Finite Group Databasebeta

← #10697119

Subgroups — #10697119

Order 768.

562 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 107 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#83288C211yesmaximal, characteristic
2#83271C211yesmaximal, characteristic
3C2×#4680C211yesmaximal, characteristic, complemented
4C3×#1492C211yesmaximal, characteristic
5#70524C211yesmaximal, characteristic, complemented
6#70506C211yesmaximal, characteristic, complemented
7#67656C211yesmaximal, characteristic, complemented
8#10172—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#951—22no—
11C2×#869C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×C3×C4×C8C2211yescharacteristic
15C2×#817—22no—
16#5092—21no—
17#5080—21no—
18#5076—21no—
19#5070—21no—
20#5060—22no—
21#5034—22no—
22#5008—22no—
23#4925C2211yescharacteristic
24C4×#208C2211yescharacteristic
25#4680—22no—
26#4680—22no—
27C3×D4⋊C8—22no—
28#4554—22no—
29#4482—22no—
30#3227—31no—
31C2×#498—31no—
32#3152—31no—
33#1997—31no—
34#1546—31no—
35#1492D311yescharacteristic
36#1278—31no—
37C2×C3×#41—21no—
38C22×Dic6—21no—
39C22×Dic6—21no—
40C2×C3×M16—21no—
41C22×C3×C8—21no—
42C3×C4⋊Q8D411yescharacteristic
43C3×#127D411yescharacteristic
44C3×#124—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C2×C3×C42C2311yescharacteristic
50#951—44no—
51#951—44no—
52C2×#216—24no—
53C2×#214—24no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3D412yes—
58C2×#208D412yes—
59#869—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869D412yes—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4×C8—21no—
70C3×C4×C8D412yes—
71#832—24no—
72#830—21no—
73#817—28no—
74#808—24no—
75C4⋊(C3⋊C8)D412yes—
76#803—21no—
77C4×C3⋊C8—21no—
78C4×C3⋊C8—21no—
79C2×#137—62no—
80#532C2×D311yescharacteristic
81C2×C4⋊Q8—31no—
82C2×C4⋊Q8—31no—
83#498—62no—
84#498—61no—
85#498—62no—
86#497—62no—
87#494—61no—
88#492—61no—
89#486—61no—
90#481—62no—
91#476—62no—
92#431—31no—
93C2×#106—62no—
94#423C2×D311yescharacteristic
95C4×M16—31no—
96C2×C4×C8C2×D311yescharacteristic
97#343—62no—
98#332—62no—
99D4⋊C8—22no—
100C3×#41—42no—
101C3×#41—42no—
102C2×C3×Q8—21no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4—21no—
106C2×C3×D4—41no—
107C22×C3×C4—41no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C22×Dic3—22no—
112C22×Dic3—41no—
113C2×Dic6—22no—
114C2×Dic6—42no—
115C2×Dic6—42no—
116C2×Dic6—41no—
117C2×Dic6—41no—
118C2×Dic6—41no—
119C2×Dic6—82no—
120C3×M16—42no—
121C2×C3×C8—24no—
122C2×C3×C8—22no—
123C2×C3×C8—41no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C22⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132#216—48no—
133#214—48no—
134C4⋊Dic3—22no—
135C4⋊Dic3—22no—
136C4⋊Dic3—22no—
137C4⋊Dic3—22no—
138C4⋊Dic3—41no—
139C4⋊Dic3—41no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142#208—22no—
143#208SD1614yes—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—22no—
146C2×#41—21no—
147C22×Q8—61no—
148C22×Q8—61no—
149#137—124no—
150#137—124no—
151C2×Q16—64no—
152C2×SD16—64no—
153C2×M16—21no—
154C2×M16—32no—
155C22×C8—21no—
156C4⋊Q8—32no—
157C4⋊Q8—61no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8D1211yescharacteristic
165#127D1211yescharacteristic
166#124—21no—
167#121—21no—
168#121—21no—
169C4×D4—21no—
170C4×D4—21no—
171C2×C4⋊C4—32no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C42C22×D311yescharacteristic
176#106—64no—
177#106—68no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—32no—
181#102—64no—
182C22⋊C8—61no—
183#97—61no—
