Finite Group Databasebeta

← #10705120

Subgroups — #10705120

Order 768.

356 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 89 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#79538C211yesmaximal, characteristic
2#69275C211yesmaximal, characteristic, complemented
3#69073C211yesmaximal, characteristic, complemented
4C3×#1330C211yesmaximal, characteristic, complemented
5#67761C211yesmaximal, characteristic, complemented
6#67557C211yesmaximal, characteristic, complemented
7#67515C211yesmaximal, characteristic, complemented
8#11635—31nomaximal, Sylow
9C2×#907—21no—
10#5548C411yescharacteristic
11C3×#436C2211yescharacteristic
12#5128—21no—
13D4×C3⋊C8C411yescharacteristic
14#4837C411yescharacteristic
15#4834C411yescharacteristic
16#4727—21no—
17#67⋊C8—21no—
18#4688C2211yescharacteristic
19C4×D3×C8C2211yescharacteristic
20C3×C4⋊C16C2211yescharacteristic
21C3×#346C2211yescharacteristic
22(C2×D3)⋊C16—21no—
23(C2×D3)⋊C16—21no—
24#4502—21no—
25Dic3⋊C16—21no—
26C4⋊(C3⋊C16)C2211yescharacteristic
27#4461C2211yescharacteristic
28#2931—31no—
29#2163—31no—
30#2163—31no—
31#1330—31no—
32#1330—31no—
33#1330—31no—
34#1330D311yescharacteristic, complemented
35C2×#235—21no—
36C2×C3×M16—21no—
37C3×C4×D4C2×C411yescharacteristic
38#940C811yescharacteristic
39#938C811yescharacteristic
40C4×#67—21no—
41C22×C3⋊C8—21no—
42#907—42no—
43#907—42no—
44C2×ΓL(1,𝔽25)—21no—
45C2×ΓL(1,𝔽25)—21no—
46C2×D3×C8—21no—
47C2×D3×C8D411yescharacteristic
48C2×D3×C8D411yescharacteristic
49#895C811yescharacteristic
50#888C811yescharacteristic
51#885—21no—
52#884—21no—
53C4×D12C2×C411yescharacteristic
54C42×D3C2×C411yescharacteristic
55C2×C3×C16—21no—
56C2×C3×C16—21no—
57C3×C4⋊C8C2×C411yescharacteristic
58C3×C22⋊C8—21no—
59C3×C4×C8C2311yescharacteristic
60#830—21no—
61(C2×D3)⋊C8—21no—
62(C2×D3)⋊C8—21no—
63Dic3⋊C8—21no—
64C8×Dic3D411yescharacteristic
65C8×Dic3D411yescharacteristic
66C2×C3⋊C16—21no—
67C2×C3⋊C16—21no—
68C4⋊(C3⋊C8)C2×C411yescharacteristic
69C4×C3⋊C8C2×C411yescharacteristic
70C2×#131—61no—
71#532—31no—
72#436—31no—
73#436—61no—
74#436C2×D311yescharacteristic
75C8×D4—31no—
76C8×D4—31no—
77C8×D4—61no—
78#423—31no—
79#408—61no—
80C2×C4×C8—31no—
81C4⋊C16—31no—
82C4⋊C16—61no—
83C4⋊C16C2×D311yescharacteristic
84C22⋊C16—61no—
85C22⋊C16—61no—
86#346—31no—
87#346—61no—
88#346C2×D311yescharacteristic
89C2×C3×D4C2×C811yescharacteristic
90C22×C3×C4—21no—
91C2×#67—21no—
92C2×#67—21no—
93C2×#67—21no—
94#235—42no—
95#235—42no—
96C2×D12—21no—
97C2×D12C2×C811yescharacteristic
98C2×C4×D3—21no—
99C2×C4×D3C22⋊C411yescharacteristic
100C2×C4×D3C2×C811yescharacteristic
101C2×C4×D3M1611yescharacteristic
102C2×Dic6—21no—
103C3×M16—41no—
104C3×M16—41no—
105C2×C3×C8—21no—
106C2×C3×C8C2×D411yescharacteristic
107C2×C3×C8C2×D411yescharacteristic
108C3×C4⋊C4C2×C811yescharacteristic
109C3×C22⋊C4—21no—
110C3×C42C22×C411yescharacteristic
111#217—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114C4⋊Dic3C2×C811yescharacteristic
115Dic3⋊C4—21no—
116C4×Dic3C22⋊C411yescharacteristic
117C4×Dic3C2×C811yescharacteristic
118C4×Dic3M1611yescharacteristic
119C2×C3⋊C8—21no—
120C2×C3⋊C8—42no—
121C2×C3⋊C8C22⋊C411yescharacteristic
122C2×C3⋊C8C22⋊C411yescharacteristic
123ΓL(1,𝔽25)—42no—
124ΓL(1,𝔽25)—41no—
125ΓL(1,𝔽25)—41no—
126D3×C8—21no—
127D3×C8—21no—
128D3×C8—41no—
129D3×C8—42no—
130C3×C16—41no—
131C3×C16—41no—
132C3⋊C16—41no—
133C3⋊C16—41no—
134C2×#41—61no—
135#131—122no—
136#131—122no—
137C2×M16—21no—
138C2×M16—61no—
139C2×M16—61no—
140C22×C8—31no—
141C22×C8—31no—
142C22×C8—61no—
143C22×C8—61no—
144C4⋊Q8—31no—
145#127—31no—
146#124—61no—
147#121—31no—
148#121—31no—
149#121—61no—
150C4×D4—31no—
151C4×D4—61no—
152C4×D4C4×D311yescharacteristic
153C2×C42—31no—
154C2×C16—21no—
155C2×C16—21no—
156C2×C16—61no—
157C2×C16—61no—
