Finite Group Databasebeta

← #10684136

Subgroups — #10684136

Order 768.

380 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 79 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#76187C211yesmaximal, characteristic
2#69605C211yesmaximal, characteristic
3#68632C211yesmaximal, characteristic
4#68631C211yesmaximal, characteristic
5C3×#1271C211yesmaximal, characteristic
6#67478C211yesmaximal, characteristic
7#67477C211yesmaximal, characteristic
8#7775—31nomaximal, Sylow
9C2×#869C2211yescharacteristic
10C3×#389C2211yescharacteristic
11C2×#818—21no—
12C2×#816—21no—
13C2×#816—21no—
14#4986—21no—
15#4985—21no—
16#4980—21no—
17#4978—21no—
18C2×#808—21no—
19#4928C2211yescharacteristic
20#4928C2211yescharacteristic
21#4659—21no—
22#4655—21no—
23C3×#336C2211yescharacteristic
24C3×#336C2211yescharacteristic
25#4551—21no—
26#4529—21no—
27#4524C2211yescharacteristic
28#2810—31no—
29#2058—31no—
30#2057—31no—
31#2043—31no—
32#1271—31no—
33#1271—31no—
34#1271D311yescharacteristic
35C22×Dic6—21no—
36C22×C3×C8—21no—
37C22×C3×C8—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C42C2311yescharacteristic
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3D411yescharacteristic
45C2×C4⋊Dic3D411yescharacteristic
46C2×C4⋊Dic3D411yescharacteristic
47C2×C4⋊Dic3D411yescharacteristic
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×C4×Dic3—21no—
51C22×C3⋊C8—21no—
52#869—42no—
53C3×#95—21no—
54C3⋊#95—21no—
55C3⋊#95—21no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63#818—42no—
64#816—42no—
65#816—42no—
66#808—42no—
67C2×C4⋊Q8—31no—
68C2×#107—61no—
69C2×#106—61no—
70C2×#103—61no—
71C2×#103—61no—
72#400—61no—
73#400—61no—
74#400—61no—
75#389—31no—
76#389—31no—
77#389C2×D311yescharacteristic
78#387—61no—
79#387—61no—
80#387—61no—
81#340—61no—
82#340—61no—
83#336—31no—
84#336C2×D311yescharacteristic
85#336C2×D311yescharacteristic
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4C2×D411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C22×Dic3—21no—
92C22×Dic3—21no—
93C22×Dic3—21no—
94C22×Dic3—21no—
95C22×Dic3—41no—
96C2×Dic6—22no—
97C2×Dic6—41no—
98C2×Dic6—41no—
99C2×Dic6—41no—
100C2×Dic6—41no—
101C2×Dic6—41no—
102C2×C3×C8—22no—
103C2×C3×C8—41no—
104C2×C3×C8—41no—
105C2×C3×C8—41no—
106C2×C3×C8—41no—
107C2×C3×C8—41no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C42—41no—
114C4⋊Dic3—22no—
115C4⋊Dic3—22no—
116C4⋊Dic3—21no—
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3—41no—
121C4⋊Dic3—41no—
122C4⋊Dic3—41no—
123C4⋊Dic3—41no—
124Dic3⋊C4—41no—
125Dic3⋊C4—41no—
126Dic3⋊C4—41no—
127Dic3⋊C4—41no—
128C4×Dic3—81no—
129C2×C3⋊C8—22no—
130C2×C3⋊C8—41no—
131C2×C3⋊C8—41no—
132C22×Q8—61no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—61no—
136C4⋊Q8—122no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4D1211yescharacteristic
147C2×C4⋊C4D1211yescharacteristic
148C2×C42—61no—
149C2×C42C22×D311yescharacteristic
150#107—122no—
151#106—122no—
152#103—122no—
153#103—122no—
154#95—21no—
155#95—61no—
156#95—61no—
157#95—61no—
158#95—61no—
159#95—61no—
160#95—61no—
161#95—61no—
162#95—61no—
163#95—61no—
164C23×C3#12011yescharacteristic
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4#13511yescharacteristic
178C2×C3×C4#13511yescharacteristic
179C2×C3×C4#13511yescharacteristic
180C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
181C2×C3×C4#13711yescharacteristic
182C2×C3×C4#13711yescharacteristic
183C2×Dic3—41no—
184C2×Dic3—41no—
185C2×Dic3—41no—
186C2×Dic3—41no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—42no—
196C2×Dic3—81no—
197C2×Dic3—81no—
198Dic6—42no—
199Dic6—42no—
200Dic6—81no—
201Dic6—81no—
202C3×C8—42no—
203C3×C8—81no—
204C3⋊C8—42no—
205C2×Q8—62no—
206C2×Q8—121no—
207C2×Q8—121no—
208C2×Q8—121no—
209C2×Q8—121no—
210C2×Q8—121no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—121no—
218C22×C4C2×D1211yescharacteristic
219C22×C4D3×D411yescharacteristic
220C22×C4D3×D411yescharacteristic
221C2×C8—22no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—62no—
228C2×C8—121no—
229C2×C8—121no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—62no—
236C4⋊C4—62no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C42—41no—
250C42—241no—
251C22×C3#44911yescharacteristic
252C22×C3#45211yescharacteristic
253C22×C3#45211yescharacteristic
254C22×C3#45411yescharacteristic
255C22×C3#45611yescharacteristic
256C22×C3#45611yescharacteristic
257C22×C3B(4,𝔽2)11yescharacteristic
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—81no—
265C3×C4—81no—
266Dic3—82no—
267Dic3—81no—
268Dic3—81no—
269Dic3—81no—
270Dic3—81no—
271C23#88211yescharacteristic
272Q8—122no—
273Q8—122no—
274Q8—241no—
275Q8—241no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4—122no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4—241no—
302C2×C4—241no—
303C2×C4#90411yescharacteristic
304C2×C4#90911yescharacteristic
305C2×C4#91211yescharacteristic
306C2×C4#91411yescharacteristic
307C2×C4#91511yescharacteristic
308C2×C4#91911yescharacteristic
309C8—42no—
310C8—81no—
311C8—122no—
312C2×C3#159711yescharacteristic
313C2×C3#160311yescharacteristic
314C2×C3#160311yescharacteristic
315C2×C3#163811yescharacteristic
316C2×C3#165011yescharacteristic
317C2×C3#165211yescharacteristic
318C2×C3#166311yescharacteristic
319C22#473311yescharacteristic
320C22#474111yescharacteristic
321C22#475211yescharacteristic
322C22#478311yescharacteristic
323C22#481011yescharacteristic
324C22#482611yescharacteristic
325C22#482711yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—81no—
333C4—81no—
334C4—241no—
335C4—241no—
336C4—241no—
337C4—241no—
338C4—242no—
339C3#777511yescharacteristic, complemented, Sylow
340C2#6813211yescharacteristic
341C2#6820511yescharacteristic
342C2#6825011yescharacteristic
343C2#6829411yescharacteristic
344C2#6832411yescharacteristic
345C2#6837611yescharacteristic
346C2#6846211yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1271C2
socle, prime radicalC23×C3#120
hypercenter#1271D3
hypocenterC3#7775
LCSG ▷ C2×C3×C42 ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
339. C3#7775semidirect

Maximal subgroups

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

subgroupindex
#761872
#696052
#686322
#686312
C3×#12712
#674782
#674772
#77753