Finite Group Databasebeta

← #10712107

Subgroups — #10712107

Order 768.

466 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83787C211yesmaximal, characteristic
2#82007C211yesmaximal, characteristic, complemented
3C3×#2117C211yesmaximal, characteristic, complemented
4#72006C211yesmaximal, characteristic, complemented
5#71435C211yesmaximal, characteristic, complemented
6#67811C211yesmaximal, characteristic, complemented
7#67811C211yesmaximal, characteristic, complemented
8#12988—31nomaximal, Sylow
9C2×C3×#131C2211yescharacteristic
10#5832C411yescharacteristic
11C2×#901C2211yescharacteristic
12C2×C3×M32C2211yescharacteristic
13C2×C3×M32C2211yescharacteristic
14#5234C411yescharacteristic
15#5129C411yescharacteristic
16#5127—22no—
17#5118C411yescharacteristic
18#5111—22no—
19C2×#815C2211yescharacteristic
20C2×#812C2211yescharacteristic
21C2×#812C2211yescharacteristic
22#4907—22no—
23D3×M32—22no—
24#4901—22no—
25C3×#350—22no—
26#4561—22no—
27#4514—22no—
28#3407—31no—
29#2932—31no—
30#2117—31no—
31#2117D311yescharacteristic, complemented
32#2115—31no—
33#1375—31no—
34#1375—31no—
35C2×C3×#41C2×C411yescharacteristic
36C2×#235C2×C411yescharacteristic
37C3×#131—21no—
38C3×#131—21no—
39C3×#131—21no—
40C3×#131—21no—
41C2×C3×M16—21no—
42C2×C3×M16C2×C411yescharacteristic
43C22×C3×C8—21no—
44C22×C3×C8C2311yescharacteristic
45#947C811yescharacteristic
46#944C811yescharacteristic
47#940C811yescharacteristic
48#939—22no—
49#936C811yescharacteristic
50C4×#67—22no—
51C2×C4×Dic3C2×C411yescharacteristic
52C2×#208C2×C411yescharacteristic
53C22×C3⋊C8C2×C411yescharacteristic
54#901—21no—
55#901—21no—
56#901—21no—
57#901—21no—
58C2×ΓL(1,𝔽25)D412yes—
59C2×D3×C8D412yes—
60C3×M32—21no—
61C3×M32—21no—
62C3×M32—21no—
63C3×M32—21no—
64C2×C3×C16—21no—
65C2×C3×C16D412yescomplemented
66#830—22no—
67(C2×D3)⋊C8—22no—
68#815—21no—
69#815—21no—
70Dic3⋊C8—22no—
71#812—21no—
72#812—21no—
73#812—21no—
74#812—21no—
75C2×C3⋊C16—21no—
76C2×C3⋊C16D412yescomplemented
77#798—24no—
78D3×C16—24no—
79C2×#131—31no—
80C2×#131C2×D311yescharacteristic
81#532—31no—
82#504—62no—
83#504—62no—
84C2×M32—31no—
85C2×M32—31no—
86C2×M32—62no—
87C2×M32C2×D311yescharacteristic
88C2×M32C2×D311yescharacteristic
89#435—62no—
90#435—62no—
91#423—31no—
92#408—31no—
93#408—31no—
94C2×#97—31no—
95#350—22no—
96#350—62no—
97#349—62no—
98C3×#41—21no—
99C3×#41—21no—
100C3×#41—21no—
101C3×#41—21no—
102C2×C3×Q8C2×C811yescharacteristic
103C2×C3×D4—21no—
104C2×C3×D4C2×C811yescharacteristic
105C22×C3×C4—21no—
106C22×C3×C4C22×C411yescharacteristic
107C2×#67—22no—
108C2×#67M1612yes—
109C22×Dic3—21no—
110#235—21no—
111#235—21no—
112#235—21no—
113#235—21no—
114C2×D12C2×C811yescharacteristic
115C2×C4×D3C22⋊C412yes—
116C2×Dic6C2×C811yescharacteristic
117C3×M16—21no—
118C3×M16—21no—
