Finite Group Databasebeta

← #10715607

Subgroups — #10715607

Order 768.

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

normal subgroups only

This group has 304 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#81744C211yesmaximal, characteristic
2#69173C211yesmaximal, characteristic
3#69171C211yesmaximal, characteristic
4C3×#1413C211yesmaximal, characteristic, complemented
5#67786C211yesmaximal, characteristic, complemented
6#67553C211yesmaximal, characteristic, complemented
7#67553C211yesmaximal, characteristic, complemented
8#13612—31nomaximal, Sylow
9#5724C2211yescharacteristic
10#5698C411yescharacteristic
11C3×#491C2211yescharacteristic
12#5092C2211yescharacteristic
13#5068C411yescharacteristic
14#4809—21no—
15#4808—21no—
16#4798—21no—
17#4795—21no—
18#4751—21no—
19D3×#101—21no—
20#4715C411yescharacteristic
21#4709C411yescharacteristic
22C3×#356C2211yescharacteristic
23C3×#356C2211yescharacteristic
24#4548C2211yescharacteristic
25#4548C2211yescharacteristic
26#4478—21no—
27#4477—21no—
28#3223—31no—
29#2135—31no—
30#2133—31no—
31#1413—31no—
32#1413D311yescharacteristic, complemented
33#1411—31no—
34#1411—31no—
35C2×#239D411yescharacteristic
36C2×D3×Q8D411yescharacteristic
37C3×C4⋊Q8C2311yescharacteristic
38C3×#125C2×C411yescharacteristic
39#947D411yescharacteristic
40#944D411yescharacteristic
41C2×#216—21no—
42C2×#216—21no—
43D3×M16—41no—
44D3×M16—41no—
45C2×Dic12—21no—
46C2×#204—21no—
47#898—41no—
48#897—21no—
49#896—41no—
50#894—41no—
51#890—41no—
52#887—21no—
53#875C2×C411yescharacteristic
54#872C2×C411yescharacteristic
55#869C2×C411yescharacteristic
56C3×#103—21no—
57C3×#103—21no—
58C3×#101—21no—
59C3×#101—21no—
60C3×#97C2×C411yescharacteristic
61#836—21no—
62#836—21no—
63#824—41no—
64#823—41no—
65(C2×D3)⋊C8—21no—
66Dic3⋊C8—21no—
67#810—21no—
68#809—21no—
69C4×C3⋊C8C2×C411yescharacteristic
70#541—31no—
71#536—31no—
72#497—31no—
73#494—31no—
74#491C2×D311yescharacteristic
75#488—31no—
76#464—61no—
77#462—61no—
78#452—61no—
79#451—61no—
80#433—31no—
81#413—61no—
82C2×#101—61no—
83#356—31no—
84#356—31no—
85#356—61no—
86#356C2×D311yescharacteristic
87#356C2×D311yescharacteristic
88#355—61no—
89C2×C3×Q8C2×D411yescharacteristic
90C2×C3×Q8C2×D411yescharacteristic
91#239—81no—
92D3×Q8—81no—
93C2×D12—21no—
94C2×D12C22⋊C411yescharacteristic
95C2×C4×D3—21no—
96C2×C4×D3—21no—
97C2×C4×D3C22⋊C411yescharacteristic
98C2×Dic6—21no—
99C2×Dic6—21no—
100C2×Dic6C22⋊C411yescharacteristic
101C3×M16—41no—
102C3×M16—41no—
103C2×C3×C8—21no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—41no—
106C3×C4⋊C4—41no—
107C3×C42C22×C411yescharacteristic
108#216—81no—
109#216—81no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—41no—
112(C2×D3)⋊C4—41no—
113(C2×D3)⋊C4—41no—
114C4⋊Dic3—41no—
115C4⋊Dic3—41no—
116Dic3⋊C4—21no—
117Dic3⋊C4—41no—
118Dic3⋊C4—41no—
119C4×Dic3—41no—
120C4×Dic3C22⋊C411yescharacteristic
121#208—41no—
122#208—41no—
123C2×C3⋊C8—21no—
124Dic12—81no—
125#204—81no—
126ΓL(1,𝔽25)—81no—
127ΓL(1,𝔽25)—81no—
128D3×C8—81no—
129D3×C8—81no—
130C2×#41—31no—
131C22×Q8—31no—
132C2×Q16—61no—
133C2×Q16—61no—
134C2×Q16—61no—
135C2×SD16—61no—
136C2×M16—121no—
137C2×M16—121no—
138C4⋊Q8—31no—
139C4⋊Q8—31no—
140C4⋊Q8C22×D311yescharacteristic
141#126—121no—
142#125—121no—
