Finite Group Databasebeta

← #10712068

Subgroups — #10712068

Order 768.

488 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83734C211yesmaximal, characteristic
2#82007C211yesmaximal, characteristic, complemented
3C3×#2112C211yesmaximal, characteristic, complemented
4#71963C211yesmaximal, characteristic, complemented
5#71435C211yesmaximal, characteristic, complemented
6#67814C211yesmaximal, characteristic, complemented
7#67813C211yesmaximal, characteristic, complemented
8#12981—31nomaximal, Sylow
9C23×C3×C8C2211yescharacteristic
10#5791C411yescharacteristic
11C2×#901C2211yescharacteristic
12C2×C3×M32C2211yescharacteristic
13C2×C3×M32C2211yescharacteristic
14#5207C411yescharacteristic
15#5111—22no—
16#5110C411yescharacteristic
17C8×#67—22no—
18#5102C411yescharacteristic
19#5101C2211yescharacteristic
20C2×#812C2211yescharacteristic
21C2×#812C2211yescharacteristic
22#4907—22no—
23D3×M32—22no—
24#4901—22no—
25C3×C22⋊C16—22no—
26#4549—22no—
27#4514—22no—
28#3407—31no—
29#2929—31no—
30#2115—31no—
31#2112—31no—
32#2112D311yescharacteristic, complemented
33#1378—31no—
34#1378—31no—
35C23×C3×C4C2×C411yescharacteristic
36C2×#235C2×C411yescharacteristic
37C22×C3×C8—21no—
38C22×C3×C8—21no—
39C22×C3×C8—21no—
40C22×C3×C8—21no—
41C22×C3×C8—21no—
42C22×C3×C8—21no—
43C22×C3×C8C2×C411yescharacteristic
44C22×C3×C8C2311yescharacteristic
45#953C812yes—
46#930—21no—
47#929C812yes—
48C4×#67—22no—
49#926C2×C411yescharacteristic
50#924—21no—
51C2×#208C2×C411yescharacteristic
52C2×#208C2×C411yescharacteristic
53#901—21no—
54#901—21no—
55#901—21no—
56#901—21no—
57C2×ΓL(1,𝔽25)D412yes—
58C2×D3×C8D412yes—
59C3×M32—21no—
60C3×M32—21no—
61C3×M32—21no—
62C3×M32—21no—
63C2×C3×C16—21no—
64C2×C3×C16D412yescomplemented
65#830—22no—
66(C2×D3)⋊C8—22no—
67#815—21no—
68Dic3⋊C8—22no—
69C8×Dic3—21no—
70#812—21no—
71#812—21no—
72#812—21no—
73#812—21no—
74C2×C3⋊C16—21no—
75C2×C3⋊C16D412yescomplemented
76#798—24no—
77D3×C16—24no—
78C2×#131—31no—
79C23×C8C2×D311yescharacteristic
80#525—31no—
81#504—62no—
82#504—62no—
83C2×M32—31no—
84C2×M32—31no—
85C2×M32—62no—
86C2×M32C2×D311yescharacteristic
87C2×M32C2×D311yescharacteristic
88#435—62no—
89C8×D4—62no—
90#424—31no—
91#408—31no—
92#407—31no—
93#405—31no—
94#349—62no—
95C22⋊C16—22no—
96C22⋊C16—62no—
97C24×C3C2×C811yescharacteristic
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4C2×C811yescharacteristic
105C22×C3×C4C22×C411yescharacteristic
106C2×#67—22no—
107C2×#67C2×C812yes—
108#235—21no—
109#235—21no—
110#235—21no—
111#235—21no—
112C2×D12M1611yescharacteristic
113C2×C4×D3C22⋊C412yes—
114C2×Dic6M1611yescharacteristic
115C2×C3×C8—21no—
116C2×C3×C8—21no—
117C2×C3×C8—21no—
118C2×C3×C8—41no—
119C2×C3×C8—41no—
120C2×C3×C8—41no—
121C2×C3×C8—41no—
122C2×C3×C8C2×D411yescharacteristic
123C2×C3×C8C2×D411yescharacteristic
124C2×C3×C8C2×D411yescharacteristic
125C2×C3×C8C2×D411yescharacteristic
126C2×C3×C8#4111yescharacteristic
127C2×C3×C8#4111yescharacteristic
128#217—22no—
129#217C2×C812yes—
130(C2×D3)⋊C4—22no—
131C4⋊Dic3—21no—
132Dic3⋊C4—22no—
133C4×Dic3—21no—
134#208—21no—
135#208—21no—
136#208—21no—
137#208—21no—
138C2×C3⋊C8—21no—
139C2×C3⋊C8C22⋊C412yes—
140ΓL(1,𝔽25)—22no—
141ΓL(1,𝔽25)—22no—
142D3×C8—22no—
143D3×C8—22no—
144C3×C16—22no—
145C3×C16—41no—
146C3⋊C16—22no—
147C3⋊C16—41no—
148C2×#41—31no—
149C23×C4C4×D311yescharacteristic
150#131—61no—
151#131—61no—
152#131—61no—
153#131—61no—
154C2×M16—31no—
155C2×M16—32no—
156C2×M16—31no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—32no—
164C22×C8C4×D311yescharacteristic
165C22×C8C22×D311yescharacteristic
166#123—32no—
167C22⋊Q8—61no—
168#121—61no—
169#120—32no—
170C4×D4—62no—
171#117—31no—
172M32—21no—
173M32—21no—
174M32—21no—
175M32—21no—
176M32—64no—
177M32—61no—
178M32—61no—
179M32—61no—
180M32—61no—
181C2×C16—21no—
182C2×C16—32no—
183C2×C16—64no—
184C2×C16—61no—
185C2×C16#6712yescomplemented
186C4⋊C8—62no—
187C22⋊C8—62no—
188C22⋊C8—62no—
