Finite Group Databasebeta

← #10708476

Subgroups — #10708476

Order 768.

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

normal subgroups only

This group has 316 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#79030C211yesmaximal, characteristic
2C3×#2087C211yesmaximal, characteristic, complemented
3#70603C211yesmaximal, characteristic, complemented
4#70600C211yesmaximal, characteristic
5#67709C211yesmaximal, characteristic, complemented
6#67486C211yesmaximal, characteristic, complemented
7#67485C211yesmaximal, characteristic
8#12293—31nomaximal, Sylow
9#5479—21no—
10C2×C3×#107C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C3×#389—21no—
13#5116C2211yescharacteristic
14#5115C2211yescharacteristic
15#5114—21no—
16#4847—21no—
17#4845—21no—
18#4844—21no—
19#4829—21no—
20C3×#340—21no—
21C3×#340C2211yescharacteristic
22#4549C2211yescharacteristic
23#4549C2211yescharacteristic
24#4492—21no—
25#4491—21no—
26#4490—21no—
27#4489—21no—
28#3250—31no—
29#2236—31no—
30#2233—31no—
31#2087D311yescharacteristic, complemented
32#1348—31no—
33#1347—31no—
34#1322—31no—
35C22×C3×C8D411yescharacteristic
36C22×C3×C8C2311yescharacteristic
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—41no—
40C2×C3×C42D411yescharacteristic
41#930—21no—
42#929—41no—
43C4×#67—41no—
44#924—21no—
45C2×Dic12D411yescharacteristic
46C2×D24D411yescharacteristic
47C2×#204—21no—
48#876—41no—
49#875—21no—
50C4×D12—21no—
51#872—41no—
52#870—21no—
53C4×Dic6—21no—
54C3×#107—21no—
55C3×#107—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊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#529—61no—
69#479—61no—
70#478—61no—
71#467—31no—
72#466—31no—
73#465—61no—
74#464—61no—
75#463—61no—
76C2×#107C2×D311yescharacteristic
77C2×C4⋊C8C2×D311yescharacteristic
78#389—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
87C22×C3×C4—41no—
88C22×C3×C4—41no—
89C22×C3×C4D811yescharacteristic
90C22×C3×C4D811yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C2×#67—41no—
93C2×D12—21no—
94C2×C4×D3—41no—
95C2×Dic6—21no—
96C2×C3×C8—21no—
97C2×C3×C8—41no—
98C2×C3×C8—41no—
99C2×C3×C8C2×D411yescharacteristic
100C2×C3×C8C2×D411yescharacteristic
101C3×C4⋊C4—22no—
102C3×C4⋊C4—41no—
103C3×C4⋊C4—41no—
104C3×C4⋊C4—41no—
105C3×C4⋊C4—82no—
106C3×C42—21no—
107C3×C42—21no—
108#217—41no—
109(C2×D3)⋊C4—41no—
110(C2×D3)⋊C4—41no—
111(C2×D3)⋊C4—41no—
112(C2×D3)⋊C4—41no—
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115Dic3⋊C4—41no—
116Dic3⋊C4—41no—
117Dic3⋊C4—41no—
118Dic3⋊C4—41no—
119C4×Dic3—41no—
120Dic12—41no—
121D24—41no—
122#204—81no—
123C3⋊C16—41no—
124C3⋊C16—41no—
125C2×Q16—31no—
126C2×SD16—61no—
127C2×D8—31no—
128C22×C8#6711yescharacteristic
129C22×C8C22×D311yescharacteristic
130#126—121no—
131#125—61no—
132#124—61no—
133#123—121no—
134C22⋊Q8—61no—
135#121—61no—
136C4×Q8—61no—
137C4×D4—61no—
138C4×D4—121no—
139#117—121no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—41no—
143C2×C42#6711yescharacteristic
144C2×C16—61no—
145C2×C16—61no—
146#107—21no—
147#107—21no—
148#107—31no—
149#107—31no—
150#106—61no—
151C4⋊C8—21no—
152C4⋊C8—21no—
153#103—61no—
154#103—61no—
155#102—61no—
156#102—61no—
157#95—41no—