143#125C4×D311yescharacteristic
144#124—31no—
145#124—31no—
146#123—121no—
147C22⋊Q8—61no—
148C22⋊Q8—121no—
149C4×Q8—61no—
150#117—31no—
151C4⋊C8—61no—
152#103—21no—
153#103—21no—
154#103—61no—
155#102—61no—
156#101—21no—
157#101—21no—
158#101—61no—
159#101—61no—
160#101—121no—
161#100—121no—
162C22⋊C8—61no—
163#97C4×D311yescharacteristic
164C4×C8—31no—
165C22×D3—21no—
166C22×D3#9911yescharacteristic
167C3×Q8—41no—
168C3×Q8—41no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—41no—
173C2×C3×C4C2×C22⋊C411yescharacteristic
174C2×Dic3—21no—
175C2×Dic3—21no—
176C2×Dic3—41no—
177C2×Dic3#9911yescharacteristic
178D12—41no—
179D12—81no—
180C4×D3—41no—
181C4×D3—41no—
182C4×D3—41no—
183C4×D3—41no—
184C4×D3—41no—
185Dic6—41no—
186Dic6—41no—
187Dic6—41no—
188Dic6—81no—
189C3×C8—41no—
190C3×C8—81no—
191C3×C8—81no—
192C3⋊C8—41no—
193C3⋊C8—81no—
194C3⋊C8—81no—
195#41—241no—
196C2×Q8—31no—
197C2×Q8—61no—
198C2×Q8—61no—
199C2×Q8—241no—
200C2×Q8D3×D411yescharacteristic
201C2×Q8D3×D411yescharacteristic
202C2×D4—31no—
203C2×D4—61no—
204C22×C4—31no—
205C22×C4—61no—
206C22×C4—61no—
207Q16—241no—
208Q16—241no—
209Q16—241no—
210SD16—241no—
211M16—41no—
212M16—41no—
213M16—121no—
214M16—121no—
215M16—241no—
216M16—241no—
217C2×C8—21no—
218C2×C8—61no—
219C2×C8—241no—
220C2×C8—241no—
221C4⋊C4—21no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—61no—
225C4⋊C4—121no—
226C4⋊C4—121no—
227C4⋊C4—121no—
228C4⋊C4—121no—
229C22⋊C4—61no—
230C22⋊C4—121no—
231C22⋊C4—121no—
232C22⋊C4—121no—
233C42—31no—
234C42—121no—
235C42C2×C4×D311yescharacteristic
236C22×C3C2×#9911yescharacteristic
237C2×D3—21no—
238C2×D3—41no—
239C2×D3—41no—
240C2×D3—81no—
241C3×C4—21no—
242C3×C4—41no—
243C3×C4—41no—
244C3×C4—41no—
245C3×C4—81no—
246Dic3—21no—
247Dic3—41no—
248Dic3—41no—
249Dic3—81no—
250C23—31no—
251C23—61no—
252Q8—41no—
253Q8—41no—
254Q8—121no—
255Q8—121no—
256Q8—121no—
257Q8—241no—
258D4—121no—
259D4—241no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—31no—
264C2×C4—41no—
265C2×C4—61no—
266C2×C4—61no—
267C2×C4—121no—
268C2×C4—121no—
269C2×C4—121no—
270C2×C4—121no—
271C2×C4—121no—
272C2×C4—121no—
273C2×C4D3×C22⋊C411yescharacteristic
274C8—41no—
275C8—81no—
276C8—81no—
277C8—121no—
278C8—241no—
279C8—241no—
280C2×C3—21no—
281C2×C3#212711yescharacteristic
282D3—41no—
283D3—81no—
284C22—61no—
285C22—121no—
286C22—121no—
287C22—241no—
288C22D3×#9911yescharacteristic
289C4—21no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—61no—
294C4—81no—
295C4—121no—
296C4—121no—
297C4—241no—
298C3#1361211yescharacteristic, complemented, Sylow
299C2—21no—
300C2—121no—
301C2—241no—
302C2#6915711yescharacteristic
subgroupG/H
centerC2#69157
commutatorC3×C42C22×C4
FrattiniC4⋊Q8C22×D3
FittingC3×#1413C2
socleC2×C3#2127
prime radicalC2×#239D4
cosocleC3×C4⋊Q8C23
hypercenter#1413D3
hypocenterC3#13612
LCSG ▷ C3×C42 ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #1413

Complements

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

NH ≅ G/Ntype
4. C3×#1413C2semidirect
5. #67786C2semidirect
6. #67553C2semidirect
7. #67553C2semidirect
32. #1413D3semidirect
298. C3#13612semidirect

Maximal subgroups

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

subgroupindex
#817442
#691732
#691712
C3×#14132
#677862
#675532
#136123