Finite Group Databasebeta

← #10691849

Subgroups — #10691849

Order 768.

422 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 113 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#83370C211yesmaximal, characteristic
2#83329C211yesmaximal, characteristic
3C3×#1518C211yesmaximal, characteristic
4#70780C211yesmaximal, characteristic
5#69555C211yesmaximal, characteristic
6#67740C211yesmaximal, characteristic
7#67631C211yesmaximal, characteristic
8#9290—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#382C2211yescharacteristic
13#5093C412yescomplemented
14#5092C412yescomplemented
15#5050—22no—
16#5031—22no—
17#5024—22no—
18#5024—22no—
19#4999—21no—
20#4997—21no—
21#4925C2211yescharacteristic
22#4924C2211yescharacteristic
23C2×#803C2211yescharacteristic
24C3×#328—22no—
25#4529—21no—
26#4529—21no—
27#4482—22no—
28#3209—31no—
29#2961—31no—
30#1993—31no—
31#1549—31no—
32#1518D311yescharacteristic
33#1286—31no—
34#1275—31no—
35C22×Dic6—21no—
36C3×C4⋊Q8C2×C411yescharacteristic
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×#125—21no—
39C3×C22⋊Q8—21no—
40C3×C22⋊Q8D412yes—
41C3×C4×Q8—21no—
42C3×C4×Q8D412yes—
43C3×#117—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C42C2311yescharacteristic
46C2×#216—24no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×#208—21no—
50C22×C3⋊C8D411yescharacteristic
51C22×C3⋊C8D411yescharacteristic
52#869—21no—
53#869—21no—
54#869—21no—
55#869C2×C412yes—
56C3×C4⋊C8—21no—
57C3×C22⋊C8—21no—
58C3×C4×C8—21no—
59C3×#95—21no—
60C3×#95—21no—
61#830—21no—
62#810—24no—
63C4⋊(C3⋊C8)—24no—
64C4⋊(C3⋊C8)—21no—
65C4⋊(C3⋊C8)—21no—
66C4⋊(C3⋊C8)D412yes—
67#803—22no—
68#803—21no—
69#803—21no—
70C4×C3⋊C8—22no—
71C4×C3⋊C8C2×C412yes—
72#533C2×D311yescharacteristic
73C2×C4⋊Q8—31no—
74#494—32no—
75#487—32no—
76#462—62no—
77#451—62no—
78#446—62no—
79#446—62no—
80#431—31no—
81#431C2×D311yescharacteristic
82#424—61no—
83#423—31no—
84#405—61no—
85C2×#97—31no—
86#382C2×D311yescharacteristic
87#340—61no—
88#340—61no—
89#332—62no—
90#328—22no—
91C2×C3×Q8—21no—
92C2×C3×Q8C22⋊C412yes—
93C22×C3×C4—41no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4#4111yescharacteristic
97C22×Dic3—41no—
98C2×Dic6—22no—
99C2×Dic6—41no—
100C2×Dic6—82no—
101C2×C3×C8—41no—
102C3×C4⋊C4—21no—
103C3×C4⋊C4—21no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—22no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4C22⋊C412yes—
110C3×C22⋊C4—22no—
111C3×C42—22no—
112C3×C42C22×C411yescharacteristic
113C3×C42C2×D411yescharacteristic
114C3×C42C2×D411yescharacteristic
115C3×C42#4111yescharacteristic
116#216—84no—
117C4⋊Dic3—22no—
118C4⋊Dic3—22no—
119C4⋊Dic3—41no—
120C4⋊Dic3—41no—
121#208—41no—
122#208—41no—
123C2×C3⋊C8—21no—
124C2×C3⋊C8—21no—
125C2×C3⋊C8—22no—
126C2×C3⋊C8—22no—
127C2×C3⋊C8—21no—
128C2×C3⋊C8—21no—
129C2×C3⋊C8—21no—
130C2×C3⋊C8Q1612yescomplemented
131C22×Q8—61no—
132C2×Q16—64no—
133C2×M16—61no—
134C22×C8—31no—
135C22×C8—31no—
136C4⋊Q8—32no—
137C4⋊Q8—61no—
138C4⋊Q8—61no—
139C4⋊Q8—61no—
140C4⋊Q8C4×D311yescharacteristic
141C4⋊Q8C4×D311yescharacteristic
142#125—21no—
143C22⋊Q8—21no—
144C22⋊Q8#6712yes—
145C4×Q8—21no—
146C4×Q8D1212yes—
147#117—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C42C22×D311yescharacteristic
152C4⋊C8—21no—
153C4⋊C8—32no—
154C4⋊C8—64no—
155C4⋊C8—61no—
156C4⋊C8—61no—
157#103—64no—
158C22⋊C8—21no—
159C22⋊C8—61no—
160#97—62no—
161#97—61no—
162#97—61no—
163C4×C8—21no—
164C4×C8—32no—
165C4×C8—62no—
166#95—21no—
