Finite Group Databasebeta

← #10708186

Subgroups — #10708186

Order 768.

394 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 57 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#77796C211yesmaximal, characteristic
2#69713C211yesmaximal, characteristic
3#68978C211yesmaximal, characteristic
4#68140C211yesmaximal, characteristic
5C3×#1346C211yesmaximal, characteristic
6#67495C211yesmaximal, characteristic
7#67453C211yesmaximal, characteristic
8#12247—31nomaximal, Sylow
9#5500C2211yescharacteristic
10C2×#887—21no—
11C3×#398C2211yescharacteristic
12#5121—21no—
13#4982—21no—
14#4981C2211yescharacteristic
15#4976—21no—
16#4975C2211yescharacteristic
17#4971—21no—
18#4721—21no—
19#4647—21no—
20#4646—21no—
21#4646—21no—
22C3×#343C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4555—21no—
25#4528C2211yescharacteristic
26#4467—21no—
27#4452—21no—
28#2582—31no—
29#2060—31no—
30#1611—31no—
31#1611—31no—
32#1346—31no—
33#1346D311yescharacteristic
34#1325—31no—
35C22×Dic6—21no—
36C2×C3×M16—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C4⋊C4C2311yescharacteristic
40#926—21no—
41C2×C4⋊Dic3D411yescharacteristic
42C2×C4⋊Dic3D411yescharacteristic
43#924—21no—
44#924—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×C4×Dic3—21no—
51C2×C4×Dic3—21no—
52C2×C4×Dic3D411yescharacteristic
53#890—41no—
54#888—41no—
55#887—42no—
56#887—42no—
57C3×C22⋊C8—21no—
58C3×#95—21no—
59C3×#95D411yescharacteristic
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95D411yescharacteristic
65#830—21no—
66#818—41no—
67#817—41no—
68#817—41no—
69#816—41no—
70#810—41no—
71#808—41no—
72#544—31no—
73C2×C4×Q8—61no—
74#483—61no—
75#483—61no—
76#428—61no—
77#398—31no—
78#398—61no—
79#398C2×D311yescharacteristic
80#391—61no—
81#391—61no—
82#389—61no—
83#389—61no—
84#384—31no—
85#343C2×D311yescharacteristic
86#339—61no—
87#328—61no—
88#328—61no—
89#324—31no—
90#324C2×D311yescharacteristic
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×Dic3—21no—
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C2×Dic6—24no—
101C2×Dic6—21no—
102C2×Dic6—21no—
103C2×Dic6—42no—
104C2×Dic6—42no—
105C3×M16—22no—
106C3×M16—41no—
107C2×C3×C8—41no—
108C2×C3×C8—41no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113#217—41no—
114C4⋊Dic3—21no—
115C4⋊Dic3—21no—
116C4⋊Dic3—21no—
117C4⋊Dic3—21no—
118C4⋊Dic3—41no—
119C4⋊Dic3—41no—
120Dic3⋊C4—42no—
121Dic3⋊C4—42no—
122Dic3⋊C4—42no—
123Dic3⋊C4—41no—
124Dic3⋊C4—41no—
125Dic3⋊C4—41no—
126Dic3⋊C4—41no—
127Dic3⋊C4—81no—
128C4×Dic3—21no—
129C4×Dic3—42no—
130C4×Dic3—42no—
131C4×Dic3—41no—
132C4×Dic3SD1612yes—
133C2×C3⋊C8—41no—
134C22×Q8—61no—
135C2×M16—21no—
136C4⋊Q8—121no—
137#125—121no—
138C22⋊Q8—61no—
139C22⋊Q8—61no—
140C4×Q8—122no—
141C4×Q8—122no—
142#117—61no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4D1211yescharacteristic
152C2×C4⋊C4C22×D311yescharacteristic
153C2×C42—31no—
154C2×C42—61no—
155C2×C42—61no—
156#107—121no—
157#106—121no—
158#106—121no—
159#106—121no—
160#103—121no—
161#103—121no—
162C22⋊C8—21no—
163C22⋊C8—61no—
164#95—21no—
165#95—31no—
166#95—61no—
167#95—61no—
168#95—61no—
169#95—61no—
170#95D1211yescharacteristic
171C23×C3#12011yescharacteristic
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—22no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4C2×SD1611yescharacteristic
185C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
186C2×Dic3—22no—
187C2×Dic3—22no—
188C2×Dic3—22no—
189C2×Dic3—22no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200Dic6—48no—
201Dic6—81no—
202Dic6—81no—
203C3×C8—42no—
204C3×C8—81no—
205C3⋊C8—81no—
206C2×Q8—64no—
207C2×Q8—61no—
208C2×Q8—61no—
209C2×Q8—122no—
210C2×Q8—122no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4C2×D1211yescharacteristic
218C22×C4D3×D411yescharacteristic
219C22×C4D3×D411yescharacteristic
220M16—22no—
221M16—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—121no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—121no—
234C4⋊C4—121no—
235C4⋊C4—121no—
236C4⋊C4—121no—
237C4⋊C4—122no—
238C4⋊C4—122no—
239C4⋊C4—122no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C4⋊C4—241no—
243C22⋊C4—121no—
244C42—32no—
245C42—61no—
246C42—121no—
247C42—122no—
248C42—122no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3#45011yescharacteristic
252C22×C3#45611yescharacteristic
253C22×C3B(4,𝔽2)11yescharacteristic
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—42no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—81no—
260Dic3—24no—
261Dic3—42no—
262Dic3—81no—
263Dic3—81no—
264Dic3—81no—
265Dic3—81no—
266C23#88211yescharacteristic
267Q8—128no—
268Q8—241no—
269Q8—241no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—62no—
283C2×C4—62no—
284C2×C4—62no—
285C2×C4—62no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4#91111yescharacteristic
297C2×C4D3×SD1611yescharacteristic
298C8—42no—
299C8—81no—
300C8—241no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3#160411yescharacteristic
304C2×C3#219811yescharacteristic
305C2×C3#220511yescharacteristic
306C22—21no—
307C22—21no—
308C22#474111yescharacteristic
309C22#477611yescharacteristic
310C22#482611yescharacteristic
311C4—21no—
312C4—21no—
313C4—42no—
314C4—41no—
315C4—41no—
316C4—64no—
317C4—81no—
318C4—122no—
319C4—241no—
320C4—241no—
321C4—241no—
322C4—241no—
323C3#1224711yescharacteristic, complemented, Sylow
324C2—21no—
325C2—21no—
326C2#6829511yescharacteristic
327C2#6914011yescharacteristic
328C2#6916911yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1346C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#1346D3
hypocenterC3#12247
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1346

Complements

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

NH ≅ G/Ntype
323. C3#12247semidirect

Maximal subgroups

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

subgroupindex
#777962
#697132
#689782
#681402
C3×#13462
#674952
#674532
#122473