189#97—61no—
190C4×C8—61no—
191C23×C3—21no—
192C23×C3—21no—
193C23×C3—21no—
194C23×C3—42no—
195C23×C3C22×C811yescharacteristic
196C22×D3C22⋊C812yes—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4C2×C22⋊C411yescharacteristic
205C2×C3×C4C4×D411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4C2×M1611yescharacteristic
208C2×C3×C4#13111yescharacteristic
209C2×C3×C4#13111yescharacteristic
210#67—22no—
211#67—22no—
212#67—44no—
213C2×Dic3—22no—
214C2×Dic3C22⋊C812yes—
215D12—21no—
216D12—21no—
217C4×D3—22no—
218C4×D3—22no—
219Dic6—21no—
220Dic6—21no—
221C3×C8—41no—
222C3×C8#12011yescharacteristic
223C3×C8#12111yescharacteristic
224C3×C8#12111yescharacteristic
225C3×C8#12311yescharacteristic
226C3⋊C8—22no—
227C3⋊C8—41no—
228C24D3×C811yescharacteristic
229#41—61no—
230#41—61no—
231#41—61no—
232#41—61no—
233C2×Q8—31no—
234C2×D4—32no—
235C2×D4—31no—
236C2×D4—62no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—32no—
244C22×C4D3×C811yescharacteristic
245C22×C4C2×C4×D311yescharacteristic
246M16—61no—
247M16—61no—
248M16—62no—
249M16—62no—
250M16—61no—
251M16—61no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—32no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—62no—
261C2×C8—62no—
262C2×C8—61no—
263C2×C8#23511yescharacteristic
264C2×C8D3×D411yescharacteristic
265C2×C8D3×D411yescharacteristic
266C2×C8D3×D411yescharacteristic
267C2×C8#23711yescharacteristic
268C2×C8C2×#6711yescharacteristic
269C4⋊C4—62no—
270C4⋊C4—61no—
271C22⋊C4—32no—
272C22⋊C4—62no—
273C22⋊C4—62no—
274C42—61no—
275C16—22no—
276C16—41no—
277C16—62no—
278C16—121no—
279C22×C3—22no—
280C22×C3—22no—
281C22×C3—22no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3C2×C22⋊C811yescharacteristic
288C22×C3C8×D411yescharacteristic
289C22×C3C8×D411yescharacteristic
290C2×D3—22no—
291C2×D3—42no—
292C3×C4—41no—
293C3×C4#38611yescharacteristic
294C3×C4#40811yescharacteristic
295C3×C4#43511yescharacteristic
296C3×C4#43511yescharacteristic
297Dic3—22no—
298Dic3—42no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—32no—
303C23—42no—
304C23C2×D3×C811yescharacteristic
305Q8—61no—
306Q8—61no—
307D4—62no—
308D4—62no—
309D4—61no—
310D4—61no—
311D4—124no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—32no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—62no—
321C2×C4—62no—
322C2×C4—62no—
323C2×C4D3×C22⋊C411yescharacteristic
324C2×C4#67⋊C411yescharacteristic
325C2×C4#90111yescharacteristic
326C2×C4D3×M1611yescharacteristic
327C2×C4#90711yescharacteristic
328C2×C4C4×#6711yescharacteristic
329C8—41no—
330C8—62no—
331C8—121no—
332C8#88311yescharacteristic
333C8#88411yescharacteristic
334C8#93711yescharacteristic
335C8#93811yescharacteristic
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—42no—
340C2×C3#186911yescharacteristic
341D3—42no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—62no—
351C22—122no—
352C22D3×C22⋊C811yescharacteristic
353C22#67⋊C811yescharacteristic
354C22C8×#6711yescharacteristic
355C4—41no—
356C4—62no—
357C4—122no—
358C4#472711yescharacteristic
359C4#472811yescharacteristic
360C4#495611yescharacteristic
361C4#512711yescharacteristic
362C3#1298111yescharacteristic, complemented, Sylow
363C2—21no—
364C2—21no—
365C2—21no—
366C2—42no—
367C2—122no—
368C2#6995711yescharacteristic
subgroupG/H
centerC8#937
commutatorC23×C3C22×C8
FrattiniC22×C8C22×D3
FittingC3×#2112C2
socleC2×C3#1869
prime radical#5791C4
cosocleC22×C3×C8C23
hypercenter#2112D3
hypocenterC3#12981
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
2. #82007C2semidirect
3. C3×#2112C2semidirect
4. #71963C2semidirect
5. #71435C2semidirect
6. #67814C2semidirect
7. #67813C2semidirect
32. #2112D3semidirect
64. C2×C3×C16D4semidirect
75. C2×C3⋊C16D4semidirect
185. C2×C16#67semidirect
362. C3#12981semidirect

Maximal subgroups

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

subgroupindex
#837342
#820072
C3×#21122
#719632
#714352
#678142
#678132
#129813