Finite Group Databasebeta

← #10692621

Subgroups — #10692621

Order 768.

726 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 97 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#73605C211yesmaximal, characteristic
2#73560C211yesmaximal, characteristic
3#69980C211yesmaximal, characteristic, complemented
4#69967C211yesmaximal, characteristic
5#69445C211yesmaximal, characteristic, complemented
6C3×#1297C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#9407—31nomaximal, Sylow
9C22×(C2×D3)⋊C4C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C2×D3×C22⋊C4—21no—
12C3×#407C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15C4×#217C2211yescharacteristic
16#5207C2211yescharacteristic
17#4955—21no—
18#4943—21no—
19Dic3×C22⋊C4—21no—
20#4940—22no—
21C4×(C2×D3)⋊C4—22no—
22#4674C412yescomplemented
23#4673C412yescomplemented
24#4666—22no—
25#4468—22no—
26#4467—22no—
27#4452—22no—
28#2561—31no—
29#2273—31no—
30#1508—31no—
31#1508—31no—
32#1462—31no—
33#1297—31no—
34#1297D311yescharacteristic, complemented
35C24×D3D411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C23×Dic3D411yescharacteristic
38C22×C4×D3—22no—
39C22×Dic6C2×C411yescharacteristic, complemented
40C2×C3×M16—21no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C42—21no—
44C2×C3×C42—21no—
45C2×#217—21no—
46C2×#217—21no—
47C2×(C2×D3)⋊C4—22no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—22no—
53C2×(C2×D3)⋊C4—22no—
54C2×(C2×D3)⋊C4C2×C412yes—
55C2×(C2×D3)⋊C4D412yes—
56C2×C4⋊Dic3C2×C411yescharacteristic, complemented
57#924—21no—
58#924—21no—
59#924—21no—
60#924—21no—
61C2×Dic3⋊C4—21no—
62C2×C4×Dic3—22no—
63C2×C4×Dic3—21no—
64C2×#208—21no—
65D3×C22⋊C4—28no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#95C2×C412yescomplemented
69C3×#95D412yes—
70C3⋊#95—22no—
71C3⋊#95—22no—
72C3⋊#95—22no—
73C3⋊#95D412yes—
74#830—21no—
75#830—21no—
76C2×C22⋊Q8—31no—
77C22×C22⋊C4—31no—
78C22×C22⋊C4—61no—
79#407—31no—
80#407C2×D311yescharacteristic
81#397—32no—
82#393—32no—
83#388—62no—
84#381—61no—
85#379—61no—
86C4×C22⋊C4—31no—
87C4×C22⋊C4—62no—
88C4×C22⋊C4—61no—
89C4×C22⋊C4—62no—
90C4×C22⋊C4C2×D311yescharacteristic
91C4×C22⋊C4C2×D311yescharacteristic
92#328—62no—
93#328—62no—
94#323—62no—
95C24×C3C2×D411yescharacteristic
96C23×D3—21no—
97C23×D3—21no—
98C23×D3—24no—
99C23×D3—42no—
100C23×D3—81no—
101C23×D3C22⋊C412yes—
102C22×C3×C4—22no—
103C22×C3×C4—22no—
104C22×C3×C4—81no—
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—22no—
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—42no—
117C22×Dic3C22⋊C412yes—
118C2×C4×D3—28no—
119C2×C4×D3—82no—
120C2×C4×D3—82no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6—81no—
124C3×M16—81no—
125C2×C3×C8—41no—
126C3×C22⋊C4—24no—
127C3×C22⋊C4—42no—
128C3×C42—81no—
129C3×C42—81no—
130#217—24no—
131#217—41no—
132#217—41no—
133(C2×D3)⋊C4—28no—
134(C2×D3)⋊C4—22no—
135(C2×D3)⋊C4—22no—
136(C2×D3)⋊C4—22no—
137(C2×D3)⋊C4—22no—
138(C2×D3)⋊C4—44no—
139(C2×D3)⋊C4—82no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144C4×Dic3—81no—
145C4×Dic3—82no—
146#208—81no—
147C2×C3⋊C8—41no—
148C25—31no—
149C22×Q8—31no—
150C23×C4—31no—
151C23×C4—62no—
152C23×C4C22×D311yescharacteristic
153C2×M16—21no—
154C2×M16—61no—
155C22⋊Q8—61no—
156C22⋊Q8—61no—
157C22⋊Q8—61no—
158C22⋊Q8—61no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—61no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—32no—
164C2×C22⋊C4—32no—
165C2×C22⋊C4—68no—
166C2×C22⋊C4—62no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—61no—
169C2×C22⋊C4—61no—
170C2×C22⋊C4—61no—
171C2×C22⋊C4—61no—
172C2×C22⋊C4—62no—
173C2×C22⋊C4—61no—
174C2×C22⋊C4—62no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—62no—
178C2×C42—61no—
179C22⋊C8—21no—
180C22⋊C8—21no—
181C22⋊C8—61no—
182C22⋊C8—61no—
183#95—32no—
184#95—62no—
185#95—62no—
186#95—62no—
187#95C4×D312yescomplemented
188#95D1212yes—
189C23×C3—22no—
190C23×C3—22no—
191C23×C3—41no—
192C23×C3C2×C22⋊C411yescharacteristic
193C23×C3#12011yescharacteristic