158C23×C3C2×D811yescharacteristic
159C22×D3—41no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—41no—
167C2×C3×C4—42no—
168C2×C3×C4—81no—
169C2×C3×C4—81no—
170C2×C3×C4—81no—
171C2×C3×C4—81no—
172C2×C3×C4—81no—
173C2×C3×C4#12011yescharacteristic
174C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
175#67—81no—
176#67—81no—
177C2×Dic3—41no—
178C2×Dic3—41no—
179C2×Dic3—41no—
180D12—41no—
181D12—41no—
182C4×D3—82no—
183Dic6—41no—
184Dic6—41no—
185C3×C8—21no—
186C3×C8—21no—
187C3×C8—41no—
188C2×Q8—61no—
189C2×D4—61no—
190C2×D4—121no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—121no—
194C22×C4#21311yescharacteristic
195C22×C4#21311yescharacteristic
196C22×C4C2×#6711yescharacteristic
197Q16—121no—
198SD16—241no—
199D8—121no—
200C2×C8—21no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8D3×D411yescharacteristic
204C2×C8D3×D411yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—61no—
210C4⋊C4—61no—
211C4⋊C4—82no—
212C4⋊C4—121no—
213C4⋊C4—121no—
214C4⋊C4—121no—
215C4⋊C4—121no—
216C22⋊C4—121no—
217C22⋊C4—121no—
218C22⋊C4—121no—
219C22⋊C4—121no—
220C22⋊C4—121no—
221C42—21no—
222C42—21no—
223C42—121no—
224C16—121no—
225C16—121no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3#44711yescharacteristic
229C22×C3#50811yescharacteristic
230C22×C3#51011yescharacteristic
231C2×D3—81no—
232C2×D3—81no—
233C2×D3—81no—
234C3×C4—21no—
235C3×C4—41no—
236C3×C4—41no—
237C3×C4—82no—
238C3×C4—161no—
239C3×C4#45611yescharacteristic
240C3×C4#45811yescharacteristic
241Dic3—81no—
242Dic3—81no—
243Dic3—81no—
244C23—121no—
245C23C2×#21311yescharacteristic
246Q8—121no—
247Q8—121no—
248D4—121no—
249D4—121no—
250D4—241no—
251D4—241no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—121no—
266C2×C4—121no—
267C2×C4—121no—
268C2×C4—242no—
269C2×C4#93711yescharacteristic
270C2×C4#94911yescharacteristic
271C8—21no—
272C8—21no—
273C8—41no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3#161011yescharacteristic
277C2×C3#218911yescharacteristic
278C2×C3#219011yescharacteristic
279D3—161no—
280C22—21no—
281C22—21no—
282C22—241no—
283C22—241no—
284C22—241no—
285C22#503611yescharacteristic
286C22#517911yescharacteristic
287C22#519411yescharacteristic
288C4—21no—
289C4—41no—
290C4—41no—
291C4—82no—
292C4—161no—
293C4—241no—
294C4—241no—
295C4—241no—
296C4#505311yescharacteristic
297C4#509511yescharacteristic
298C3#1229311yescharacteristic, complemented, Sylow
299C2—21no—
300C2—21no—
301C2—481no—
302C2#6971211yescharacteristic
303C2#7108611yescharacteristic
304C2#7108911yescharacteristic
subgroupG/H
centerC22#5036
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC3×#2087C2
socleC22×C3#447
prime radical#5115C22
hypercenter#2087D3
hypocenterC3#12293
LCSG ▷ C22×C3×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2087

Complements

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

NH ≅ G/Ntype
2. C3×#2087C2semidirect
3. #70603C2semidirect
5. #67709C2semidirect
6. #67486C2semidirect
31. #2087D3semidirect
298. C3#12293semidirect

Maximal subgroups

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

subgroupindex
#790302
C3×#20872
#706032
#706002
#677092
#674862
#674852
#122933