167#95—21no—
168C23×C3#12111yescharacteristic
169C3×Q8—42no—
170C3×Q8—81no—
171C2×C3×C4—22no—
172C2×C3×C4—22no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—81no—
176C2×C3×C4—81no—
177C2×C3×C4—81no—
178C2×C3×C4C2×C22⋊C411yescharacteristic
179C2×C3×C4C4×D411yescharacteristic
180C2×C3×C4C4×D411yescharacteristic
181C2×C3×C4#12011yescharacteristic
182C2×C3×C4#12111yescharacteristic
183C2×C3×C4#12311yescharacteristic
184C2×C3×C4C2×Q1611yescharacteristic
185C2×C3×C4#13511yescharacteristic
186C2×C3×C4#13711yescharacteristic
187C2×C3×C4#13711yescharacteristic
188C2×Dic3—42no—
189C2×Dic3—82no—
190Dic6—44no—
191Dic6—82no—
192C3×C8—81no—
193C3⋊C8—22no—
194C3⋊C8—42no—
195C3⋊C8—41no—
196C3⋊C8—42no—
197C2×Q8—21no—
198C2×Q8—62no—
199C2×Q8—121no—
200C2×Q8—242no—
201C2×Q8(C2×D3)⋊C412yes—
202C22×C4—41no—
203C22×C4—121no—
204C22×C4D3×D411yescharacteristic
205C22×C4#23711yescharacteristic
206C22×C4C2×#6711yescharacteristic
207Q16—244no—
208M16—121no—
209M16—121no—
210C2×C8—32no—
211C2×C8—41no—
212C2×C8—61no—
213C2×C8—61no—
214C2×C8—62no—
215C2×C8—61no—
216C2×C8—61no—
217C2×C8—61no—
218C2×C8—62no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—62no—
227C4⋊C4—62no—
228C4⋊C4—121no—
229C4⋊C4—121no—
230C4⋊C4(C2×D3)⋊C412yes—
231C22⋊C4—22no—
232C42—22no—
233C42C2×C4×D311yescharacteristic
234C42C2×D1211yescharacteristic
235C42D3×D411yescharacteristic
236C42#23711yescharacteristic
237C22×C3—21no—
238C22×C3—21no—
239C22×C3#38611yescharacteristic
240C22×C3#46211yescharacteristic
241C22×C3#46411yescharacteristic
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—21no—
245C3×C4—21no—
246C3×C4—42no—
247C3×C4—42no—
248C3×C4—161no—
249C3×C4C4×Q1611yescharacteristic
250C3×C4#44111yescharacteristic
251C3×C4#45111yescharacteristic
252C3×C4#45211yescharacteristic
253Dic3—82no—
254C23#93911yescharacteristic
255Q8—42no—
256Q8—81no—
257Q8—124no—
258Q8—242no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—122no—
267C2×C4—242no—
268C2×C4#67⋊C411yescharacteristic
269C2×C4#67⋊C411yescharacteristic
270C2×C4#88211yescharacteristic
271C2×C4#88611yescharacteristic
272C2×C4#91511yescharacteristic
273C2×C4#91611yescharacteristic
274C2×C4D3×Q1611yescharacteristic
275C2×C4#91811yescharacteristic
276C2×C4C2×(C2×D3)⋊C411yescharacteristic
277C2×C4#93911yescharacteristic
278C8—62no—
279C8—81no—
280C8—122no—
281C8—121no—
282C8—122no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3#187811yescharacteristic
286C2×C3#190211yescharacteristic
287C2×C3#191011yescharacteristic
288C22—21no—
289C22—21no—
290C22#495811yescharacteristic
291C22#516611yescharacteristic
292C22#518211yescharacteristic
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—42no—
298C4—42no—
299C4—161no—
300C4—242no—
301C4#478911yescharacteristic
302C4#479011yescharacteristic
303C4#480811yescharacteristic
304C4#480911yescharacteristic
305C3#929011yescharacteristic, complemented, Sylow
306C2—21no—
307C2—21no—
308C2#7004711yescharacteristic
309C2#7023911yescharacteristic
310C2#7039911yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1518C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1518D3
hypocenterC3#9290
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
13. #5093C4semidirect
14. #5092C4semidirect
130. C2×C3⋊C8Q16semidirect
305. C3#9290semidirect

Maximal subgroups

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

subgroupindex
#833702
#833292
C3×#15182
#707802
#695552
#677402
#676312
#92903