194C23×C3#12311yescharacteristic
195C22×D3—22no—
196C22×D3—22no—
197C22×D3—22no—
198C22×D3—24no—
199C22×D3—22no—
200C22×D3—42no—
201C22×D3—42no—
202C22×D3—42no—
203C22×D3—42no—
204C22×D3—88no—
205C22×D3—81no—
206C22×D3—81no—
207C22×D3#10414yes—
208C2×C3×C4—24no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4C4×D411yescharacteristic
221C2×C3×C4C4×D411yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×Dic3—24no—
225C2×Dic3—22no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—42no—
234C2×Dic3—81no—
235C4×D3—88no—
236Dic6—81no—
237Dic6—81no—
238C3×C8—81no—
239C3⋊C8—81no—
240C24—32no—
241C24—64no—
242C24—61no—
243C24—61no—
244C24—122no—
245C24—241no—
246C24D3×D411yescharacteristic
247C2×Q8—61no—
248C2×Q8—61no—
249C2×Q8—241no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—32no—
253C22×C4—68no—
254C22×C4—62no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—81no—
260C22×C4—122no—
261C22×C4—242no—
262C22×C4—242no—
263C22×C4C2×C4×D311yescharacteristic
264C22×C4C2×D1211yescharacteristic
265C22×C4#23511yescharacteristic
266C22×C4D3×D411yescharacteristic
267C22×C4D3×D411yescharacteristic
268C22×C4#23911yescharacteristic
269M16—81no—
270M16—241no—
271C2×C8—41no—
272C2×C8—121no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C22⋊C4—24no—
278C22⋊C4—42no—
279C22⋊C4—68no—
280C22⋊C4—64no—
281C22⋊C4—62no—
282C22⋊C4—62no—
283C22⋊C4—62no—
284C22⋊C4—62no—
285C22⋊C4—124no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C22⋊C4—242no—
289C42—81no—
290C42—81no—
291C42—241no—
292C42—242no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—44no—
301C22×C3#38611yescharacteristic
302C22×C3C2×#10412yes—
303C22×C3#45512yes—
304C2×D3—24no—
305C2×D3—44no—
306C2×D3—44no—
307C2×D3—42no—
308C2×D3—42no—
309C2×D3—44no—
310C2×D3—82no—
311C2×D3—84no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—44no—
317C3×C4—82no—
318Dic3—44no—
319Dic3—42no—
320Dic3—81no—
321Dic3—81no—
322C23—22no—
323C23—22no—
324C23—34no—
325C23—41no—
326C23—62no—
327C23—62no—
328C23—64no—
329C23—62no—
330C23—62no—
331C23—122no—
332C23—122no—
333C23—122no—
334C23—122no—
335C23—241no—
336C23—241no—
337C23—248no—
338C23D3×C22⋊C411yescharacteristic
339C23#88211yescharacteristic
340C23#89411yescharacteristic
341Q8—241no—
342Q8—241no—
343C2×C4—24no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—64no—
356C2×C4—62no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—122no—
365C2×C4—248no—
366C2×C4—241no—
367C2×C4C4×D1211yescharacteristic
368C2×C4#88411yescharacteristic
369C2×C4#89311yescharacteristic
370C2×C4#89511yescharacteristic
371C8—81no—
372C8—241no—
373C2×C3—22no—
374C2×C3—22no—
375C2×C3—41no—
376C2×C3#188511yescharacteristic
377C2×C3#189611yescharacteristic
378C2×C3#191311yescharacteristic
379D3—44no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—44no—
388C22—64no—
389C22—124no—
390C22—124no—
391C22—124no—
392C22—122no—
393C22—122no—
394C22—242no—
395C22—244no—
396C22#466511yescharacteristic
397C22D3×#10412yes—
398C22#482512yes—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—44no—
404C4—82no—
405C4—124no—
406C4—122no—
407C4—241no—
408C4—241no—
409C3#940711yescharacteristic, complemented, Sylow
410C2—22no—
411C2—22no—
412C2—41no—
413C2—124no—
414C2#6875611yescharacteristic
415C2#6875911yescharacteristic
416C2#6896411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1297C2
socleC22×C3#386
prime radicalC24×D3D4
cosocleC23×C3×C4C23
hypercenter#1297D3
hypocenterC3#9407
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
3. #69980C2semidirect
5. #69445C2semidirect
6. C3×#1297C2semidirect
7. #67618C2semidirect
22. #4674C4semidirect
23. #4673C4semidirect
34. #1297D3semidirect
39. C22×Dic6C2×C4semidirect
56. C2×C4⋊Dic3C2×C4semidirect
68. C3×#95C2×C4semidirect
187. #95C4×D3semidirect
409. C3#9407semidirect

Maximal subgroups

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

subgroupindex
#736052
#735602
#699802
#699672
#694452
C3×#12972
#676182
#94073