119C3×M16—41no—
120C3×M16—41no—
121C2×C3×C8—21no—
122C2×C3×C8—41no—
123C2×C3×C8—41no—
124C2×C3×C8C2×D411yescharacteristic
125C2×C3×C8C2×D411yescharacteristic
126C2×C3×C8C2×D411yescharacteristic
127C2×C3×C8C2×D411yescharacteristic
128C2×C3×C8#4111yescharacteristic
129C2×C3×C8#4111yescharacteristic
130#217—22no—
131(C2×D3)⋊C4—22no—
132Dic3⋊C4—22no—
133C4×Dic3—21no—
134C4×Dic3C2×C811yescharacteristic
135C4×Dic3C2×C811yescharacteristic
136#208—21no—
137#208—21no—
138C2×C3⋊C8—21no—
139C2×C3⋊C8—21no—
140C2×C3⋊C8—21no—
141C2×C3⋊C8C22⋊C412yes—
142ΓL(1,𝔽25)—22no—
143ΓL(1,𝔽25)—22no—
144D3×C8—22no—
145D3×C8—22no—
146C3×C16—22no—
147C3×C16—41no—
148C3⋊C16—22no—
149C3⋊C16—41no—
150C2×#41—31no—
151C2×#41C4×D311yescharacteristic
152#131—21no—
153#131—21no—
154#131—21no—
155#131—21no—
156#131—61no—
157#131—61no—
158#131—61no—
159#131—61no—
160C2×M16—21no—
161C2×M16—31no—
162C2×M16—32no—
163C2×M16C4×D311yescharacteristic
164C22×C8—21no—
165C22×C8—32no—
166C22×C8—31no—
167C22×C8C22×D311yescharacteristic
168C4⋊Q8—31no—
169#127—31no—
170#124—31no—
171#124—31no—
172#121—62no—
173C4×D4—62no—
174C2×C42—31no—
175M32—21no—
176M32—21no—
177M32—21no—
178M32—21no—
179M32—64no—
180M32—61no—
181M32—61no—
182M32—61no—
183M32—61no—
184C2×C16—21no—
185C2×C16—32no—
186C2×C16—64no—
187C2×C16—61no—
188C2×C16#6712yescomplemented
189C4⋊C8—62no—
190C22⋊C8—62no—
191C22⋊C8—62no—
192#97—61no—
193#97—61no—
194C23×C3—21no—
195C23×C3C2×M1611yescharacteristic
196C22×D3C22⋊C812yes—
197C3×Q8—21no—
198C3×Q8—21no—
199C3×D4—21no—
200C3×D4—21no—
201C3×D4—41no—
202C3×D4—41no—
203C2×C3×C4—21no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4C2×C22⋊C411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4C22×C811yescharacteristic
210C2×C3×C4#13111yescharacteristic
211C2×C3×C4#13111yescharacteristic
212#67—22no—
213#67—22no—
214#67—82no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—81no—
218C2×Dic3C22⋊C812yes—
219D12—21no—
220D12—21no—
221C4×D3—22no—
222C4×D3—22no—
223Dic6—21no—
224Dic6—21no—
225C3×C8—41no—
226C3×C8#12011yescharacteristic
227C3×C8#12111yescharacteristic
228C3×C8#12111yescharacteristic
229C3×C8#12311yescharacteristic
230C3⋊C8—22no—
231C3⋊C8—41no—
232#41—21no—
233#41—21no—
234#41—21no—
235#41—21no—
236#41—61no—
237#41—61no—
238#41—61no—
239#41—61no—
240C2×Q8—31no—
241C2×Q8D3×C811yescharacteristic
242C2×D4—21no—
243C2×D4—32no—
244C2×D4—31no—
245C2×D4—62no—
246C2×D4D3×C811yescharacteristic
247C22×C4—21no—
248C22×C4—32no—
249C22×C4—61no—
250C22×C4C2×C4×D311yescharacteristic
251M16—21no—
252M16—21no—
253M16—41no—
254M16—41no—
255M16—61no—
256M16—61no—
257M16—62no—
258M16—62no—
259C2×C8—21no—
260C2×C8—32no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—62no—
264C2×C8—62no—
265C2×C8—61no—
266C2×C8—61no—
267C2×C8—61no—
