Finite Group Databasebeta

← #10708397

Subgroups — #10708397

Order 768.

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

normal subgroups only

This group has 348 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#78614C211yesmaximal, characteristic
2C3×#2075C211yesmaximal, characteristic, complemented
3#70602C211yesmaximal, characteristic, complemented
4#70599C211yesmaximal, characteristic, complemented
5#67673C211yesmaximal, characteristic, complemented
6#67486C211yesmaximal, characteristic, complemented
7#67485C211yesmaximal, characteristic, complemented
8#12280—31nomaximal, Sylow
9#5791—21no—
10C2×C3×#107C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C3×#380—21no—
13#5116C2211yescharacteristic
14#5115C2211yescharacteristic
15#5114—21no—
16#4737—21no—
17#4734—21no—
18#4733—21no—
19#4723—21no—
20C3×#340—21no—
21C3×#340C2211yescharacteristic
22#4549C2211yescharacteristic
23#4549C2211yescharacteristic
24#4492—21no—
25#4491—21no—
26#4490—21no—
27#4489—21no—
28#3191—31no—
29#2235—31no—
30#2232—31no—
31#2075D311yescharacteristic, complemented
32#1348—31no—
33#1347—31no—
34#1314—31no—
35C23×C3×C4D411yescharacteristic
36C2×#235—21no—
37C22×C3×C8D411yescharacteristic
38C22×C3×C8C2311yescharacteristic
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C22⋊C4—41no—
42#953—41no—
43#930—21no—
44#930—21no—
45#929—41no—
46C4×#67—41no—
47C4×#67—41no—
48#926—21no—
49#924—21no—
50#924—21no—
51C2×Dic12D411yescharacteristic
52C2×D24D411yescharacteristic
53C2×#204—21no—
54C3×#107—21no—
55C3×#107—21no—
56C3×C22⋊C8—21no—
57C3×C22⋊C8—21no—
58C3×#95—41no—
59#821—21no—
60#821—21no—
61#818D411yescharacteristic
62#818D411yescharacteristic
63#817—21no—
64#816—21no—
65#816—21no—
66C2×C3⋊C16—21no—
67C2×C3⋊C16—21no—
68#525—61no—
69#485—61no—
70#482—61no—
71#467—31no—
72#466—31no—
73#465—61no—
74#452—61no—
75#449—61no—
76C2×#107C2×D311yescharacteristic
77C2×C22⋊C8C2×D311yescharacteristic
78#380—21no—
79#367—61no—
80#366—61no—
81#358—61no—
82#357—61no—
83C22⋊C16—31no—
84C22⋊C16—31no—
85#340—21no—
86#340C2×D311yescharacteristic
87C24×C3D811yescharacteristic
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—41no—
91C22×C3×C4—41no—
92C22×C3×C4—41no—
93C22×C3×C4—41no—
94C22×C3×C4D811yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C2×#67—41no—
97C2×#67—41no—
98C2×#67—41no—
99#235—81no—
100#235—81no—
101C2×D12—21no—
102C2×C4×D3—41no—
103C2×Dic6—21no—
104C2×C3×C8—21no—
105C2×C3×C8—41no—
106C2×C3×C8—41no—
107C2×C3×C8C2×D411yescharacteristic
108C2×C3×C8C2×D411yescharacteristic
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C22⋊C4—82no—
114#217—41no—
115#217—41no—
116#217—41no—
117(C2×D3)⋊C4—41no—
118(C2×D3)⋊C4—41no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121Dic3⋊C4—41no—
122Dic3⋊C4—41no—
123C4×Dic3—41no—
124Dic12—41no—
125D24—41no—
126#204—81no—
127C3⋊C16—41no—
128C3⋊C16—41no—
129C2×#41—61no—
130C23×C4#6711yescharacteristic
131C2×Q16—31no—
132C2×SD16—61no—
133C2×D8—31no—
134C22×C8#6711yescharacteristic
135C22×C8C22×D311yescharacteristic
136#123—121no—
137C22⋊Q8—61no—
138C22⋊Q8—61no—
139#121—61no—
140#121—61no—
141#120—121no—
142C4×D4—121no—
143C4×D4—121no—
144#117—61no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C22⋊C4—41no—
148C2×C16—61no—
149C2×C16—61no—
150#107—21no—
151#107—21no—
152#107—31no—
153#107—31no—
154#106—61no—
155#103—61no—
156#103—61no—
157#102—61no—
158#102—61no—
159C22⋊C8—21no—
160C22⋊C8—21no—
161#95—41no—
162C23×C3—21no—
