Finite Group Databasebeta

← #70406

Subgroups — #70406

Order 384.

394 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 69 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#5828C211yesmaximal, characteristic, complemented
2#5822C211yesmaximal, characteristic, complemented
3#5524C211yesmaximal, characteristic, complemented
4C3×#421C211yesmaximal, characteristic, complemented
5#5139C211yesmaximal, characteristic, complemented
6#5126C211yesmaximal, characteristic, complemented
7#4531C211yesmaximal, characteristic, complemented
8#2010—31nomaximal, Sylow
9C2×C3×#41C2211yescharacteristic
10C2×#235C2211yescharacteristic
11C22×C4×D3C2211yescharacteristic
12C2×C3×M16C2211yescharacteristic
13C3×#117C2211yescharacteristic
14#951—21no—
15#950—22no—
16#949—21no—
17#946—21no—
18#942—21no—
19#938—21no—
20#937—21no—
21#935—21no—
22#932—21no—
23C4×#67—22no—
24#926C2211yescharacteristic
25C2×#208C2211yescharacteristic
26#897—21no—
27#895—21no—
28#886—21no—
29#882—21no—
30C4×D12—22no—
31C3×#104—22no—
32#825—22no—
33(C2×D3)⋊C8—22no—
34#575—31no—
35#525—31no—
36#525—31no—
37#421—31no—
38#421D311yescharacteristic, complemented
39#407—31no—
40H(3,ℤ/4)—31no—
41C23×D3D411yescharacteristic
42C3×#41—21no—
43C3×#41—21no—
44C3×#41—21no—
45C3×#41—21no—
46C2×C3×Q8D411yescharacteristic, complemented
47C2×C3×D4—21no—
48C2×C3×D4D411yescharacteristic, complemented
49C22×C3×C4—21no—
50C22×C3×C4C2311yescharacteristic
51C2×#67—22no—
52C2×#67—21no—
53C22×Dic3D411yescharacteristic
54#235—21no—
55#235—21no—
56#235—21no—
57#235—21no—
58C2×D12—22no—
59C2×D12D411yescharacteristic, complemented
60C2×C4×D3—21no—
61C2×C4×D3—21no—
62C2×C4×D3—21no—
63C2×C4×D3—21no—
64C2×C4×D3—21no—
65C2×C4×D3—21no—
66C2×C4×D3—21no—
67C2×Dic6D411yescharacteristic, complemented
68C3×M16—21no—
69C3×M16—21no—
70C2×C3×C8—21no—
71C3×C4⋊C4—21no—
72C3×C22⋊C4—21no—
73C3×C42D412yescomplemented
74#217—22no—
75#217—21no—
76#216—22no—
77#215—22no—
78#214—22no—
79#213—22no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4—22no—
82(C2×D3)⋊C4—22no—
83C4⋊Dic3—22no—
84C4⋊Dic3—21no—
85Dic3⋊C4—22no—
86C4×Dic3D412yescomplemented
87#208—21no—
88#208—21no—
89C2×C3⋊C8D412yescomplemented
90C2×#41—31no—
91C2×#41C2×D311yescharacteristic
92C23×C4—31no—
93#137—61no—
94#137—61no—
95AGL(1,ℤ/8)—61no—
96AGL(1,ℤ/8)—61no—
97#135—62no—
98C2×M16—31no—
99C2×M16C2×D311yescharacteristic
100#123—61no—
101#123—61no—
102C22⋊Q8—61no—
103C22⋊Q8—61no—
104#121—61no—
105#121—61no—
106#120—61no—
107#120—61no—
108C4×D4—62no—
109C4×D4—62no—
110#117—31no—
111#117C2×D311yescharacteristic
112#104—22no—
113#104—62no—
114C22⋊C8—62no—
115C23×C3—21no—
116C23×C3C2×D411yescharacteristic
117C22×D3—21no—
118C22×D3—21no—
119C22×D3—21no—
120C22×D3—21no—
121C22×D3—42no—
122C3×Q8—21no—
123C3×Q8—21no—
124C3×D4—21no—
125C3×D4—21no—
126C3×D4—41no—
127C3×D4—41no—
128C2×C3×C4—21no—
129C2×C3×C4—22no—
130C2×C3×C4—41no—
131C2×C3×C4—41no—
132C2×C3×C4C2×D411yescharacteristic
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2×D411yescharacteristic
136C2×C3×C4C2×D411yescharacteristic
137C2×C3×C4#4111yescharacteristic
138#67—44no—
139#67—41no—
140#67—41no—
141C2×Dic3—22no—
142C2×Dic3—21no—
143C2×Dic3—21no—
144C2×Dic3—21no—
145C2×Dic3—21no—
146D12—21no—
147D12—21no—
148D12—44no—
149C4×D3—41no—
150C4×D3—41no—
151C4×D3—41no—
152C4×D3—41no—
153C4×D3—41no—
154C4×D3—41no—
155Dic6—21no—
156Dic6—21no—
157C3×C8—41no—
158C3⋊C8—22no—
159C24—31no—
160#41—21no—
161#41—21no—
162#41—21no—
163#41—21no—
