Finite Group Databasebeta

← #10699152

Subgroups — #10699152

Order 768.

348 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 73 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#83019C211yesmaximal, characteristic
2C3×#1998C211yesmaximal, characteristic, complemented
3#71735C211yesmaximal, characteristic, complemented
4#70504C211yesmaximal, characteristic
5#70492C211yesmaximal, characteristic, complemented
6#69859C211yesmaximal, characteristic
7#67670C211yesmaximal, characteristic, complemented
8#10497—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13#5067—21no—
14#5066—21no—
15#5066—21no—
16#5064—21no—
17#4968—21no—
18#4967—21no—
19#4926C2211yescharacteristic
20#4926C2211yescharacteristic
21C4×#208C2211yescharacteristic
22C3×#343—21no—
23C3×#343—21no—
24#4543—21no—
25#4543—21no—
26#4486—21no—
27#4485—21no—
28#4480—21no—
29#4479—21no—
30#4464—21no—
31#4464—21no—
32#3234—31no—
33#1998D311yescharacteristic, complemented
34#1569—31no—
35#1557—31no—
36#1528—31no—
37#1525—31no—
38#1291—31no—
39C2×#235—21no—
40C2×C3×M16—21no—
41C2×C3×M16—21no—
42C3×#125—21no—
43C3×#125—21no—
44C3×#125—21no—
45C3×#125—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C42C2311yescharacteristic
51#930—21no—
52#930—21no—
53C2×#208—21no—
54#875—21no—
55#874D411yescharacteristic
56C4×D12—21no—
57C4×D12—21no—
58#869D411yescharacteristic
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×#97—21no—
62C3×C4×C8—21no—
63#830—21no—
64#830—21no—
65#809—21no—
66#809—21no—
67#809—21no—
68#809—21no—
69#808—21no—
70#808—21no—
71#807—21no—
72#807—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)D411yescharacteristic
75C4⋊(C3⋊C8)D411yescharacteristic
76#803—21no—
77#803—21no—
78#803D411yescharacteristic
79#803D411yescharacteristic
80C4×C3⋊C8—21no—
81#532—31no—
82C2×#125C2×D311yescharacteristic
83#501—61no—
84#490—61no—
85#481—61no—
86#480—61no—
87#478—61no—
88#478—61no—
89#432—31no—
90#432—31no—
91#423C2×D311yescharacteristic
92C4×M16—31no—
93C4×M16C2×D311yescharacteristic
94#343—21no—
95#343—21no—
96#335—61no—
97#334—61no—
98#332—61no—
99#331—61no—
100#330—61no—
101#330—61no—
102#329—61no—
103#329—61no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C2×#67—41no—
110#235—82no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12—21no—
114C2×C4×D3—41no—
115C2×Dic6—21no—
116C3×M16—22no—
117C3×M16—41no—
118C3×M16—42no—
119C2×C3×C8—41no—
120C2×C3×C8—41no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133(C2×D3)⋊C4—41no—
134(C2×D3)⋊C4—41no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137#208—42no—
138C2×C3⋊C8—21no—
139C2×C3⋊C8—21no—
140C2×C3⋊C8—21no—
141C2×C3⋊C8—21no—
142C2×C3⋊C8—41no—
143C2×#41—61no—
144C2×M16—21no—
145C2×M16—21no—
146C2×M16—61no—
147C4⋊Q8—31no—
148#127—31no—
149#125—21no—
150#125—21no—
151#125—21no—
152#125—21no—
153#124—61no—
154#121—61no—
155#121—61no—
156C4×D4—61no—
157C4×D4—61no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4#6711yescharacteristic
161C2×C4⋊C4#6711yescharacteristic
162C2×C42C22×D311yescharacteristic
163#107—61no—
164#107—61no—
165#106—61no—
166#106—61no—
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—31no—
170C4⋊C8—31no—
171C4⋊C8—61no—
172#102—61no—
173#102—61no—
174#102—61no—
175#102—61no—
176C22⋊C8—61no—
177C22⋊C8—61no—
178#97—21no—
179#97—31no—
180#97—31no—
181#97—61no—
182#97—61no—
183C4×C8—21no—
184C4×C8—61no—
185C23×C3#12311yescharacteristic
186C22×D3—41no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—81no—
194C2×C3×C4—81no—
195C2×C3×C4#12011yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12311yescharacteristic
200C2×C3×C4#12411yescharacteristic
201#67—82no—
202C2×Dic3—41no—
203D12—41no—
204D12—41no—
205D12—81no—
206D12—81no—
207C4×D3—82no—
208Dic6—81no—
209C3×C8—42no—
210C3×C8—81no—
211C3⋊C8—22no—
212C3⋊C8—41no—
213C3⋊C8—41no—
214C3⋊C8—41no—
215C3⋊C8—81no—
216#41—242no—
217C2×Q8—61no—
218C2×D4—61no—
219C2×D4—61no—
220C2×D4—61no—
221C2×D4—121no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—121no—
225C22×C4#23511yescharacteristic
226C22×C4#23511yescharacteristic
227C22×C4C2×#6711yescharacteristic
228M16—22no—
229M16—41no—
230M16—42no—
231M16—122no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—61no—
235C2×C8—61no—
236C2×C8—61no—
237C2×C8—61no—
238C2×C8—121no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C22⋊C4—121no—
250C22⋊C4—121no—
251C42D3×D411yescharacteristic
252C42D3×D411yescharacteristic
253C42D3×D411yescharacteristic
254C42#23711yescharacteristic
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3#39411yescharacteristic
259C2×D3—81no—
260C2×D3—81no—
261C2×D3—81no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—81no—
268C3×C4—81no—
269C3×C4#45411yescharacteristic
270C3×C4#45611yescharacteristic
271Dic3—81no—
272C23—121no—
273C23#92911yescharacteristic
274Q8—241no—
275D4—121no—
276D4—121no—
277D4—241no—
278D4—241no—
279D4—242no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—121no—
289C2×C4—242no—
290C2×C4#88311yescharacteristic
291C2×C4#88411yescharacteristic
292C2×C4#88411yescharacteristic
293C2×C4#88511yescharacteristic
294C2×C4#93711yescharacteristic
295C2×C4#93911yescharacteristic
296C8—42no—
297C8—62no—
298C8—81no—
299C8—121no—
300C8—121no—
301C8—121no—
302C8—241no—
303C2×C3—41no—
304C2×C3#203811yescharacteristic
305C2×C3#204811yescharacteristic
306C2×C3#205011yescharacteristic
307D3—161no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—241no—
312C22—241no—
313C22—241no—
314C22#495911yescharacteristic
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—81no—
321C4—81no—
322C4—241no—
323C4#506211yescharacteristic
324C4#509511yescharacteristic
325C3#1049711yescharacteristic, complemented, Sylow
326C2—41no—
327C2—481no—
328C2#6969511yescharacteristic
329C2#7014011yescharacteristic
330C2#7014411yescharacteristic
subgroupG/H
centerC22#4959
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1998C2
socleC22×C3#394
prime radical#5477C22
hypercenter#1998D3
hypocenterC3#10497
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1998

Complements

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

NH ≅ G/Ntype
2. C3×#1998C2semidirect
3. #71735C2semidirect
5. #70492C2semidirect
7. #67670C2semidirect
33. #1998D3semidirect
325. C3#10497semidirect

Maximal subgroups

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

subgroupindex
#830192
C3×#19982
#717352
#705042
#704922
#698592
#676702
#104973