Finite Group Databasebeta

← #10691838

Subgroups — #10691838

Order 768.

410 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 107 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#79272C211yesmaximal, characteristic
2#79045C211yesmaximal, characteristic
3#70507C211yesmaximal, characteristic
4#69555C211yesmaximal, characteristic
5#69546C211yesmaximal, characteristic
6C3×#1286C211yesmaximal, characteristic
7#67740C211yesmaximal, characteristic
8#9288—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×#382C2211yescharacteristic
14#5122—21no—
15#5119—21no—
16#4925C2211yescharacteristic
17C2×#803C2211yescharacteristic
18#4850—22no—
19#4828—22no—
20#4828—22no—
21#4722—22no—
22#4682C412yescomplemented
23#4681C412yescomplemented
24#4529—21no—
25#4529—21no—
26#4481—22no—
27#4452—22no—
28#3230—31no—
29#2961—31no—
30#1993—31no—
31#1548—31no—
32#1518—31no—
33#1286D311yescharacteristic
34#1275—31no—
35C22×Dic6—21no—
36C2×C3×M16—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C42C2311yescharacteristic
39#926—21no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3—21no—
42#924—21no—
43#924D412yes—
44C22×C3⋊C8D411yescharacteristic
45C22×C3⋊C8D411yescharacteristic
46#870—21no—
47#869—21no—
48#869—21no—
49#869—21no—
50#869C2×C412yes—
51#869C2×C411yescharacteristic
52#869C2×C411yescharacteristic
53C4×Dic6—21no—
54C4×Dic6D412yes—
55C3×C4⋊C8—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊C8D412yes—
58C3×C22⋊C8—21no—
59C3×C4×C8C2×C412yes—
60C3×#95—21no—
61C3×#95—21no—
62#830—21no—
63#818—24no—
64#816—24no—
65#815—22no—
66Dic3⋊C8—24no—
67C8×Dic3—22no—
68C4⋊(C3⋊C8)—21no—
69#803—21no—
70#803—21no—
71C4×C3⋊C8—21no—
72#533—31no—
73C2×C4⋊Q8—31no—
74#500—32no—
75#487—32no—
76#484—62no—
77#477—62no—
78#446—62no—
79#446—62no—
80#431—31no—
81#431C2×D311yescharacteristic
82#424—61no—
83#423C2×D311yescharacteristic
84#405—61no—
85C2×#97—31no—
86#382C2×D311yescharacteristic
87#340—61no—
88#340—61no—
89#333—62no—
90#328—62no—
91C22×C3×C4—41no—
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4#4111yescharacteristic
95C22×Dic3—41no—
96C2×Dic6—21no—
97C2×Dic6—22no—
98C2×Dic6—41no—
99C2×Dic6—82no—
100C2×Dic6C22⋊C412yes—
101C3×M16—41no—
102C3×M16—41no—
103C2×C3×C8—22no—
104C2×C3×C8—22no—
105C3×C4⋊C4—41no—
106C3×C4⋊C4—41no—
107C3×C42C22×C411yescharacteristic
108C3×C42C2×D411yescharacteristic
109C3×C42C2×D411yescharacteristic
110C3×C42#4111yescharacteristic
111#217—22no—
112C4⋊Dic3—21no—
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3—22no—
116C4⋊Dic3—22no—
117C4⋊Dic3—41no—
118C4⋊Dic3—41no—
119C4⋊Dic3C22⋊C412yes—
120Dic3⋊C4—22no—
121Dic3⋊C4—22no—
122C4×Dic3—22no—
123C2×C3⋊C8—21no—
124C2×C3⋊C8—21no—
125C2×C3⋊C8—21no—
126C2×C3⋊C8—21no—
127C2×C3⋊C8—21no—
128C2×C3⋊C8—41no—
129C2×C3⋊C8Q1612yes—
130C22×Q8—61no—
131C2×M16—21no—
132C22×C8—31no—
133C22×C8—31no—
134C4⋊Q8—31no—
135C4⋊Q8—31no—
136C4⋊Q8—32no—
137C4⋊Q8—61no—
138C4⋊Q8—61no—
139C4⋊Q8—61no—
140#125—61no—
141C22⋊Q8—32no—
142C22⋊Q8—61no—
143C4×Q8—32no—
144C4×Q8—61no—
145#117—61no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C42C22×D311yescharacteristic
150#107—64no—
151C4⋊C8—21no—
152C4⋊C8—21no—
153C4⋊C8—64no—
154C4⋊C8—61no—
155C4⋊C8D1212yes—
156#103—64no—
157C22⋊C8—21no—
158C22⋊C8—61no—
159#97—62no—
160#97—61no—
161#97—61no—
162C4×C8—62no—
163C4×C8—61no—
164C4×C8C4×D312yes—
165#95—21no—
166#95—21no—