164#41—61no—
165#41—61no—
166#41—61no—
167#41—61no—
168C2×Q8—31no—
169C2×Q8D1211yescharacteristic, complemented
170C2×D4—21no—
171C2×D4—31no—
172C2×D4—62no—
173C2×D4—62no—
174C2×D4—61no—
175C2×D4D1211yescharacteristic, complemented
176C22×C4—21no—
177C22×C4—31no—
178C22×C4—61no—
179C22×C4—61no—
180C22×C4—61no—
181C22×C4—61no—
182C22×C4—61no—
183C22×C4—61no—
184C22×C4—61no—
185C22×C4C22×D311yescharacteristic
186Q16—62no—
187SD16—62no—
188SD16—62no—
189D8—62no—
190M16—21no—
191M16—21no—
192M16—61no—
193M16—61no—
194C2×C8—21no—
195C2×C8—32no—
196C4⋊C4—21no—
197C4⋊C4—62no—
198C4⋊C4—62no—
199C4⋊C4—61no—
200C22⋊C4—21no—
201C22⋊C4—62no—
202C22⋊C4—62no—
203C22⋊C4—62no—
204C22⋊C4—62no—
205C22⋊C4—61no—
206C42—32no—
207C42#6712yescomplemented
208C22×C3—41no—
209C22×C3—42no—
210C22×C3—41no—
211C22×C3#12011yescharacteristic
212C22×C3#12011yescharacteristic
213C22×C3#12111yescharacteristic
214C2×D3—22no—
215C2×D3—22no—
216C2×D3—22no—
217C2×D3—41no—
218C2×D3—41no—
219C2×D3—41no—
220C2×D3—41no—
221C2×D3—42no—
222C2×D3—41no—
223C3×C4—42no—
224C3×C4—41no—
225C3×C4#12011yescharacteristic
226C3×C4#12111yescharacteristic
227C3×C4#12111yescharacteristic
228C3×C4#12711yescharacteristic
229Dic3—42no—
230Dic3—41no—
231Dic3—41no—
232C23—21no—
233C23—61no—
234C23—61no—
235C23—61no—
236C23—61no—
237C23—122no—
238C23D3×D411yescharacteristic
239Q8—21no—
240Q8—21no—
241Q8—61no—
242Q8—61no—
243D4—21no—
244D4—21no—
245D4—41no—
246D4—41no—
247D4—61no—
248D4—61no—
249D4—124no—
250D4—124no—
251D4—121no—
252D4—121no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—62no—
258C2×C4—61no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—61no—
262C2×C4—121no—
263C2×C4—121no—
264C2×C4—121no—
265C2×C4—121no—
266C2×C4—121no—
267C2×C4—121no—
268C2×C4C2×D1211yescharacteristic
269C2×C4#23511yescharacteristic
270C2×C4D3×D411yescharacteristic
271C2×C4D3×D411yescharacteristic
272C2×C4D3×D411yescharacteristic
273C2×C4C2×#6711yescharacteristic
274C8—41no—
275C8—62no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—81no—
280C2×C3#39211yescharacteristic
281D3—42no—
282D3—81no—
283C22—41no—
284C22—42no—
285C22—41no—
286C22—62no—
287C22—62no—
288C22—62no—
289C22—121no—
290C22—121no—
291C22—121no—
292C22—122no—
293C22—121no—
294C22—121no—
295C22#88211yescharacteristic
296C22#88411yescharacteristic
297C22#93711yescharacteristic
298C4—42no—
299C4—41no—
300C4—122no—
301C4—121no—
302C4—121no—
303C4#88211yescharacteristic
304C4#88411yescharacteristic
305C4#93011yescharacteristic
306C4#94011yescharacteristic
307C3#201011yescharacteristic, complemented, Sylow
308C2—21no—
309C2—21no—
310C2—21no—
311C2—81no—
312C2—122no—
313C2—241no—
314C2#496011yescharacteristic
subgroupG/H
centerC4#940
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#421C2
socleC2×C3#392
hypercenter#421D3
hypocenterC3#2010
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. #5828C2semidirect
2. #5822C2semidirect
3. #5524C2semidirect
4. C3×#421C2semidirect
5. #5139C2semidirect
6. #5126C2semidirect
7. #4531C2semidirect
38. #421D3semidirect
46. C2×C3×Q8D4semidirect
48. C2×C3×D4D4semidirect
59. C2×D12D4semidirect
67. C2×Dic6D4semidirect
73. C3×C42D4semidirect
86. C4×Dic3D4semidirect
89. C2×C3⋊C8D4semidirect
169. C2×Q8D12semidirect
175. C2×D4D12semidirect
207. C42#67semidirect
307. C3#2010semidirect

Maximal subgroups

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

subgroupindex
#58282
#58222
#55242
C3×#4212
#51392
#51262
#45312
#20103