Finite Group Databasebeta

← #10708451

Subgroups — #10708451

Order 768.

354 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 63 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#80702C211yesmaximal, characteristic
2#71068C211yesmaximal, characteristic, complemented
3#69095C211yesmaximal, characteristic, complemented
4#68838C211yesmaximal, characteristic, complemented
5C3×#1347C211yesmaximal, characteristic, complemented
6#67524C211yesmaximal, characteristic, complemented
7#67485C211yesmaximal, characteristic, complemented
8#12288—31nomaximal, Sylow
9#5596—21no—
10C3×#466C2211yescharacteristic
11#5150—21no—
12#5115C2211yescharacteristic
13C2×#818C2211yescharacteristic
14C2×Dic3⋊C8C2211yescharacteristic
15#5038—21no—
16#4984—21no—
17#4890—21no—
18#4786—21no—
19#4759—21no—
20C3×C22⋊C16C2211yescharacteristic
21C3×#340C2211yescharacteristic
22#4562—21no—
23#4549C2211yescharacteristic
24#4529—21no—
25#4509—21no—
26#4504—21no—
27#4489—21no—
28#3245—31no—
29#2232—31no—
30#2232—31no—
31#2086—31no—
32#1347—31no—
33#1347D311yescharacteristic, complemented
34#1322—31no—
35C2×#237—41no—
36C2×C3×D8D411yescharacteristic
37C22×C3×C8C2311yescharacteristic
38C3×#121—21no—
39C2×C3×C4⋊C4—21no—
40#940—21no—
41#939—41no—
42D4×Dic3—21no—
43C2×#213—21no—
44#930—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—41no—
47C2×C4×Dic3D411yescharacteristic
48C22×C3⋊C8D411yescharacteristic
49C2×D24D411yescharacteristic
50#893—21no—
51#888—21no—
52#885—41no—
53#67⋊C4—41no—
54C2×C3×C16—21no—
55C3×#107D411yescharacteristic
56C3×#102—21no—
57#832—21no—
58C3⋊#95—41no—
59#821—21no—
60#818—21no—
61#818—21no—
62#818D411yescharacteristic
63Dic3⋊C8—21no—
64Dic3⋊C8—21no—
65C2×C3⋊C16—21no—
66#809—21no—
67#807—21no—
68#532—61no—
69#474—61no—
70#474—61no—
71#466—31no—
72#466—61no—
73#466C2×D311yescharacteristic
74#459—61no—
75#459—61no—
76C2×#107—31no—
77C2×C4⋊C8—31no—
78#389—61no—
79#366—61no—
80#366—61no—
81#357—61no—
82#357—61no—
83C22⋊C16—31no—
84C22⋊C16C2×D311yescharacteristic
85#340—61no—
86#340C2×D311yescharacteristic
87C2×C3×D4—21no—
88C2×C3×D4—41no—
89C22×C3×C4—41no—
90C22×C3×C4C2×D411yescharacteristic
91C2×#67—41no—
92C2×#67—41no—
93C2×#67—41no—
94C22×Dic3—41no—
95C22×Dic3—41no—
96C22×Dic3D811yescharacteristic
97C22×Dic3D811yescharacteristic
98#237—82no—
99#237—82no—
100C2×D12—21no—
101C2×C4×D3—41no—
102C2×Dic6—41no—
103C3×D8—41no—
104C2×C3×C8—41no—
105C2×C3×C8C2×D411yescharacteristic
106C2×C3×C8C2×D411yescharacteristic
107C3×C4⋊C4—22no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—41no—
110C3×C22⋊C4—41no—
111#217—41no—
112#213—81no—
113(C2×D3)⋊C4—41no—
114(C2×D3)⋊C4—41no—
115C4⋊Dic3—21no—
116C4⋊Dic3—41no—
117C4⋊Dic3—41no—
118C4⋊Dic3—82no—
119Dic3⋊C4—41no—
120C4×Dic3—21no—
121C4×Dic3—21no—
122C2×C3⋊C8—21no—
123C2×C3⋊C8—41no—
124D24—41no—
125C3×C16—41no—
126C3⋊C16—41no—
127C2×#41—121no—
128C2×D8—31no—
129C2×D8—61no—
130C2×D8D1211yescharacteristic
131C22×C8—31no—
132C22×C8C22×D311yescharacteristic
133C4⋊Q8—61no—
134#127—61no—
135#124—121no—
136#121—21no—
137#121—61no—
138#121—121no—
139C4×D4—61no—
140C4×D4—61no—
141C4×D4—121no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—121no—
145C2×C42—31no—
146C2×C16—21no—
147C2×C16—61no—
148#107—31no—
149#107—61no—
150#107—61no—
151#107—61no—
152#107D1211yescharacteristic
153C4⋊C8—61no—
154C4⋊C8—61no—
155#102—21no—
156#102—61no—
157#102—61no—
158#102—61no—
159#95—121no—
160C23×C3—41no—
161C23×C3C2×D811yescharacteristic
162C22×D3—41no—
163C3×D4—41no—
164C3×D4—41no—
165C3×D4—81no—
166C3×D4—81no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—42no—
