Finite Group Databasebeta

← #10698136

Subgroups — #10698136

Order 768.

574 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 87 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#82179C211yesmaximal, characteristic
2#82178C211yesmaximal, characteristic
3#78646C211yesmaximal, characteristic
4C3×#1821C211yesmaximal, characteristic
5#69980C211yesmaximal, characteristic
6#69980C211yesmaximal, characteristic
7#67683C211yesmaximal, characteristic
8#10335—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C3×#407C2211yescharacteristic
14#5211—22no—
15#5210—22no—
16C4×#217C2211yescharacteristic
17C4×#217C2211yescharacteristic
18#5137—22no—
19#4985—22no—
20#4983—22no—
21#4977—21no—
22#4975—21no—
23#4973—21no—
24#4973—21no—
25#4739—21no—
26#4739—21no—
27#4721—21no—
28#4721—21no—
29C3×#343—22no—
30#4467—22no—
31#4467—22no—
32#3195—31no—
33#2503—31no—
34#2502—31no—
35#1821D311yescharacteristic
36#1508—31no—
37#1508—31no—
38#1297—31no—
39C23×C3×C4D412yes—
40C23×C3×C4C2311yescharacteristic
41C22×Dic6D411yescharacteristic
42C22×Dic6D411yescharacteristic
43C2×C3×M16—21no—
44C2×C3×M16D412yes—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×#217—21no—
52C2×#217—21no—
53C2×#217—21no—
54C2×#217—21no—
55C2×C4⋊Dic3D411yescharacteristic
56C2×C4⋊Dic3D411yescharacteristic
57#924—21no—
58#924—21no—
59#924—21no—
60#924—21no—
61#924—21no—
62#924—21no—
63#924—21no—
64#924—21no—
65C2×Dic3⋊C4—21no—
66C2×Dic3⋊C4—21no—
67C2×Dic3⋊C4—24no—
68C2×C4×Dic3—21no—
69C2×C4×Dic3—21no—
70C2×Dic12—22no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75C3⋊#95—24no—
76C3⋊#95—24no—
77C3⋊#95—24no—
78C3⋊#95D412yes—
79C3⋊#95D412yes—
80#817—22no—
81#816—22no—
82#816—22no—
83C2×C22⋊Q8—31no—
84C2×C22⋊Q8—31no—
85C22×C4⋊C4C2×D311yescharacteristic
86#483—61no—
87#483—61no—
88#470—62no—
89#451—61no—
90#451—61no—
91#407C2×D311yescharacteristic
92#407C2×D311yescharacteristic
93#400—62no—
94#398—62no—
95#391—61no—
96#391—61no—
97#385—62no—
98#384—61no—
99#382—61no—
100#381—62no—
101C4×C22⋊C4—31no—
102C4×C22⋊C4—31no—
103#343—22no—
104#328—62no—
105#328—62no—
106C24×C3C2×D411yescharacteristic
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—42no—
111C22×C3×C4—44no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C22×Dic3—22no—
119C22×Dic3—22no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6—42no—
129C2×Dic6—42no—
130C3×M16—42no—
131C3×M16—42no—
132C2×C3×C8—22no—
133C2×C3×C8—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138#217—24no—
139#217—42no—
140#217—41no—
141#217—41no—
142#217—41no—
143#217—41no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152Dic3⋊C4—84no—
153C4×Dic3—42no—
154C4×Dic3—42no—
155Dic12—82no—
156C22×Q8—31no—
157C22×Q8—31no—
158C23×C4D1212yes—
159C23×C4C22×D311yescharacteristic
160C2×Q16—62no—
161C2×M16—21no—
162C2×M16#6712yes—
163C22⋊Q8—61no—
164C22⋊Q8—61no—
165C22⋊Q8—61no—
166C22⋊Q8—61no—
167C22⋊Q8—61no—
168C22⋊Q8—61no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—64no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C42—61no—
187C2×C42—61no—
188#106—62no—
189#103—62no—
190#103—62no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195#95—32no—
196#95—32no—
197#95—64no—
198#95—64no—
199#95—64no—
200C23×C3—22no—
201C23×C3—22no—
202C23×C3—41no—
203C23×C3#12011yescharacteristic
204C23×C3#12011yescharacteristic
205C23×C3#12111yescharacteristic
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—44no—
213C2×C3×C4—42no—
214C2×C3×C4—44no—
215C2×C3×C4—82no—
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12711yescharacteristic
220C2×Dic3—24no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—42no—
234Dic6—42no—
235Dic6—42no—
236Dic6—81no—
237Dic6—81no—
238C3×C8—42no—
239C3×C8—81no—
240C24C2×D1211yescharacteristic
241C2×Q8—61no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×Q8—61no—
245C2×Q8—122no—
246C2×Q8—122no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—62no—
253C22×C4—62no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4#23511yescharacteristic
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261C22×C4D3×D411yescharacteristic
262C22×C4D3×D411yescharacteristic
263C22×C4C2×#6711yescharacteristic
264Q16—242no—
265M16—42no—
266M16—42no—
267C2×C8—22no—
268C2×C8—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C4⋊C4—121no—
280C4⋊C4—121no—
281C4⋊C4—244no—
282C22⋊C4—64no—
283C22⋊C4—122no—
284C22⋊C4—121no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C42—122no—
289C42—122no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—44no—
298C22×C3#39211yescharacteristic
299C22×C3#45512yes—
300C22×C3#45612yes—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—82no—
305C3×C4#45812yes—
306Dic3—44no—
307Dic3—82no—
308Dic3—81no—
309Dic3—81no—
310Dic3—81no—
311Dic3—81no—
312C23—22no—
313C23—22no—
314C23—41no—
315C23#88211yescharacteristic
316C23#88211yescharacteristic
317C23#93011yescharacteristic
318Q8—122no—
319Q8—122no—
320Q8—241no—
321Q8—241no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—64no—
332C2×C4—82no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—122no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4#88411yescharacteristic
347C2×C4#88411yescharacteristic
348C2×C4#93711yescharacteristic
349C2×C4#94011yescharacteristic
350C8—42no—
351C8—81no—
352C2×C3—22no—
353C2×C3—22no—
354C2×C3—41no—
355C2×C3#200411yescharacteristic
356C2×C3#200511yescharacteristic
357C2×C3#202211yescharacteristic
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—44no—
366C22#474612yes—
367C22#475212yes—
368C22#496011yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—82no—
373C4—124no—
374C4—242no—
375C4—241no—
376C4—241no—
377C4—241no—
378C4—241no—
379C4#505312yes—
380C3#1033511yescharacteristic, complemented, Sylow
381C2—22no—
382C2—22no—
383C2—41no—
384C2#6953911yescharacteristic
385C2#7008411yescharacteristic
386C2#7011911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1821C2
socleC22×C3#392
prime radicalC24×C3C2×D4
hypercenter#1821D3
hypocenterC3#10335
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
380. C3#10335semidirect

Maximal subgroups

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

subgroupindex
#821792
#821782
#786462
C3×#18212
#699802
#676832
#103353