268C2×C8#23511yescharacteristic
269C2×C8D3×D411yescharacteristic
270C2×C8D3×D411yescharacteristic
271C2×C8D3×D411yescharacteristic
272C2×C8#23711yescharacteristic
273C2×C8C2×#6711yescharacteristic
274C4⋊C4—62no—
275C22⋊C4—62no—
276C22⋊C4—62no—
277C42—31no—
278C42—31no—
279C42—61no—
280C16—22no—
281C16—41no—
282C16—62no—
283C16—121no—
284C22×C3—41no—
285C22×C3—42no—
286C22×C3—41no—
287C22×C3C2×C22⋊C811yescharacteristic
288C22×C3#43511yescharacteristic
289C22×C3#43511yescharacteristic
290C2×D3—22no—
291C2×D3—42no—
292C3×C4—41no—
293C3×C4#38611yescharacteristic
294C3×C4#40811yescharacteristic
295C3×C4C8×D411yescharacteristic
296C3×C4C8×D411yescharacteristic
297Dic3—22no—
298Dic3—41no—
299Dic3—41no—
300C23—21no—
301C23—32no—
302C23D3×M1611yescharacteristic
303Q8—21no—
304Q8—21no—
305Q8—61no—
306Q8—61no—
307D4—21no—
308D4—21no—
309D4—41no—
310D4—41no—
311D4—62no—
312D4—62no—
313D4—61no—
314D4—61no—
315D4—242no—
316C2×C4—21no—
317C2×C4—32no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—62no—
321C2×C4—62no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—241no—
325C2×C4D3×C22⋊C411yescharacteristic
326C2×C4#67⋊C411yescharacteristic
327C2×C4C2×D3×C811yescharacteristic
328C2×C4#90111yescharacteristic
329C2×C4#90711yescharacteristic
330C2×C4C4×#6711yescharacteristic
331C8—41no—
332C8—62no—
333C8—121no—
334C8#88311yescharacteristic
335C8#88411yescharacteristic
336C8#93711yescharacteristic
337C8#93811yescharacteristic
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—81no—
342C2×C3#186911yescharacteristic
343D3—42no—
344C22—41no—
345C22—42no—
346C22—41no—
347C22—62no—
348C22—122no—
349C22D3×C22⋊C811yescharacteristic
350C22#472811yescharacteristic
351C22#512711yescharacteristic
352C4—41no—
353C4—62no—
354C4—121no—
355C4—121no—
356C4#67⋊C811yescharacteristic
357C4#472711yescharacteristic
358C4#495611yescharacteristic
359C4C8×#6711yescharacteristic
360C3#1298811yescharacteristic, complemented, Sylow
361C2—21no—
362C2—21no—
363C2—21no—
364C2—81no—
365C2—122no—
366C2#6995711yescharacteristic
subgroupG/H
centerC4#4956
commutatorC2×C3×C4C22×C8
FrattiniC22×C8C22×D3
FittingC3×#2117C2
socleC2×C3#1869
prime radical#5832C4
cosocleC22×C3×C8C23
hypercenter#2117D3
hypocenterC3#12988
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C8 ◁ #2117

Complements

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

NH ≅ G/Ntype
2. #82007C2semidirect
3. C3×#2117C2semidirect
4. #72006C2semidirect
5. #71435C2semidirect
6. #67811C2semidirect
7. #67811C2semidirect
31. #2117D3semidirect
65. C2×C3×C16D4semidirect
76. C2×C3⋊C16D4semidirect
188. C2×C16#67semidirect
360. C3#12988semidirect

Maximal subgroups

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

subgroupindex
#837872
#820072
C3×#21172
#720062
#714352
#678112
#129883