dic3—21no—
71C2×C4×Dic3C2311yescharacteristic
72C2×C4×Dic3C2311yescharacteristic
73C2×#208—42no—
74C2×#208C2312yes—
75C2×#208C2312yes—
76C2×#208C2311yescharacteristic
77C2×#208C2311yescharacteristic
78C3×#104—42no—
79C3×C22⋊C8—22no—
80#837—22no—
81#837—22no—
82#837—24no—
83#836—22no—
84#833—22no—
85#830—22no—
86#830—42no—
87#822—42no—
88#819—22no—
89#815—21no—
90#815D412yescomplemented
91Dic3⋊C8—22no—
92Dic3⋊C8—22no—
93C8×Dic3—21no—
94C8×Dic3D412yescomplemented
95#575C2×D311yescharacteristic
96#568—32no—
97#525C2×D312yes—
98C2×#117—31no—
99#444—62no—
100#443—62no—
101#433—62no—
102#433—62no—
103#430—31no—
104#430—32no—
105#423—31no—
106#423—31no—
107#421C2×D312yes—
108C2×#104—32no—
109C2×#104—32no—
110#413—32no—
111#407—31no—
112#407—32no—
113#407C2×D311yescharacteristic
114C4×M16—31no—
115C4×M16—31no—
116#344—32no—
117#344—32no—
118#339—32no—
119#339—32no—
120#338—31no—
121H(3,ℤ/4)C2×D311yescharacteristic
122C24×C3#4111yescharacteristic
123C3×#41—22no—
124C3×#41—22no—
125C3×#41—22no—
126C3×#41—22no—
127C2×C3×Q8#4112yes—
128C2×C3×D4—22no—
129C2×C3×D4—42no—
130C2×C3×D4#4112yes—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4C2411yescharacteristic
138C22×Dic3—21no—
139C22×Dic3—21no—
140C3×Q16—22no—
141C3×SD16—24no—
142C3×D8—22no—
143C3×M16—22no—
144C3×M16—22no—
145C2×C3×C8—21no—
146C2×C3×C8C2×D412yes—
147C3×C4⋊C4—22no—
148C3×C4⋊C4—42no—
149C3×C22⋊C4—22no—
150C3×C22⋊C4—42no—
151C3×C22⋊C4—42no—
152C3×C42—22no—
153#217—22no—
154#217—22no—
155C4⋊Dic3—22no—
156C4⋊Dic3—22no—
157C4×Dic3—21no—
158C4×Dic3—21no—
159C4×Dic3C2×D411yescharacteristic
160C4×Dic3C2×D411yescharacteristic
161C4×Dic3#4112yes—
162#208—22no—
163#208—22no—
164#208—22no—
165#208—22no—
166#208—22no—
167#208—22no—
168#208—44no—
169C2×C3⋊C8—21no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—44no—
173C2×C3⋊C8C2×D412yes—
174C2×#41C22×D312yes—
175C23×C4C22×D311yescharacteristic
176#137—22no—
177AGL(1,ℤ/8)—22no—
178#135—22no—
179#131—64no—
180#131—122no—
181C2×M16—31no—
182C2×M16—32no—
183C2×M16—32no—
184C2×M16—31no—
185C2×M16—122no—
186C2×M16C22×D311yescharacteristic
187C2×M16C22×D311yescharacteristic
188#123—22no—
189C22⋊Q8—22no—
190#121—22no—
191#120—22no—
192C4×D4—42no—
193#117—62no—
194#117—61no—
195#117—61no—
196#117C22×D312yes—
197C2×C4⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C42—31no—
200C2×C42—31no—
201#108—62no—
202#108—122no—
203C4⋊C8—62no—
204C4⋊C8—62no—
205#104—42no—
206#104—64no—
207#104—62no—
208#104—62no—
209#101—62no—
210#100—62no—
211C22⋊C8—22no—
212C22⋊C8—62no—
213C22⋊C8—122no—
214#97—32no—
215#97—61no—
216C4×C8—32no—
217C4×C8—61no—
218C23×C3—21no—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—81no—
222C23×C3C2×#4111yescharacteristic
223C3×Q8—22no—
224C3×Q8—22no—
225C3×D4—22no—
226C3×D4—22no—
227C3×D4—42no—
228C3×D4—42no—
229C3×D4—84no—
230C2×C3×C4—21no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4C22×D411yescharacteristic
240C2×C3×C4C2×#4112yes—
241C2×C3×C4C2×#4111yescharacteristic
