Finite Group Databasebeta

← #10684145

Subgroups — #10684145

Order 768.

392 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 81 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#76181C211yesmaximal, characteristic
2C3×#2043C211yesmaximal, characteristic
3#71794C211yesmaximal, characteristic
4#71782C211yesmaximal, characteristic
5#67638C211yesmaximal, characteristic
6#67480C211yesmaximal, characteristic
7#67478C211yesmaximal, characteristic
8#7777—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#103—21no—
11C2×C3×#103—21no—
12C3×#389C2211yescharacteristic
13#4985—21no—
14#4980—21no—
15#4979—21no—
16#4976—21no—
17C2×#810—21no—
18C2×#807—21no—
19#4929C2211yescharacteristic
20#4928C2211yescharacteristic
21#4660—21no—
22#4653—21no—
23C3×#336C2211yescharacteristic
24#4529—21no—
25#4529—21no—
26#4524C2211yescharacteristic
27#4524C2211yescharacteristic
28#2805—31no—
29#2057—31no—
30#2043D311yescharacteristic
31#2024—31no—
32#1272—31no—
33#1271—31no—
34#1271—31no—
35C22×C3×Q8—21no—
36C22×Dic6D411yescharacteristic
37C22×C3×C8—21no—
38C3×C4⋊Q8—42no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C42C2311yescharacteristic
45C2×C4⋊Dic3D411yescharacteristic
46C2×C4⋊Dic3D411yescharacteristic
47C2×C4⋊Dic3D411yescharacteristic
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3—21no—
52C22×C3⋊C8—21no—
53C22×C3⋊C8—21no—
54C3×#103—42no—
55C3×#103—42no—
56C3×#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65#810—42no—
66#807—42no—
67C2×C4⋊Q8C2×D311yescharacteristic
68C2×#107—61no—
69C2×#103—21no—
70C2×#103—21no—
71C2×#103—61no—
72#400—61no—
73#400—61no—
74#395—61no—
75#391—31no—
76#389—31no—
77#389—61no—
78#389C2×D311yescharacteristic
79#387—61no—
80#387—61no—
81#340—61no—
82#340—61no—
83#336—31no—
84#336—31no—
85#336C2×D311yescharacteristic
86C2×C3×Q8—22no—
87C2×C3×Q8—41no—
88C2×C3×Q8—41no—
89C2×C3×Q8—41no—
90C2×C3×Q8—41no—
91C2×C3×Q8—41no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—41no—
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C2×Dic6—21no—
103C2×Dic6—21no—
104C2×Dic6—41no—
105C2×Dic6—41no—
106C2×C3×C8—41no—
107C2×C3×C8—41no—
108C2×C3×C8—41no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C42—41no—
119C4⋊Dic3—21no—
120C4⋊Dic3—41no—
121C4⋊Dic3—41no—
122C4⋊Dic3Q1612yes—
123Dic3⋊C4—41no—
124Dic3⋊C4—41no—
125Dic3⋊C4—81no—
126Dic3⋊C4—81no—
127C4×Dic3—81no—
128C2×C3⋊C8—22no—
129C2×C3⋊C8—41no—
130C2×C3⋊C8—41no—
131C2×C3⋊C8—41no—
132C2×C3⋊C8—41no—
133C2×C3⋊C8—41no—
134C22×Q8—21no—
135C22×Q8—31no—
136C22×C8—21no—
137C22×C8—61no—
138C22×C8—61no—
139C4⋊Q8—42no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4#6711yescharacteristic
150C2×C4⋊C4#6711yescharacteristic
151C2×C42—61no—
152C2×C42C22×D311yescharacteristic
153#107—122no—
154#103—42no—
155#103—42no—
156#103—122no—
157#95—21no—
158#95—61no—
159#95—61no—
160#95—61no—
161#95—61no—
162#95—61no—
163#95—61no—
164#95—61no—
165#95—61no—
166C23×C3#12011yescharacteristic
167C3×Q8—42no—
168C3×Q8—42no—
169C3×Q8—81no—
170C3×Q8—81no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—42no—
184C2×C3×C4—81no—
185C2×C3×C4—81no—
186C2×C3×C4C2×Q1611yescharacteristic
187C2×C3×C4#13511yescharacteristic
188C2×C3×C4#13511yescharacteristic
189C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
190C2×C3×C4#13711yescharacteristic
191C2×C3×C4#13711yescharacteristic
192C2×Dic3—22no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204Dic6—42no—
205Dic6—81no—
206C3×C8—81no—
207C3⋊C8—42no—
208C3⋊C8—81no—
209C2×Q8—22no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—61no—
216C2×Q8—61no—
217C2×Q8—121no—
218C2×Q8—121no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—41no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4D3×D411yescharacteristic
227C22×C4D3×D411yescharacteristic
228C22×C4C2×#6711yescharacteristic
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—62no—
233C2×C8—121no—
234C2×C8—121no—
235C2×C8—121no—
236C2×C8—121no—
237C2×C8—121no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—32no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—61no—
249C4⋊C4—121no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C4⋊C4—121no—
253C4⋊C4—241no—
254C4⋊C4—241no—
255C42—41no—
256C42—241no—
257C22×C3#44911yescharacteristic
258C22×C3#45111yescharacteristic
259C22×C3#45211yescharacteristic
260C22×C3#45411yescharacteristic
261C22×C3#45511yescharacteristic
262C22×C3#45611yescharacteristic
263C22×C3B(4,𝔽2)11yescharacteristic
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—82no—
271C3×C4—81no—
272C3×C4—81no—
273Dic3—42no—
274Dic3—81no—
275Dic3—81no—
276Dic3—81no—
277C23#93711yescharacteristic
278Q8—42no—
279Q8—42no—
280Q8—81no—
281Q8—81no—
282Q8—122no—
283Q8—241no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—62no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4#91511yescharacteristic
312C2×C4#91611yescharacteristic
313C2×C4D3×Q1611yescharacteristic
314C2×C4#91811yescharacteristic
315C2×C4#93211yescharacteristic
316C2×C4#95011yescharacteristic
317C8—81no—
318C8—122no—
319C8—241no—
320C2×C3#159711yescharacteristic
321C2×C3#160511yescharacteristic
322C2×C3#160611yescharacteristic
323C2×C3#163611yescharacteristic
324C2×C3#164411yescharacteristic
325C2×C3#165011yescharacteristic
326C2×C3#166411yescharacteristic
327C22#503211yescharacteristic
328C22#503711yescharacteristic
329C22#506211yescharacteristic
330C22#517311yescharacteristic
331C22#518611yescharacteristic
332C22#520211yescharacteristic
333C22#520511yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—81no—
341C4—82no—
342C4—81no—
343C4—122no—
344C4—241no—
345C4—241no—
346C4—241no—
347C3#777711yescharacteristic, complemented, Sylow
348C2#6970211yescharacteristic
349C2#7097311yescharacteristic
350C2#7098311yescharacteristic
351C2#7104911yescharacteristic
352C2#7106211yescharacteristic
353C2#7123611yescharacteristic
354C2#7125311yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2043C2
socle, prime radicalC23×C3#120
hypercenter#2043D3
hypocenterC3#7777
LCSG ▷ C2×C3×C42 ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2043

Complements

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

NH ≅ G/Ntype
347. C3#7777semidirect

Maximal subgroups

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

subgroupindex
#761812
C3×#20432
#717942
#717822
#676382
#674802
#674782
#77773