158C4⋊C8—31no—
159C4⋊C8—61no—
160C4⋊C8C4×D311yescharacteristic
161C22⋊C8—21no—
162C22⋊C8—61no—
163C22⋊C8—61no—
164C22⋊C8—61no—
165C4×C8—31no—
166C4×C8—31no—
167C4×C8—31no—
168C4×C8C22×D311yescharacteristic
169C23×C3—21no—
170C22×D3—21no—
171C22×D3C22⋊C811yescharacteristic
172C3×D4—41no—
173C2×C3×C4—21no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4C2×C22⋊C411yescharacteristic
177C2×C3×C4C22×C811yescharacteristic
178C2×C3×C4C2×M1611yescharacteristic
179#67—42no—
180#67—42no—
181#67—42no—
182C2×Dic3—21no—
183C2×Dic3C22⋊C811yescharacteristic
184C2×Dic3C22⋊C811yescharacteristic
185C2×Dic3C22⋊C811yescharacteristic
186D12—41no—
187D12—42no—
188C4×D3—21no—
189C4×D3—21no—
190C4×D3—41no—
191C4×D3—41no—
192C4×D3—41no—
193C4×D3—41no—
194Dic6—42no—
195C3×C8—21no—
196C3×C8—41no—
197C3×C8AGL(1,ℤ/8)11yescharacteristic
198C3×C8AGL(1,ℤ/8)11yescharacteristic
199C3⋊C8—21no—
200C3⋊C8—21no—
201C3⋊C8—41no—
202#41—122no—
203#41—122no—
204C2×Q8—61no—
205C2×D4—31no—
206C2×D4—61no—
207C2×D4—61no—
208C2×D4—61no—
209C2×D4—61no—
210C2×D4D3×C811yescharacteristic
211C22×C4—21no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—61no—
216M16—41no—
217M16—41no—
218M16—122no—
219M16—121no—
220M16—121no—
221C2×C8—21no—
222C2×C8—31no—
223C2×C8—31no—
224C2×C8—61no—
225C2×C8—61no—
226C2×C8—61no—
227C2×C8—121no—
228C2×C8—122no—
229C2×C8—122no—
230C2×C8D3×D411yescharacteristic
231C2×C8D3×D411yescharacteristic
232C4⋊C4—31no—
233C4⋊C4—61no—
234C4⋊C4D3×C811yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—61no—
237C22⋊C4—61no—
238C22⋊C4—61no—
239C42—31no—
240C42—31no—
241C42—31no—
242C42C2×C4×D311yescharacteristic
243C16—41no—
244C16—41no—
245C16—121no—
246C16—121no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3C2×C22⋊C811yescharacteristic
251C2×D3—21no—
252C2×D3—21no—
253C2×D3—21no—
254C2×D3—41no—
255C2×D3—41no—
256C2×D3—41no—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—41no—
260C3×C4#41811yescharacteristic
261C3×C4#42111yescharacteristic
262Dic3—21no—
263Dic3—21no—
264Dic3—21no—
265Dic3—41no—
266C23—21no—
267C23—31no—
268C23—61no—
269Q8—122no—
270D4—41no—
271D4—121no—
272D4—122no—
273D4—122no—
274D4—122no—
275D4—122no—
276C2×C4—21no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4D3×C22⋊C411yescharacteristic
290C2×C4C2×D3×C811yescharacteristic
291C2×C4D3×M1611yescharacteristic
292C8—21no—
293C8—41no—
294C8—61no—
295C8—61no—
296C8—121no—
297C8#91111yescharacteristic
298C8#91411yescharacteristic
299C2×C3—81no—
300C2×C3#147811yescharacteristic
301C2×C3#211511yescharacteristic
302C2×C3#211711yescharacteristic
303D3—41no—
304D3—81no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—61no—
309C22—61no—
310C22—61no—
311C22—121no—
312C22—121no—
313C22—121no—
314C22D3×C22⋊C811yescharacteristic
315C4—21no—
316C4—21no—
317C4—41no—
318C4—61no—
319C4—61no—
320C4—61no—
321C4—121no—
322C4#477011yescharacteristic
323C4#482111yescharacteristic
324C3#1163511yescharacteristic, complemented, Sylow
325C2—81no—
326C2—121no—
327C2—241no—
328C2#6816711yescharacteristic
329C2#6911911yescharacteristic
330C2#6912511yescharacteristic
subgroupG/H
centerC2×C4D3×C22⋊C4
commutatorC2×C3×C4C22×C8
FrattiniC4×C8C22×D3
FittingC3×#1330C2
socleC22×C3C2×C22⋊C8
prime radical#5548C4
cosocleC3×C4×C8C23
hypercenter#1330D3
hypocenterC3#11635
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1330

Complements

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

NH ≅ G/Ntype
2. #69275C2semidirect
3. #69073C2semidirect
4. C3×#1330C2semidirect
5. #67761C2semidirect
6. #67557C2semidirect
7. #67515C2semidirect
34. #1330D3semidirect
324. C3#11635semidirect

Maximal subgroups

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

subgroupindex
#795382
#692752
#690732
C3×#13302
#677612
#675572
#675152
#116353