Finite Group Databasebeta

← #10691898

Subgroups — #10691898

Order 768.

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

normal subgroups only

This group has 454 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#83228C211yesmaximal, characteristic
2#83222C211yesmaximal, characteristic
3C3×#1548C211yesmaximal, characteristic
4#70793C211yesmaximal, characteristic
5#70515C211yesmaximal, characteristic
6#67737C211yesmaximal, characteristic
7#67713C211yesmaximal, characteristic
8#9297—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13#5229—22no—
14#5228—22no—
15#5092C412yescomplemented
16#5090—22no—
17#5079C412yescomplemented
18#4984—21no—
19#4980—22no—
20Dic3×C4⋊C4—21no—
21#4936C2211yescharacteristic
22#4932C2211yescharacteristic
23#4925C2211yescharacteristic
24C3×#333—22no—
25#4555—21no—
26#4555—21no—
27#4467—22no—
28#3209—31no—
29#2331—31no—
30#1995—31no—
31#1548D311yescharacteristic
32#1518—31no—
33#1286—31no—
34#1274—31no—
35C22×C3×Q8D412yes—
36C2×C3×M16—21no—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8C2×C412yes—
40C3×C4⋊Q8D412yes—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C42C2311yescharacteristic
44C2×#216—24no—
45#926—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—22no—
48#924—21no—
49#924—21no—
50C2×C4×Dic3—22no—
51C2×C4×Dic3D411yescharacteristic
52C2×C4×Dic3D411yescharacteristic
53#870—21no—
54#869C2×C411yescharacteristic
55#869C2×C411yescharacteristic
56C4×Dic6—21no—
57C4×Dic6—21no—
58C3×C4⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×C4×C8—21no—
63#835—24no—
64C3⋊#95—22no—
65C3⋊#95—22no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95D412yes—
70#830—21no—
71C4⋊(C3⋊C8)—21no—
72C4×C3⋊C8C2×C412yes—
73#533—31no—
74C2×C4⋊Q8C2×D311yescharacteristic
75#494—32no—
76#487—32no—
77#462—62no—
78#451—62no—
79#431—31no—
80#431C2×D311yescharacteristic
81#423C2×D311yescharacteristic
82#391—62no—
83#389—61no—
84#387—62no—
85#382—31no—
86C4×C4⋊C4—61no—
87#376—31no—
88#339—61no—
89#339—61no—
90#333—22no—
91#328—62no—
92C2×C3×Q8—22no—
93C2×C3×Q8—82no—
94C2×C3×Q8C22⋊C414yes—
95C22×C3×C4—22no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—22no—
102C2×Dic6—21no—
103C2×Dic6—21no—
104C3×M16—41no—
105C3×M16—41no—
106C2×C3×C8—41no—
107C2×C3×C8—41no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—22no—
112C3×C42C22×C411yescharacteristic
113C3×C42C2×D411yescharacteristic
114C3×C42#4111yescharacteristic
115C3×C42#4111yescharacteristic
116#217—41no—
117#216—84no—
118C4⋊Dic3—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3—41no—
123C4⋊Dic3—41no—
124C4⋊Dic3—82no—
125Dic3⋊C4—41no—
126Dic3⋊C4—41no—
127C4×Dic3—21no—
128C4×Dic3—21no—
129C4×Dic3—21no—
130C4×Dic3—41no—
131C4×Dic3—82no—
132C4×Dic3Q1612yescomplemented
133C2×C3⋊C8—22no—
134C22×Q8D1212yes—
135C2×Q16—64no—
136C2×M16—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—31no—
140C4⋊Q8—31no—
141C4⋊Q8C4×D312yes—
142C4⋊Q8#6712yes—
143#125—61no—
144C22⋊Q8—61no—
145C22⋊Q8—61no—
146C4×Q8—61no—
147C4×Q8—61no—
148#117—61no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4—62no—
153C2×C42—31no—
154C2×C42—31no—
155C2×C42—62no—
156C2×C42C22×D311yescharacteristic
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—21no—
160C4⋊C8—61no—
161#103—64no—
162C22⋊C8—21no—
163C22⋊C8—61no—
164C4×C8—21no—
165C4×C8—32no—
166#95—32no—
167#95—62no—
168#95—62no—
169#95—61no—
170#95—61no—
171#95—61no—
172C23×C3#12011yescharacteristic
173C3×Q8—44no—
174C3×Q8—82no—
