Finite Group Databasebeta

← #10693013

Subgroups — #10693013

Order 768.

614 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 93 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#82282C211yesmaximal, characteristic
2#73567C211yesmaximal, characteristic
3C3×#1826C211yesmaximal, characteristic, complemented
4#71590C211yesmaximal, characteristic, complemented
5#69967C211yesmaximal, characteristic
6#67682C211yesmaximal, characteristic, complemented
7#67616C211yesmaximal, characteristic, complemented
8#9481—31nomaximal, Sylow
9C2×C3×#117C2211yescharacteristic
10C22×(C2×D3)⋊C4C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×C4×C22⋊C4C2211yescharacteristic
13#5211C2211yescharacteristic
14C4×#217C2211yescharacteristic
15#5207C2211yescharacteristic
16#4993C412yescomplemented
17#4992C412yescomplemented
18#4955—21no—
19#4955—21no—
20#4946—21no—
21#4942—21no—
22#4940—22no—
23#4940—22no—
24#4669—22no—
25C3×#339—22no—
26#4532—22no—
27#4468—22no—
28#2482—31no—
29#2312—31no—
30#1826—31no—
31#1826D311yescharacteristic, complemented
32#1462—31no—
33#1284—31no—
34#1284—31no—
35C24×D3D411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C23×Dic3D411yescharacteristic
38C2×C3×M16—21no—
39C3×#117—21no—
40C3×#117—21no—
41C3×#117—21no—
42C3×#117—21no—
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C42—21no—
48C2×C3×C42D412yes—
49C2×#217—21no—
50C2×#217—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—22no—
56C2×(C2×D3)⋊C4—22no—
57C2×(C2×D3)⋊C4—22no—
58C2×(C2×D3)⋊C4—22no—
59C2×(C2×D3)⋊C4C2×C412yes—
60C2×(C2×D3)⋊C4D412yes—
61C2×Dic3⋊C4—22no—
62C2×Dic3⋊C4—22no—
63C2×C4×Dic3D412yes—
64C2×#208—21no—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×#95—21no—
68C3⋊#95—22no—
69C3⋊#95—21no—
70C3⋊#95—21no—
71C3⋊#95—22no—
72C3⋊#95C2×C412yes—
73#830—21no—
74#830—21no—
75C2×#117C2×D311yescharacteristic
76C22×C22⋊C4—31no—
77#407—31no—
78#407C2×D311yescharacteristic
79#399—32no—
80#397—32no—
81#388—62no—
82#388—62no—
83#388—62no—
84#385—61no—
85#381—31no—
86#380—61no—
87#379—61no—
88#379—61no—
89C4×C22⋊C4—31no—
90C4×C22⋊C4C2×D311yescharacteristic
91#339—22no—
92#339—62no—
93#323—62no—
94C24×C3C2×D411yescharacteristic
95C23×D3—21no—
96C23×D3—21no—
97C23×D3—42no—
98C23×D3—42no—
99C23×D3—81no—
100C23×D3C22⋊C412yes—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4—22no—
104C22×C3×C4—42no—
105C22×C3×C4C22×C411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—22no—
114C22×Dic3—22no—
115C22×Dic3—81no—
116C22×Dic3C22⋊C412yes—
117C3×M16—42no—
118C2×C3×C8—41no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C22⋊C4—42no—
124C3×C22⋊C4—41no—
125C3×C22⋊C4—41no—
126C3×C42—42no—
127C3×C42—41no—
128C3×C42—41no—
129#217—42no—
130#217—42no—
131(C2×D3)⋊C4—22no—
132(C2×D3)⋊C4—22no—
133(C2×D3)⋊C4—22no—
134(C2×D3)⋊C4—22no—
135(C2×D3)⋊C4—44no—
136(C2×D3)⋊C4—82no—
137(C2×D3)⋊C4—82no—
138(C2×D3)⋊C4—82no—
139Dic3⋊C4—82no—
140Dic3⋊C4—82no—
141C4×Dic3—42no—
142#208—42no—
143C2×C3⋊C8—41no—
144C25—31no—
145C23×C4—31no—
146C23×C4C22×D311yescharacteristic
147C2×M16—21no—
148C2×M16—61no—
149#117—21no—
150#117—21no—
151#117—21no—
152#117—21no—
153C2×C4⋊C4—62no—
154C2×C4⋊C4—62no—
155C2×C4⋊C4C4×D311yescharacteristic
156C2×C4⋊C4C4×D311yescharacteristic
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—32no—
160C2×C22⋊C4—32no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—62no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—62no—
169C2×C22⋊C4—62no—
170C2×C22⋊C4—62no—
171C2×C42—21no—
172C2×C42—32no—
173C2×C42#6712yes—
174C22⋊C8—21no—
175C22⋊C8—21no—
176C22⋊C8—61no—
177C22⋊C8—61no—
178#95—21no—
179#95—32no—
180#95—62no—
181#95—62no—
182#95—61no—
183#95—61no—
184C23×C3—22no—
185C23×C3—22no—