242C2×C3×C4H(5,𝔽2)12yes—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—22no—
246C2×Dic3—81no—
247C2×Dic3—81no—
248C3×C8—22no—
249C3×C8—41no—
250C3⋊C8—22no—
251C3⋊C8—41no—
252C3⋊C8—42no—
253C3⋊C8—42no—
254C24#23711yescharacteristic
255#41—22no—
256#41—22no—
257#41—22no—
258#41—22no—
259C2×Q8#23712yes—
260C2×D4—22no—
261C2×D4—42no—
262C2×D4#23712yes—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4#23711yescharacteristic
271C22×C4C23×D311yescharacteristic
272Q16—22no—
273SD16—24no—
274D8—22no—
275M16—22no—
276M16—22no—
277M16—62no—
278M16—62no—
279M16—62no—
280M16—62no—
281M16—62no—
282M16—62no—
283M16—124no—
284C2×C8—21no—
285C2×C8—32no—
286C2×C8—61no—
287C2×C8—62no—
288C2×C8—62no—
289C2×C8—124no—
290C2×C8D3×D412yes—
291C4⋊C4—22no—
292C4⋊C4—42no—
293C4⋊C4—62no—
294C4⋊C4—62no—
295C22⋊C4—22no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—62no—
299C22⋊C4—62no—
300C42—22no—
301C42—31no—
302C42—31no—
303C42—32no—
304C42—61no—
305C42—61no—
306C22×C3—41no—
307C22×C3—42no—
308C22×C3—42no—
309C22×C3—42no—
310C22×C3—41no—
311C22×C3—42no—
312C22×C3—82no—
313C22×C3#52511yescharacteristic
314C22×C3#55112yes—
315C3×C4—41no—
316C3×C4—42no—
317C3×C4—82no—
318C3×C4#52511yescharacteristic
319C3×C4#53511yescharacteristic
320C3×C4#55312yes—
321Dic3—41no—
322Dic3—41no—
323Dic3—42no—
324C23—22no—
325C23—21no—
326C23—22no—
327C23—81no—
328C23C2×#23711yescharacteristic
329Q8—22no—
330Q8—22no—
331D4—22no—
332D4—22no—
333D4—42no—
334D4—42no—
335D4—84no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—62no—
348C2×C4—241no—
349C2×C4—241no—
350C2×C4C2×D3×D411yescharacteristic
351C2×C4C2×#23712yes—
352C2×C4#100412yes—
353C2×C4D3×#4111yescharacteristic
354C8—22no—
355C8—41no—
356C8—62no—
357C8—122no—
358C8—121no—
359C8—122no—
360C2×C3—21no—
361C2×C3—22no—
362C2×C3—81no—
363C2×C3—82no—
364C2×C3#259811yescharacteristic
365C22—41no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—41no—
370C22—42no—
371C22—82no—
372C22#558712yes—
373C22#560611yescharacteristic
374C4—41no—
375C4—42no—
376C4—82no—
377C4—121no—
378C4—121no—
379C4—122no—
380C4#558611yescharacteristic
381C4#560112yes—
382C4#571511yescharacteristic
383C3#2938511yescharacteristic, complemented, Sylow
384C2—22no—
385C2—21no—
386C2—81no—
387C2—82no—
388C2#7669211yescharacteristic
subgroupG/H
centerC4#5715
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2010C2
socleC2×C3#2598
hypercenter#2010D3
hypocenterC3#29385
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. #78620C2semidirect
4. #72117C2semidirect
5. #71714C2semidirect
6. #71712C2semidirect
7. #70401C2semidirect
8. #70398C2semidirect
9. #69987C2semidirect
10. #68701C2semidirect
11. #68696C2semidirect
90. #815D4semidirect
94. C8×Dic3D4semidirect
383. C3#29385semidirect

Maximal subgroups

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

subgroupindex
#840002
#786202
C3×#20102
#721172
#717142
#717122
#704012
#703982
#699872
#687012
#686962
#293853