167C23×C3#12111yescharacteristic
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4—81no—
171C2×C3×C4—81no—
172C2×C3×C4—81no—
173C2×C3×C4C2×C22⋊C411yescharacteristic
174C2×C3×C4C4×D411yescharacteristic
175C2×C3×C4C4×D411yescharacteristic
176C2×C3×C4#12011yescharacteristic
177C2×C3×C4#12111yescharacteristic
178C2×C3×C4#12311yescharacteristic
179C2×C3×C4C2×Q1611yescharacteristic
180C2×C3×C4#13511yescharacteristic
181C2×C3×C4#13711yescharacteristic
182C2×C3×C4#13711yescharacteristic
183C2×Dic3—22no—
184C2×Dic3—22no—
185C2×Dic3—42no—
186C2×Dic3—82no—
187Dic6—42no—
188Dic6—81no—
189Dic6—82no—
190Dic6—82no—
191C3×C8—42no—
192C3×C8—42no—
193C3⋊C8—22no—
194C3⋊C8—41no—
195C3⋊C8—81no—
196C2×Q8—32no—
197C2×Q8—61no—
198C2×Q8—62no—
199C2×Q8—121no—
200C2×Q8—242no—
201C22×C4—41no—
202C22×C4—121no—
203C22×C4#23511yescharacteristic
204C22×C4D3×D411yescharacteristic
205C22×C4D3×D411yescharacteristic
206M16—41no—
207M16—41no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—32no—
211C2×C8—61no—
212C2×C8—61no—
213C2×C8—61no—
214C2×C8—61no—
215C2×C8—61no—
216C2×C8—121no—
217C4⋊C4—32no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—61no—
223C4⋊C4—62no—
224C4⋊C4—62no—
225C4⋊C4—62no—
226C4⋊C4—62no—
227C4⋊C4—121no—
228C4⋊C4—121no—
229C22⋊C4—62no—
230C42—62no—
231C42C2×C4×D311yescharacteristic
232C42C2×D1211yescharacteristic
233C42D3×D411yescharacteristic
234C42#23911yescharacteristic
235C22×C3—21no—
236C22×C3—21no—
237C22×C3#38611yescharacteristic
238C22×C3#46211yescharacteristic
239C22×C3#46411yescharacteristic
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—161no—
245C3×C4C4×Q1611yescharacteristic
246C3×C4#44111yescharacteristic
247C3×C4#45111yescharacteristic
248C3×C4#45211yescharacteristic
249Dic3—42no—
250Dic3—42no—
251Dic3—82no—
252C23#88411yescharacteristic
253Q8—122no—
254Q8—241no—
255Q8—242no—
256Q8—242no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—122no—
265C2×C4—242no—
266C2×C4C4×D1211yescharacteristic
267C2×C4D3×C22⋊C411yescharacteristic
268C2×C4#88211yescharacteristic
269C2×C4#89311yescharacteristic
270C2×C4#89411yescharacteristic
271C2×C4#89511yescharacteristic
272C2×C4#91211yescharacteristic
273C2×C4#91511yescharacteristic
274C2×C4#91511yescharacteristic
275C2×C4D3×Q1611yescharacteristic
276C8—42no—
277C8—42no—
278C8—62no—
279C8—121no—
280C8—241no—
281C2×C3—21no—
282C2×C3—21no—
283C2×C3#187811yescharacteristic
284C2×C3#190311yescharacteristic
285C2×C3#190911yescharacteristic
286C22—21no—
287C22—21no—
288C22#466511yescharacteristic
289C22#476711yescharacteristic
290C22#479511yescharacteristic
291C4—21no—
292C4—21no—
293C4—21no—
294C4—21no—
295C4—122no—
296C4—122no—
297C4—161no—
298C4—242no—
299C4#478311yescharacteristic
300C4#480911yescharacteristic
301C4#485311yescharacteristic
302C4#487311yescharacteristic
303C3#928811yescharacteristic, complemented, Sylow
304C2—21no—
305C2—21no—
306C2#6871211yescharacteristic
307C2#6890811yescharacteristic
308C2#6903811yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1286C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1286D3
hypocenterC3#9288
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
22. #4682C4semidirect
23. #4681C4semidirect
303. C3#9288semidirect

Maximal subgroups

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

subgroupindex
#792722
#790452
#705072
#695552
#695462
C3×#12862
#677402
#92883