170C2×C3×C4—41no—
171C2×C3×C4—81no—
172C2×C3×C4—81no—
173C2×C3×C4#12011yescharacteristic
174C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
175#67—82no—
176#67—82no—
177#67—81no—
178#67—81no—
179C2×Dic3—21no—
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3—41no—
183C2×Dic3—41no—
184C2×Dic3—82no—
185C2×Dic3—81no—
186C2×Dic3—81no—
187C2×Dic3—81no—
188C2×Dic3D1612yes—
189D12—41no—
190D12—41no—
191C4×D3—82no—
192Dic6—82no—
193C3×C8—21no—
194C3×C8—21no—
195C3⋊C8—41no—
196#41—242no—
197#41—242no—
198C2×Q8—121no—
199C2×D4—21no—
200C2×D4—41no—
201C2×D4—61no—
202C2×D4—121no—
203C2×D4—121no—
204C2×D4—121no—
205C22×C4—31no—
206C22×C4—31no—
207C22×C4—41no—
208C22×C4—121no—
209C22×C4—121no—
210C22×C4—121no—
211C22×C4D3×D411yescharacteristic
212D8—41no—
213D8—121no—
214D8—241no—
215C2×C8—41no—
216C2×C8—61no—
217C2×C8—121no—
218C2×C8C2×D1211yescharacteristic
219C2×C8D3×D411yescharacteristic
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4—41no—
223C4⋊C4—61no—
224C4⋊C4—121no—
225C4⋊C4—121no—
226C4⋊C4—121no—
227C4⋊C4—242no—
228C22⋊C4—41no—
229C22⋊C4—121no—
230C22⋊C4—121no—
231C22⋊C4—121no—
232C42—61no—
233C42—61no—
234C16—41no—
235C16—121no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—81no—
239C22×C3—81no—
240C22×C3—81no—
241C22×C3#44711yescharacteristic
242C22×C3C2×D1611yescharacteristic
243C22×C3#50911yescharacteristic
244C2×D3—81no—
245C2×D3—81no—
246C2×D3—81no—
247C3×C4—21no—
248C3×C4—82no—
249C3×C4—81no—
250C3×C4#45611yescharacteristic
251C3×C4#45811yescharacteristic
252Dic3—22no—
253Dic3—41no—
254Dic3—81no—
255Dic3—161no—
256C23—41no—
257C23—121no—
258C23D3×D811yescharacteristic
259Q8—242no—
260D4—41no—
261D4—41no—
262D4—81no—
263D4—81no—
264D4—121no—
265D4—121no—
266D4—242no—
267D4—242no—
268D4—241no—
269D4—241no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—32no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—61no—
276C2×C4—61no—
277C2×C4—61no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—121no—
281C2×C4—121no—
282C2×C4—242no—
283C2×C4—242no—
284C2×C4—241no—
285C2×C4—241no—
286C2×C4—241no—
287C2×C4#88211yescharacteristic
288C2×C4#91411yescharacteristic
289C8—21no—
290C8—21no—
291C8—121no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—161no—
295C2×C3#161011yescharacteristic
296C2×C3#218411yescharacteristic
297C2×C3#219111yescharacteristic
298D3—161no—
299C22—21no—
300C22—21no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—241no—
305C22—241no—
306C22—241no—
307C22#477311yescharacteristic
308C22D3×D1611yescharacteristic
309C22#491411yescharacteristic
310C4—21no—
311C4—62no—
312C4—82no—
313C4—81no—
314C4—121no—
315C4—241no—
316C4—481no—
317C4#474211yescharacteristic
318C4#482611yescharacteristic
319C3#1228811yescharacteristic, complemented, Sylow
320C2—21no—
321C2—21no—
322C2—161no—
323C2—481no—
324C2#6829711yescharacteristic
325C2#6919111yescharacteristic
326C2#6925311yescharacteristic
subgroupG/H
centerC22#4773
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC3×#1347C2
socleC22×C3#447
prime radical#80702C2
hypercenter#1347D3
hypocenterC3#12288
LCSG ▷ C22×C3×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1347

Complements

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

NH ≅ G/Ntype
2. #71068C2semidirect
3. #69095C2semidirect
4. #68838C2semidirect
5. C3×#1347C2semidirect
6. #67524C2semidirect
7. #67485C2semidirect
33. #1347D3semidirect
319. C3#12288semidirect

Maximal subgroups

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

subgroupindex
#807022
#710682
#690952
#688382
C3×#13472
#675242
#674852
#122883