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—41no—
166C23×C3—41no—
167C23×C3C2×D811yescharacteristic
168C22×D3—41no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—81no—
176C2×C3×C4—81no—
177C2×C3×C4—81no—
178C2×C3×C4—81no—
179C2×C3×C4—81no—
180C2×C3×C4—81no—
181C2×C3×C4—81no—
182C2×C3×C4#12011yescharacteristic
183C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
184#67—82no—
185#67—82no—
186#67—81no—
187#67—81no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191D12—41no—
192D12—41no—
193C4×D3—82no—
194Dic6—41no—
195Dic6—41no—
196C3×C8—21no—
197C3×C8—21no—
198C3×C8—41no—
199C24#21311yescharacteristic
200#41—241no—
201#41—241no—
202C2×Q8—61no—
203C2×D4—61no—
204C2×D4—121no—
205C2×D4—121no—
206C2×D4—121no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—121no—
214C22×C4#21311yescharacteristic
215C22×C4C2×#6711yescharacteristic
216Q16—121no—
217SD16—241no—
218D8—121no—
219C2×C8—21no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8D3×D411yescharacteristic
223C2×C8D3×D411yescharacteristic
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—121no—
231C4⋊C4—121no—
232C22⋊C4—82no—
233C22⋊C4—121no—
234C22⋊C4—121no—
235C22⋊C4—121no—
236C22⋊C4—121no—
237C22⋊C4—121no—
238C42—121no—
239C16—121no—
240C16—121no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—42no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3—41no—
247C22×C3—81no—
248C22×C3#44711yescharacteristic
249C22×C3#50811yescharacteristic
250C22×C3#50911yescharacteristic
251C2×D3—81no—
252C2×D3—81no—
253C2×D3—81no—
254C3×C4—21no—
255C3×C4—41no—
256C3×C4—161no—
257C3×C4—161no—
258C3×C4#45511yescharacteristic
259C3×C4#45811yescharacteristic
260Dic3—81no—
261Dic3—81no—
262Dic3—81no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23—121no—
269C23C2×#21311yescharacteristic
270Q8—121no—
271Q8—121no—
272D4—121no—
273D4—121no—
274D4—242no—
275D4—242no—
276D4—241no—
277D4—241no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—242no—
295C2×C4#93711yescharacteristic
296C2×C4#94911yescharacteristic
297C8—21no—
298C8—21no—
299C8—41no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—81no—
303C2×C3#160711yescharacteristic
304C2×C3#218611yescharacteristic
305C2×C3#219111yescharacteristic
306D3—161no—
307C22—21no—
308C22—21no—
309C22—42no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—81no—
314C22—241no—
315C22—241no—
316C22—241no—
317C22#503611yescharacteristic
318C22#514711yescharacteristic
319C22#519411yescharacteristic
320C4—21no—
321C4—41no—
322C4—161no—
323C4—161no—
324C4—241no—
325C4—241no—
326C4—241no—
327C4#505311yescharacteristic
328C4#506111yescharacteristic
329C3#1228011yescharacteristic, complemented, Sylow
330C2—21no—
331C2—21no—
332C2—81no—
333C2—481no—
334C2#6970811yescharacteristic
335C2#7107611yescharacteristic
336C2#7109111yescharacteristic
subgroupG/H
centerC22#5036
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC3×#2075C2
socleC22×C3#447
prime radical#78614C2
hypercenter#2075D3
hypocenterC3#12280
LCSG ▷ C22×C3×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #2075

Complements

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

NH ≅ G/Ntype
2. C3×#2075C2semidirect
3. #70602C2semidirect
4. #70599C2semidirect
5. #67673C2semidirect
6. #67486C2semidirect
7. #67485C2semidirect
31. #2075D3semidirect
329. C3#12280semidirect

Maximal subgroups

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

subgroupindex
#786142
C3×#20752
#706022
#705992
#676732
#674862
#674852
#122803