Finite Group Databasebeta

← #10695908

Subgroups — #10695908

Order 768.

510 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 89 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×#3195C211yesmaximal, characteristic
2C2×#5044C211yesmaximal, characteristic
3C2×#5043C211yesmaximal, characteristic
4#71691C211yesmaximal, characteristic
5#71691C211yesmaximal, characteristic
6#69973C211yesmaximal, characteristic
7#67688C211yesmaximal, characteristic
8#9995—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C22×C4⋊Dic3C2211yescharacteristic
12C3×#483—21no—
13C3×#483—21no—
14C3×#470—22no—
15C3×#451—21no—
16C3×#451—21no—
17C3×#407C2211yescharacteristic
18C2×#835—21no—
19C2×#835—21no—
20#5207C2211yescharacteristic
21C2×#830C2211yescharacteristic
22C2×#830C2211yescharacteristic
23#5044—22no—
24#5044—22no—
25#5043—22no—
26#5043—22no—
27C2×#808—21no—
28C2×#807—21no—
29#4554—22no—
30#4528—22no—
31#4528—22no—
32#3195D311yescharacteristic
33C2×#484—31no—
34C2×#483—31no—
35#1821—31no—
36#1472—31no—
37#1472—31no—
38#1298—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×Q8D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C23×Dic3—21no—
43C2×C3×Q16—22no—
44C2×C3×M16D412yes—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3D412yes—
65C2×#208—21no—
66C22×C3⋊C8—21no—
67C22×C3⋊C8—21no—
68C3×#106—22no—
69C3×#103—22no—
70C3×#103—22no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73#835—24no—
74#835—24no—
75#830—21no—
76#830—21no—
77#830—21no—
78#830—21no—
79#830D412yes—
80#830D412yes—
81#808—24no—
82#807—24no—
83C2×C22⋊Q8C2×D311yescharacteristic
84C2×C22⋊Q8C2×D311yescharacteristic
85C22×C4⋊C4—31no—
86#484—62no—
87#484—62no—
88#483—21no—
89#483—21no—
90#483—62no—
91#483—62no—
92#470—22no—
93#451—21no—
94#451—21no—
95C2×#107—61no—
96C2×#106—61no—
97C2×#103—61no—
98C2×#103—61no—
99#407—31no—
100#407C2×D311yescharacteristic
101C2×C22⋊C8—31no—
102C2×C22⋊C8—31no—
103#343—62no—
104#324—62no—
105#324—62no—
106C24×C3#4111yescharacteristic
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—42no—
112C2×C3×Q8—42no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—42no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×Dic3—22no—
125C22×Dic3—41no—
126C22×Dic3—41no—
127C22×Dic3—82no—
128C3×Q16—82no—
129C3×M16—42no—
130C2×C3×C8—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C4⋊Dic3—22no—
144C4⋊Dic3—22no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147#208—42no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—42no—
151C2×C3⋊C8—42no—
152C2×C3⋊C8—41no—
153C22×Q8#6711yescharacteristic
154C22×Q8#6711yescharacteristic
155C23×C4—61no—
156C23×C4C22×D311yescharacteristic
157C2×Q16—22no—
158C2×M16—61no—
159C2×M16#6712yes—
160C22×C8—61no—
161C22×C8—61no—
162C22⋊Q8—21no—
163C22⋊Q8—21no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167C22⋊Q8—21no—
168C22⋊Q8—21no—
169C22⋊Q8—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—32no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4#6711yescharacteristic
179C2×C4⋊C4#6711yescharacteristic
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182#107—64no—
183#106—22no—
184#106—64no—
185#103—22no—
186#103—22no—
187#103—64no—
188#103—64no—
189C22⋊C8—21no—
190C22⋊C8—21no—
191C22⋊C8—32no—
192C22⋊C8—32no—
193C22⋊C8—61no—
194C22⋊C8—61no—
195C22⋊C8—61no—
196C22⋊C8—61no—
197C23×C3—22no—
198C23×C3—22no—
199C23×C3—41no—
200C23×C3#12111yescharacteristic
201C23×C3#12111yescharacteristic
202C23×C3#12111yescharacteristic
203C3×Q8—42no—
204C3×Q8—42no—
205C3×Q8—42no—
206C3×Q8—42no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—81no—
220C2×C3×C4—81no—
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12711yescharacteristic
225C2×Dic3—42no—
226C2×Dic3—82no—
227C2×Dic3—82no—
228C2×Dic3—82no—
229C3×C8—42no—
230C3⋊C8—42no—
231C3⋊C8—42no—
232C3⋊C8—81no—
233C24#23711yescharacteristic
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—42no—
239C2×Q8—42no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—42no—
245C22×C4—62no—
246C22×C4—121no—
247C22×C4—121no—
248C22×C4—242no—
249C22×C4D3×D411yescharacteristic
250C22×C4D3×D411yescharacteristic
251C22×C4D3×D411yescharacteristic
252C22×C4C2×#6711yescharacteristic
253C22×C4C2×#6711yescharacteristic
254C22×C4C2×#6711yescharacteristic
255Q16—82no—
256M16—42no—
257M16—122no—
258C2×C8—22no—
259C2×C8—62no—
260C2×C8—62no—
261C2×C8—122no—
262C2×C8—122no—
263C2×C8—121no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—62no—
273C4⋊C4—62no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—44no—
288C22×C3#39211yescharacteristic
289C22×C3#47112yes—
290C22×C3#47312yes—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—81no—
295C3×C4—81no—
296C3×C4—81no—
297C3×C4—81no—
298C3×C4#45812yes—
299Dic3—82no—
300C23—22no—
301C23—22no—
302C23—41no—
303C23#93811yescharacteristic
304C23#93911yescharacteristic
305C23#93911yescharacteristic
306Q8—42no—
307Q8—42no—
308Q8—42no—
309Q8—42no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—122no—
325C2×C4—242no—
326C2×C4—242no—
327C2×C4—242no—
328C2×C4#93711yescharacteristic
329C2×C4#93711yescharacteristic
330C2×C4#94011yescharacteristic
331C2×C4#95311yescharacteristic
332C8—42no—
333C8—122no—
334C8—122no—
335C8—241no—
336C2×C3—22no—
337C2×C3—22no—
338C2×C3—41no—
339C2×C3#201911yescharacteristic
340C2×C3#202011yescharacteristic
341C2×C3#202211yescharacteristic
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—44no—
350C22#520312yes—
351C22#520412yes—
352C22#522311yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—81no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4—242no—
361C4#524312yes—
362C3#999511yescharacteristic, complemented, Sylow
363C2—22no—
364C2—22no—
365C2—41no—
366C2#7184911yescharacteristic
367C2#7210711yescharacteristic
368C2#7210911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3195C2
socleC22×C3#392
prime radicalC24×C3#41
hypercenter#3195D3
hypocenterC3#9995
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3195

Complements

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

NH ≅ G/Ntype
362. C3#9995semidirect

Maximal subgroups

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

subgroupindex
C3×#31952
C2×#50442
C2×#50432
#716912
#699732
#676882
#99953