Finite Group Databasebeta

← #10695910

Subgroups — #10695910

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
1C2×C3×#484C211yesmaximal, characteristic
2#82742C211yesmaximal, characteristic
3C2×#4971C211yesmaximal, characteristic
4#71691C211yesmaximal, characteristic
5#71562C211yesmaximal, characteristic
6#69923C211yesmaximal, characteristic
7#67680C211yesmaximal, characteristic
8#9995—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C3×#484—22no—
13C3×#484—22no—
14C2×C3×#107—21no—
15C2×C3×#103—21no—
16C2×C3×C22⋊C8C2211yescharacteristic
17#5228—21no—
18#5207C2211yescharacteristic
19#5207C2211yescharacteristic
20C2×#830C2211yescharacteristic
21#5052—22no—
22#5050—21no—
23#5043—21no—
24#4971—21no—
25#4971—22no—
26#4971—22no—
27C2×#810—21no—
28C2×#808—21no—
29#4530—22no—
30#4528—22no—
31#4469—22no—
32#3195—31no—
33C2×#484D311yescharacteristic
34C2×#483—31no—
35#1821—31no—
36#1472—31no—
37#1472—31no—
38#1298—31no—
39C22×C3×Q8D411yescharacteristic
40C23×C3×C4—21no—
41C23×C3×C4C2311yescharacteristic
42C22×Dic6D411yescharacteristic
43C22×C3×C8—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C4⋊C4D412yes—
56C2×C3×C22⋊C4—21no—
57C2×#217—21no—
58C2×#216—22no—
59C2×C4⋊Dic3D411yescharacteristic
60#924—21no—
61#924—21no—
62#924—21no—
63#924—21no—
64C2×Dic3⋊C4—21no—
65C2×#208—21no—
66C2×#208D412yes—
67C22×C3⋊C8—21no—
68C3×#107—24no—
69C3×#103—24no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8D412yes—
72#835—22no—
73#830—21no—
74#830—21no—
75#830—21no—
76#830—21no—
77#830—21no—
78#830D412yes—
79#810—24no—
80#810—22no—
81#808—24no—
82#808—22no—
83C2×C22⋊Q8—31no—
84C2×C22⋊Q8C2×D311yescharacteristic
85C22×C4⋊C4C2×D311yescharacteristic
86#484—22no—
87#484—22no—
88#483—62no—
89#483—61no—
90#483—62no—
91#483—61no—
92#470—62no—
93#451—61no—
94#451—61no—
95C2×#107—21no—
96C2×#106—61no—
97C2×#103—21no—
98C2×#103—61no—
99#407—31no—
100#407—31no—
101C2×C22⋊C8—31no—
102C2×C22⋊C8C2×D311yescharacteristic
103#343—62no—
104#324—62no—
105#324—62no—
106C24×C3#4111yescharacteristic
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—42no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—22no—
113C22×C3×C4—42no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—82no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×Dic3—21no—
124C22×Dic3—21no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—42no—
128C2×C3×C8—22no—
129C2×C3×C8—42no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140#217—41no—
141#217—41no—
142#216—82no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147#208—42no—
148#208—42no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—42no—
152C2×C3⋊C8—41no—
153C22×Q8—31no—
154C22×Q8#6711yescharacteristic
155C23×C4—21no—
156C23×C4C22×D311yescharacteristic
157C2×Q16—62no—
158C2×M16—32no—
159C2×M16—61no—
160C22×C8—21no—
161C22×C8—61no—
162C22⋊Q8—21no—
163C22⋊Q8—21no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166C22⋊Q8—61no—
167C22⋊Q8—61no—
168C22⋊Q8—61no—
169C22⋊Q8—61no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4#6712yes—
179C2×C4⋊C4#6711yescharacteristic
180C2×C22⋊C4—21no—
181C2×C22⋊C4—61no—
182#107—24no—
183#106—64no—
184#106—62no—
185#103—24no—
186#103—64no—
187#103—62no—
188#103—62no—
189C22⋊C8—21no—
190C22⋊C8—32no—
191C22⋊C8—61no—
192C22⋊C8—61no—
193C22⋊C8—61no—
194C22⋊C8—61no—
195C22⋊C8—61no—
196C22⋊C8#6712yes—
197C23×C3—22no—
198C23×C3—22no—
199C23×C3—41no—
200C23×C3#12111yescharacteristic
201C23×C3#12111yescharacteristic
202C23×C3#12111yescharacteristic
203C3×Q8—42no—
204C3×Q8—42no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—82no—
216C2×C3×C4—82no—
217C2×C3×C4—82no—
218C2×C3×C4—81no—
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12711yescharacteristic
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—42no—
226C2×Dic3—81no—
227Dic6—42no—
228Dic6—42no—
229C3×C8—42no—
230C3⋊C8—42no—
231C3⋊C8—42no—
232C3⋊C8—81no—
233C24#23711yescharacteristic
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—42no—
237C2×Q8—61no—
238C2×Q8—61no—
239C2×Q8—122no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—82no—
249C22×C4D3×D411yescharacteristic
250C22×C4D3×D411yescharacteristic
251C22×C4D3×D411yescharacteristic
252C22×C4C2×#6711yescharacteristic
253C22×C4C2×#6711yescharacteristic
254C22×C4C2×#6711yescharacteristic
255Q16—242no—
256M16—122no—
257M16—122no—
258C2×C8—22no—
259C2×C8—42no—
260C2×C8—62no—
261C2×C8—62no—
262C2×C8—122no—
263C2×C8—121no—
264C4⋊C4—22no—
265C4⋊C4—21no—
266C4⋊C4—22no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—121no—
279C22⋊C4—121no—
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—82no—
296C3×C4—81no—
297C3×C4#45812yes—
298Dic3—81no—
299Dic3—81no—
300C23—22no—
301C23—22no—
302C23—41no—
303C23#93811yescharacteristic
304C23#93911yescharacteristic
305C23#93911yescharacteristic
306Q8—42no—
307Q8—42no—
308Q8—122no—
309Q8—122no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—81no—
324C2×C4—122no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—241no—
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#519212yes—
351C22#520012yes—
352C22#522311yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—81no—
357C4—81no—
358C4—82no—
359C4—241no—
360C4—241no—
361C4#505312yes—
362C3#999511yescharacteristic, complemented, Sylow
363C2—22no—
364C2—22no—
365C2—41no—
366C2#7170611yescharacteristic
367C2#7207411yescharacteristic
368C2#7209711yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#484C2
socleC22×C3#392
prime radicalC24×C3#41
hypercenterC2×#484D3
hypocenterC3#9995
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#484

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
C2×C3×#4842
#827422
C2×#49712
#716912
#715622
#699232
#676802
#99953