175C2×C3×C4—24no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4C2×C22⋊C411yescharacteristic
181C2×C3×C4C4×D411yescharacteristic
182C2×C3×C4C4×D411yescharacteristic
183C2×C3×C4#12111yescharacteristic
184C2×C3×C4#12111yescharacteristic
185C2×C3×C4#12311yescharacteristic
186C2×C3×C4C2×Q1611yescharacteristic
187C2×C3×C4#13511yescharacteristic
188C2×C3×C4#13711yescharacteristic
189C2×C3×C4#13711yescharacteristic
190C2×Dic3—22no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—42no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—42no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—42no—
201Dic6—42no—
202Dic6—81no—
203C3×C8—81no—
204C3×C8—81no—
205C3⋊C8—42no—
206C2×Q8—22no—
207C2×Q8—61no—
208C2×Q8—61no—
209C2×Q8—82no—
210C2×Q8(C2×D3)⋊C414yes—
211C22×C4—22no—
212C22×C4—61no—
213C22×C4—62no—
214C22×C4—61no—
215C22×C4C2×D1211yescharacteristic
216C22×C4D3×D411yescharacteristic
217C22×C4D3×D411yescharacteristic
218Q16—244no—
219M16—41no—
220M16—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—62no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—121no—
233C4⋊C4—121no—
234C4⋊C4—121no—
235C4⋊C4—121no—
236C4⋊C4—242no—
237C22⋊C4—121no—
238C42—32no—
239C42—61no—
240C42—61no—
241C42—61no—
242C42—121no—
243C42—242no—
244C42C2×C4×D311yescharacteristic
245C42#23711yescharacteristic
246C42#23711yescharacteristic
247C42C2×#6711yescharacteristic
248C22×C3—21no—
249C22×C3—21no—
250C22×C3#38611yescharacteristic
251C22×C3#45111yescharacteristic
252C22×C3#45211yescharacteristic
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—44no—
258C3×C4C4×Q1611yescharacteristic
259C3×C4#44111yescharacteristic
260C3×C4#46211yescharacteristic
261C3×C4#46411yescharacteristic
262Dic3—42no—
263Dic3—82no—
264Dic3—81no—
265Dic3—81no—
266Dic3—81no—
267C23#88211yescharacteristic
268Q8—44no—
269Q8—82no—
270Q8—122no—
271Q8—241no—
272C2×C4—24no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—62no—
278C2×C4—121no—
279C2×C4—121no—
280C2×C4—122no—
281C2×C4—121no—
282C2×C4—122no—
283C2×C4—121no—
284C2×C4—121no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4—122no—
288C2×C4#67⋊C411yescharacteristic
289C2×C4#67⋊C411yescharacteristic
290C2×C4#88611yescharacteristic
291C2×C4#91511yescharacteristic
292C2×C4#91611yescharacteristic
293C2×C4D3×Q1611yescharacteristic
294C2×C4#91811yescharacteristic
295C2×C4C2×(C2×D3)⋊C411yescharacteristic
296C2×C4#93911yescharacteristic
297C2×C4#93911yescharacteristic
298C8—81no—
299C8—81no—
300C8—122no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3#187811yescharacteristic
304C2×C3#190511yescharacteristic
305C2×C3#191111yescharacteristic
306C22—21no—
307C22—21no—
308C22#480811yescharacteristic
309C22#480911yescharacteristic
310C22#495811yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—44no—
316C4—122no—
317C4—242no—
318C4—241no—
319C4—241no—
320C4—241no—
321C4#478911yescharacteristic
322C4#479011yescharacteristic
323C4#516611yescharacteristic
324C4#518211yescharacteristic
325C3#929711yescharacteristic, complemented, Sylow
326C2—21no—
327C2—21no—
328C2#7007111yescharacteristic
329C2#7023911yescharacteristic
330C2#7033411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1548D3
hypocenterC3#9297
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
15. #5092C4semidirect
17. #5079C4semidirect
132. C4×Dic3Q16semidirect
325. C3#9297semidirect

Maximal subgroups

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

subgroupindex
#832282
#832222
C3×#15482
#707932
#705152
#677372
#677132
#92973