Finite Group Databasebeta

← #10684113

Subgroups — #10684113

Order 768.

416 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C3×#2726C211yesmaximal, characteristic
2#69633C211yesmaximal, characteristic
3#69605C211yesmaximal, characteristic
4#69604C211yesmaximal, characteristic
5#67638C211yesmaximal, characteristic
6#67638C211yesmaximal, characteristic
7#67638C211yesmaximal, characteristic
8#7772—31nomaximal, Sylow
9C2×C3×#125—21no—
10C2×#869C2211yescharacteristic
11C3×#398—21no—
12C3×#398—21no—
13C3×#398—21no—
14C3×#389—21no—
15C3×#389C2211yescharacteristic
16C3×#389C2211yescharacteristic
17C3×#389C2211yescharacteristic
18C3×#387—21no—
19C2×#810—21no—
20C2×#810—21no—
21C2×#808—21no—
22C2×#807—21no—
23#4529—21no—
24#4529—21no—
25#4524C2211yescharacteristic
26#4524C2211yescharacteristic
27#4524C2211yescharacteristic
28#2726D311yescharacteristic
29#2058—31no—
30#2057—31no—
31#2043—31no—
32#1271—31no—
33#1271—31no—
34#1271—31no—
35C22×Dic6—21no—
36C3×#125—42no—
37C3×#125—42no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4D411yescharacteristic
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C22×C3⋊C8—21no—
56C22×C3⋊C8—21no—
57C22×C3⋊C8—21no—
58#869—42no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95—21no—
63C3×#95—21no—
64C3×#95—21no—
65#810—42no—
66#810—42no—
67#808—42no—
68#807—42no—
69C2×C4⋊Q8—31no—
70C2×#125—21no—
71C2×#107—61no—
72C2×#106—61no—
73C2×#103—61no—
74C2×#103—61no—
75#398—21no—
76#398—21no—
77#398—21no—
78#389—21no—
79#389C2×D311yescharacteristic
80#389C2×D311yescharacteristic
81#389C2×D311yescharacteristic
82#387—21no—
83#340—61no—
84#340—61no—
85#336—31no—
86#336—31no—
87#336—31no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×Dic3—41no—
98C2×Dic6—22no—
99C2×Dic6—41no—
100C2×Dic6—41no—
101C2×Dic6—41no—
102C2×Dic6—41no—
103C2×Dic6—41no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—21no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—42no—
112C3×C4⋊C4—42no—
113C3×C4⋊C4—42no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—81no—
121C3×C4⋊C4SD1612yes—
122C3×C4⋊C4Q1612yes—
123C3×C42—41no—
124C3×C42—42no—
125C4⋊Dic3—22no—
126C4⋊Dic3—22no—
127C4⋊Dic3—41no—
128C4⋊Dic3—41no—
129C2×C3⋊C8—22no—
130C2×C3⋊C8—22no—
131C2×C3⋊C8—41no—
132C2×C3⋊C8—41no—
133C2×C3⋊C8—41no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8—41no—
136C2×C3⋊C8—41no—
137C2×C3⋊C8—41no—
138C22×Q8—61no—
139C22×C8—61no—
140C22×C8—61no—
141C22×C8—61no—
142C4⋊Q8—122no—
143#125—42no—
144#125—42no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4#6711yescharacteristic
155C2×C4⋊C4#6711yescharacteristic
156C2×C4⋊C4#6711yescharacteristic
157C2×C4⋊C4#6711yescharacteristic
158C2×C4⋊C4#6711yescharacteristic
159C2×C4⋊C4#6711yescharacteristic
160C2×C42—21no—
161C2×C42C22×D311yescharacteristic
162#107—122no—
163#106—122no—
164#103—122no—
165#103—122no—
166#95—21no—
167#95—21no—
168#95—21no—
169#95—21no—
170#95—21no—
171#95—21no—
172C23×C3#12011yescharacteristic
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4C2×SD1611yescharacteristic
198C2×C3×C4C2×Q1611yescharacteristic
199C2×C3×C4#13511yescharacteristic
200C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
201C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
202C2×C3×C4#13711yescharacteristic
203C2×Dic3—42no—
204C2×Dic3—81no—
205C2×Dic3—81no—
206Dic6—42no—
207Dic6—42no—
208Dic6—81no—
209Dic6—81no—
210C3⋊C8—42no—
211C3⋊C8—42no—
212C3⋊C8—81no—
213C2×Q8—62no—
214C2×Q8—121no—
215C2×Q8—121no—
216C2×Q8—121no—
217C2×Q8—121no—
218C2×Q8—121no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—121no—
226C22×C4C2×#6711yescharacteristic
227C22×C4C2×#6711yescharacteristic
228C22×C4C2×#6711yescharacteristic
229C2×C8—62no—
230C2×C8—62no—
231C2×C8—121no—
232C2×C8—121no—
233C2×C8—121no—
234C2×C8—121no—
235C2×C8—121no—
236C2×C8—121no—
237C2×C8—121no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—62no—
255C4⋊C4—62no—
256C4⋊C4—81no—
257C4⋊C4—121no—
258C4⋊C4—121no—
259C4⋊C4#21412yes—
260C4⋊C4#21612yes—
261C42—41no—
262C42—42no—
263C22×C3#44911yescharacteristic
264C22×C3#45011yescharacteristic
265C22×C3#45111yescharacteristic
266C22×C3#45411yescharacteristic
267C22×C3#45611yescharacteristic
268C22×C3#45611yescharacteristic
269C22×C3B(4,𝔽2)11yescharacteristic
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—42no—
277C3×C4—42no—
278C3×C4—81no—
279C3×C4—81no—
280C3×C4—81no—
281C3×C4—81no—
282Dic3—82no—
283C23#95311yescharacteristic
284Q8—122no—
285Q8—122no—
286Q8—241no—
287Q8—241no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—122no—
313C2×C4—241no—
314C2×C4—241no—
315C2×C4#93211yescharacteristic
316C2×C4C2×#21411yescharacteristic
317C2×C4#94211yescharacteristic
318C2×C4C2×#21611yescharacteristic
319C2×C4#94911yescharacteristic
320C2×C4#95011yescharacteristic
321C8—122no—
322C8—122no—
323C8—241no—
324C2×C3#159711yescharacteristic
325C2×C3#160411yescharacteristic
326C2×C3#160511yescharacteristic
327C2×C3#163711yescharacteristic
328C2×C3#164811yescharacteristic
329C2×C3#165411yescharacteristic
330C2×C3#166311yescharacteristic
331C22#521811yescharacteristic
332C22#522811yescharacteristic
333C22#523811yescharacteristic
334C22#524411yescharacteristic
335C22#524511yescharacteristic
336C22#524611yescharacteristic
337C22#524611yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—42no—
345C4—42no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—242no—
351C3#777211yescharacteristic, complemented, Sylow
352C2#7188011yescharacteristic
353C2#7188811yescharacteristic
354C2#7190411yescharacteristic
355C2#7191311yescharacteristic
356C2#7191511yescharacteristic
357C2#7193711yescharacteristic
358C2#7196111yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2726C2
socle, prime radicalC23×C3#120
hypercenter#2726D3
hypocenterC3#7772
LCSG ▷ C2×C3×C42 ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2726

Complements

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

NH ≅ G/Ntype
351. C3#7772semidirect

Maximal subgroups

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

subgroupindex
C3×#27262
#696332
#696052
#696042
#676382
#77723