184C4×C8—21no—
185C4×C8—61no—
186C4×C8—61no—
187C4×C8#6712yes—
188C23×C3—41no—
189C23×C3#12111yescharacteristic
190C3×Q8—42no—
191C3×D4—42no—
192C3×D4—42no—
193C3×D4—42no—
194C3×D4—82no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—41no—
198C2×C3×C4—81no—
199C2×C3×C4—81no—
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12711yescharacteristic
206C2×C3×C4C2×SD1612yes—
207C2×C3×C4AGL(1,ℤ/8)12yes—
208C2×Dic3—42no—
209C2×Dic3—42no—
210C2×Dic3—82no—
211C2×Dic3—82no—
212Dic6—44no—
213Dic6—82no—
214Dic6—82no—
215Dic6—81no—
216Dic6—81no—
217C3×C8—24no—
218C3×C8—81no—
219C3⋊C8—24no—
220C3⋊C8—42no—
221#41—42no—
222#41—42no—
223C2×Q8—21no—
224C2×Q8—62no—
225C2×Q8—121no—
226C2×Q8—121no—
227C2×Q8—121no—
228C2×Q8—122no—
229C2×Q8—122no—
230C2×Q8—242no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—41no—
235C22×C4—41no—
236C22×C4—62no—
237C22×C4—121no—
238C22×C4#23511yescharacteristic
239C22×C4D3×D411yescharacteristic
240C22×C4D3×D411yescharacteristic
241Q16—128no—
242SD16—128no—
243M16—34no—
244M16—42no—
245M16—62no—
246C2×C8—24no—
247C2×C8—22no—
248C2×C8—41no—
249C2×C8—62no—
250C2×C8—62no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—62no—
254C4⋊C4—62no—
255C4⋊C4—62no—
256C4⋊C4—62no—
257C4⋊C4—121no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C42C2×D1211yescharacteristic
264C42D3×D411yescharacteristic
265C42D3×D411yescharacteristic
266C42C2×#6711yescharacteristic
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—81no—
270C22×C3—81no—
271C22×C3—81no—
272C22×C3#39211yescharacteristic
273C22×C3#46012yes—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—81no—
277C3×C4#45012yes—
278C3×C4#45612yes—
279C3×C4#47112yes—
280C3×C4#49211yescharacteristic
281C3×C4#49611yescharacteristic
282Dic3—82no—
283Dic3—82no—
284C23—41no—
285C23#88411yescharacteristic
286Q8—42no—
287Q8—124no—
288Q8—242no—
289Q8—242no—
290Q8—241no—
291Q8—241no—
292D4—42no—
293D4—42no—
294D4—42no—
295D4—82no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—122no—
302C2×C4—122no—
303C2×C4—242no—
304C2×C4—242no—
305C2×C4#88211yescharacteristic
306C2×C4#88211yescharacteristic
307C2×C4#88411yescharacteristic
308C2×C4#91112yes—
309C2×C4D3×SD1612yes—
310C2×C4#93011yescharacteristic
311C2×C4#93711yescharacteristic
312C2×C4#94011yescharacteristic
313C8—24no—
314C8—64no—
315C8—81no—
316C8—122no—
317C2×C3—22no—
318C2×C3—161no—
319C2×C3#200511yescharacteristic
320C2×C3#201411yescharacteristic
321C2×C3#201911yescharacteristic
322C22—21no—
323C22—21no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22#476212yes—
328C22#496011yescharacteristic
329C4—21no—
330C4—21no—
331C4—81no—
332C4—242no—
333C4—242no—
334C4#477612yes—
335C4#482612yes—
336C4#514112yes—
337C4#515411yescharacteristic
338C4#517511yescharacteristic
339C3#1017211yescharacteristic, complemented, Sylow
340C2—22no—
341C2—161no—
342C2#7000711yescharacteristic
343C2#7021011yescharacteristic
344C2#7038911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1492C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1492D3
hypocenterC3#10172
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1492

Complements

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

NH ≅ G/Ntype
3. C2×#4680C2semidirect
5. #70524C2semidirect
6. #70506C2semidirect
7. #67656C2semidirect
339. C3#10172semidirect

Maximal subgroups

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

subgroupindex
#832882
#832712
C2×#46802
C3×#14922
#705242
#705062
#676562
#101723