Finite Group Databasebeta

← #10697461

Subgroups — #10697461

Order 768.

462 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 105 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×#3162C211yesmaximal, characteristic
2#83370C211yesmaximal, characteristic
3#83324C211yesmaximal, characteristic
4#71838C211yesmaximal, characteristic
5#71838C211yesmaximal, characteristic
6#69828C211yesmaximal, characteristic
7#67661C211yesmaximal, characteristic
8#10224—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#533C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#501—21no—
13C3×#501—21no—
14C3×#501—21no—
15C3×#501—21no—
16C3×#427—22no—
17C2×C3×#97C2211yescharacteristic
18#5093—21no—
19#5092—21no—
20#5079—21no—
21#5078—21no—
22#5050—22no—
23#5044—22no—
24#5031—22no—
25#5021—22no—
26#4936C2211yescharacteristic
27#4925C2211yescharacteristic
28#4925C2211yescharacteristic
29#4551—22no—
30#4538—22no—
31#4538—22no—
32#3230—31no—
33#3209—31no—
34#3162D311yescharacteristic
35#1993—31no—
36#1518—31no—
37#1518—31no—
38#1281—31no—
39C22×Dic6—21no—
40C22×C3×C8D412yes—
41C3×C4⋊Q8D411yescharacteristic
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×#117—21no—
56C3×#117—21no—
57C2×C3×C42C2311yescharacteristic
58C2×#216—24no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869D412yes—
66C3×#107—22no—
67C3×#107—22no—
68C3×#106—22no—
69C3×#106—22no—
70C3×#97—21no—
71C3×#97—21no—
72#835—24no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75#830D412yes—
76#830D412yes—
77#810—24no—
78#807—24no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84#533C2×D311yescharacteristic
85C2×C4⋊Q8—31no—
86#501—21no—
87#501—21no—
88#501—21no—
89#501—21no—
90#500—61no—
91#494—61no—
92#487—61no—
93#487—61no—
94#484—62no—
95#477—62no—
96#462—62no—
97#451—62no—
98#431—31no—
99#431—31no—
100#427—22no—
101C2×#97C2×D311yescharacteristic
102#382—31no—
103#340—62no—
104#328—62no—
105#328—62no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×Dic3—41no—
114C22×Dic3—41no—
115C2×Dic6—22no—
116C2×Dic6—41no—
117C2×Dic6—82no—
118C2×C3×C8—22no—
119C2×C3×C8—22no—
120C2×C3×C8Q1614yescomplemented
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C42—41no—
136C3×C42—41no—
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42#4111yescharacteristic
141#216—84no—
142C4⋊Dic3—22no—
143C4⋊Dic3—22no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146C4⋊Dic3—42no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—22no—
149C22×Q8—61no—
150C2×Q16—64no—
151C22×C8#6712yes—
152C4⋊Q8—32no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—61no—
156C4⋊Q8#6711yescharacteristic
157C4⋊Q8#6711yescharacteristic
158C4⋊Q8#6711yescharacteristic
159C4⋊Q8#6711yescharacteristic
160#125—21no—
161#125—21no—
162C22⋊Q8—21no—
163C22⋊Q8—21no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170#117—21no—
171#117—21no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C42C22×D311yescharacteristic
176#107—22no—
177#107—22no—
178#107—64no—
179#106—22no—
180#106—22no—
181C4⋊C8—61no—
182C4⋊C8—61no—
183#103—64no—
184#103—64no—
185C22⋊C8—32no—
186C22⋊C8—32no—
187#97—21no—
188#97—21no—
189C4×C8—61no—
190C4×C8—61no—
191#95—61no—
192#95—61no—
193C23×C3#12711yescharacteristic
194C3×Q8—42no—
195C3×Q8—42no—
196C3×Q8—81no—
197C3×Q8—81no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4C2×Q1612yes—
211C2×C3×C4#13712yes—
212C2×Dic3—42no—
213C2×Dic3—82no—
214C2×Dic3—81no—
215C2×Dic3—81no—
216C2×Dic3—81no—
217Dic6—44no—
218Dic6—82no—
219C3×C8—24no—
220C3⋊C8—42no—
221C3⋊C8—42no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—62no—
227C2×Q8—121no—
228C2×Q8—242no—
229C22×C4—121no—
230C22×C4—121no—
231C22×C4D3×D411yescharacteristic
232C22×C4D3×D411yescharacteristic
233C22×C4C2×#6711yescharacteristic
234Q16—244no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—62no—
238C2×C8—62no—
239C2×C8#21614yescomplemented
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—62no—
253C4⋊C4—62no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4—122no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C42—41no—
260C42—41no—
261C42D3×D411yescharacteristic
262C42#23711yescharacteristic
263C42C2×#6711yescharacteristic
264C42C2×#6711yescharacteristic
265C22×C3—21no—
266C22×C3—21no—
267C22×C3#39211yescharacteristic
268C22×C3#49411yescharacteristic
269C22×C3#49711yescharacteristic
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—81no—
275C3×C4—81no—
276C3×C4—81no—
277C3×C4—81no—
278C3×C4#45112yes—
279C3×C4#46212yes—
280Dic3—82no—
281Dic3—161no—
282C23#94011yescharacteristic
283Q8—42no—
284Q8—42no—
285Q8—81no—
286Q8—81no—
287Q8—124no—
288Q8—242no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—122no—
296C2×C4—242no—
297C2×C4—241no—
298C2×C4—241no—
299C2×C4—241no—
300C2×C4#93711yescharacteristic
301C2×C4#93711yescharacteristic
302C2×C4#93811yescharacteristic
303C2×C4#93911yescharacteristic
304C2×C4#93911yescharacteristic
305C2×C4#94212yes—
306C2×C4C2×#21612yes—
307C2×C4#95311yescharacteristic
308C8—24no—
309C8—122no—
310C8—122no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3#200811yescharacteristic
314C2×C3#200911yescharacteristic
315C2×C3#201611yescharacteristic
316C22—21no—
317C22—21no—
318C22#508611yescharacteristic
319C22#509211yescharacteristic
320C22#522311yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—81no—
326C4—81no—
327C4—81no—
328C4—81no—
329C4—242no—
330C4—481no—
331C4#509012yes—
332C4#522812yes—
333C3#1022411yescharacteristic, complemented, Sylow
334C2—21no—
335C2—21no—
336C2#7179311yescharacteristic
337C2#7211911yescharacteristic
338C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3162C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#3162D3
hypocenterC3#10224
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
120. C2×C3×C8Q16semidirect
239. C2×C8#216semidirect
333. C3#10224semidirect

Maximal subgroups

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

subgroupindex
C3×#31622
#833702
#833242
#718382
#698282
#676612
#102243