186C23×C3—41no—
187C23×C3C2×C22⋊C411yescharacteristic
188C23×C3#12011yescharacteristic
189C23×C3#12311yescharacteristic
190C22×D3—22no—
191C22×D3—22no—
192C22×D3—22no—
193C22×D3—42no—
194C22×D3—42no—
195C22×D3—42no—
196C22×D3—42no—
197C22×D3—42no—
198C22×D3—42no—
199C22×D3—81no—
200C22×D3—81no—
201C22×D3—88no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4C4×D411yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×Dic3—22no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—81no—
229C3×C8—81no—
230C3⋊C8—81no—
231C24—32no—
232C24—61no—
233C24—61no—
234C24—122no—
235C24—122no—
236C24—241no—
237C24D3×D411yescharacteristic
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—32no—
242C22×C4—42no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—62no—
246C22×C4—62no—
247C22×C4—241no—
248C22×C4C2×C4×D311yescharacteristic
249C22×C4#23511yescharacteristic
250C22×C4D3×D411yescharacteristic
251C22×C4D3×D411yescharacteristic
252C22×C4#23711yescharacteristic
253C22×C4C2×#6711yescharacteristic
254M16—42no—
255M16—122no—
256C2×C8—41no—
257C2×C8—121no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—242no—
263C4⋊C4—242no—
264C22⋊C4—42no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—62no—
268C22⋊C4—62no—
269C22⋊C4—62no—
270C22⋊C4—62no—
271C22⋊C4—124no—
272C22⋊C4—122no—
273C22⋊C4—122no—
274C22⋊C4—242no—
275C22⋊C4—242no—
276C22⋊C4—242no—
277C42—42no—
278C42—41no—
279C42—41no—
280C42—122no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—44no—
289C22×C3#38611yescharacteristic
290C22×C3#42112yes—
291C22×C3#45812yes—
292C2×D3—44no—
293C2×D3—42no—
294C2×D3—42no—
295C2×D3—42no—
296C2×D3—84no—
297C2×D3—84no—
298C2×D3—82no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—82no—
304C3×C4—81no—
305C3×C4—81no—
306Dic3—42no—
307Dic3—82no—
308Dic3—82no—
309C23—22no—
310C23—22no—
311C23—41no—
312C23—62no—
313C23—62no—
314C23—62no—
315C23—122no—
316C23—122no—
317C23—122no—
318C23—122no—
319C23—122no—
320C23—122no—
321C23—241no—
322C23—241no—
323C23—248no—
324C23D3×C22⋊C411yescharacteristic
325C23#88311yescharacteristic
326C23#93711yescharacteristic
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—62no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—122no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4—241no—
350C2×C4#67⋊C411yescharacteristic
351C2×C4#88411yescharacteristic
352C2×C4C4×#6711yescharacteristic
353C2×C4#93811yescharacteristic
354C8—81no—
355C8—241no—
356C2×C3—22no—
357C2×C3—22no—
358C2×C3—41no—
359C2×C3#189011yescharacteristic
360C2×C3#189711yescharacteristic
361C2×C3#191111yescharacteristic
362D3—82no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—44no—
371C22—124no—
372C22—122no—
373C22—122no—
374C22—122no—
375C22—242no—
376C22—244no—
377C22—244no—
378C22#482112yes—
379C22#495611yescharacteristic
380C22#505312yes—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—82no—
386C4—81no—
387C4—81no—
388C4—122no—
389C4—242no—
390C4—242no—
391C3#948111yescharacteristic, complemented, Sylow
392C2—22no—
393C2—22no—
394C2—41no—
395C2—242no—
396C2#6953811yescharacteristic
397C2#7038311yescharacteristic
398C2#7040811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1826C2
socleC22×C3#386
prime radicalC24×D3D4
cosocleC23×C3×C4C23
hypercenter#1826D3
hypocenterC3#9481
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
3. C3×#1826C2semidirect
4. #71590C2semidirect
6. #67682C2semidirect
7. #67616C2semidirect
16. #4993C4semidirect
17. #4992C4semidirect
31. #1826D3semidirect
391. C3#9481semidirect

Maximal subgroups

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

subgroupindex
#822822
#735672
C3×#18262
#715902
#699672
#676822
#676162
#94813