Finite Group Databasebeta

← #10691232

Subgroups — #10691232

Order 768.

560 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 113 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#82788C211yesmaximal, characteristic
2#73869C211yesmaximal, characteristic
3C3×#1520C211yesmaximal, characteristic
4#71690C211yesmaximal, characteristic
5C2×#4469C211yesmaximal, characteristic
6#67676C211yesmaximal, characteristic
7#67625C211yesmaximal, characteristic
8#9200—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×Dic3⋊C4C2211yescharacteristic
11C2×#877—21no—
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14#5230C412yescomplemented
15#5212C2211yescharacteristic
16#5212C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#4985C412yescomplemented
19#4982—22no—
20#4976—22no—
21#4946—21no—
22#4943—21no—
23#4942—21no—
24#4647—22no—
25C3×#328—22no—
26#4538—22no—
27#4469—22no—
28#2627—31no—
29#2316—31no—
30#1520—31no—
31#1520D311yescharacteristic
32C2×#324—31no—
33#1289—31no—
34#1289—31no—
35C22×C3×Q8C2×C411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C23×Dic3D411yescharacteristic
38C23×Dic3D411yescharacteristic
39C22×C3×C8—21no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4C2×C411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×#217—21no—
50C2×#217—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—22no—
58C2×Dic3⋊C4—22no—
59C2×Dic3⋊C4—22no—
60C2×Dic3⋊C4C2×C412yes—
61C2×Dic3⋊C4D412yes—
62C2×C4×Dic3—22no—
63C22×C3⋊C8—21no—
64#877—42no—
65#877—42no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#95D412yes—
69C3⋊#95—22no—
70C3⋊#95—22no—
71C3⋊#95—22no—
72C3⋊#95—22no—
73C3⋊#95C2×C412yes—
74C3⋊#95D412yes—
75#830—21no—
76#830—21no—
77C2×C22⋊Q8C2×D311yescharacteristic
78C2×#117—61no—
79C22×C4⋊C4—31no—
80C2×C22⋊C8—31no—
81C2×C22⋊C8C2×D311yescharacteristic
82#400—32no—
83#398—62no—
84#395—32no—
85#389—62no—
86#389—62no—
87#385—61no—
88#381—61no—
89#380—31no—
90#380—31no—
91#380—61no—
92#380C2×D311yescharacteristic
93#328—22no—
94#328—62no—
95#324—62no—
96C24×C3C2×D411yescharacteristic
97C2×C3×Q8—21no—
98C2×C3×Q8—21no—
99C2×C3×Q8—81no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4—22no—
103C22×C3×C4—81no—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—22no—
117C22×Dic3—22no—
118C22×Dic3—81no—
119C22×Dic3C22⋊C412yes—
120C22×Dic3C22⋊C412yes—
121C22×Dic3SD1612yes—
122C22×Dic3Q1612yes—
123C2×C3×C8—41no—
124C2×C3×C8—81no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—81no—
130C3×C22⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—41no—
134#217—41no—
135#217—41no—
136#217—41no—
137#217—41no—
138C4⋊Dic3—81no—
139C4⋊Dic3—81no—
140Dic3⋊C4—22no—
141Dic3⋊C4—22no—
142Dic3⋊C4—22no—
143Dic3⋊C4—22no—
144Dic3⋊C4—44no—
145Dic3⋊C4—82no—
146Dic3⋊C4—82no—
147C4×Dic3—44no—
148C2×C3⋊C8—41no—
149C2×C3⋊C8—81no—
150C22×Q8C4×D311yescharacteristic
151C23×C4—31no—
152C23×C4—31no—
153C23×C4C22×D311yescharacteristic
154C22×C8—21no—
155C22×C8—61no—
156C22⋊Q8—21no—
157C22⋊Q8—21no—
158C22⋊Q8—21no—
159C22⋊Q8—21no—
160#117—122no—
161#117—122no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—32no—
165C2×C4⋊C4—32no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—62no—
171C2×C4⋊C4—62no—
172C2×C4⋊C4—62no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4C4×D311yescharacteristic
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—61no—
179C2×C22⋊C4—61no—
180C2×C42—62no—
181C22⋊C8—21no—
182C22⋊C8—21no—
183C22⋊C8—61no—
184C22⋊C8—61no—
185#95—32no—
186#95—32no—
187#95—62no—
188#95—62no—
189#95—62no—
190#95—62no—
191#95#6712yes—
192C23×C3—41no—
193C23×C3—41no—
194C23×C3C2×C22⋊C411yescharacteristic
195C23×C3#12011yescharacteristic
196C23×C3#12311yescharacteristic
197C23×C3C2×SD1611yescharacteristic
198C23×C3C2×Q1611yescharacteristic
199C23×C3AGL(1,ℤ/8)11yescharacteristic
200C23×C3#13711yescharacteristic
201C3×Q8—81no—
202C3×Q8—81no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4C4×D411yescharacteristic
217C2×C3×C4C4×D411yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×Dic3—22no—
221C2×Dic3—22no—
222C2×Dic3—22no—
223C2×Dic3—22no—
224C2×Dic3—22no—
225C2×Dic3—22no—
226C2×Dic3—22no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—81no—
236C2×Dic3#10314yes—
237C3×C8—81no—
238C3⋊C8—81no—
239C24D3×D411yescharacteristic
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—81no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—32no—
247C22×C4—32no—
248C22×C4—32no—
249C22×C4—32no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4—62no—
257C22×C4—62no—
258C22×C4—81no—
259C22×C4—241no—
260C22×C4C2×C4×D311yescharacteristic
261C22×C4#23511yescharacteristic
262C22×C4D3×D411yescharacteristic
263C22×C4D3×D411yescharacteristic
264C22×C4#23711yescharacteristic
265C22×C4C2×#6711yescharacteristic
266C2×C8—41no—
267C2×C8—81no—
268C2×C8—121no—
269C2×C8—241no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—62no—
275C4⋊C4—62no—
276C4⋊C4—62no—
277C4⋊C4—62no—
278C4⋊C4—81no—
279C4⋊C4—124no—
280C4⋊C4—242no—
281C4⋊C4—242no—
282C4⋊C4—241no—
283C4⋊C4—241no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C22⋊C4—121no—
292C42—124no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3#38611yescharacteristic
304C22×C3C2×#10311yescharacteristic
305C22×C3#41711yescharacteristic
306C22×C3#45011yescharacteristic
307C22×C3#45111yescharacteristic
308C22×C3#48311yescharacteristic
309C22×C3#48411yescharacteristic
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—82no—
315C3×C4—81no—
316C3×C4—81no—
317Dic3—24no—
318Dic3—42no—
319Dic3—82no—
320Dic3—82no—
321C23—41no—
322C23—41no—
323C23D3×C22⋊C411yescharacteristic
324C23#88311yescharacteristic
325C23D3×SD1611yescharacteristic
326C23#91411yescharacteristic
327C23D3×Q1611yescharacteristic
328C23#91811yescharacteristic
329C23#93711yescharacteristic
330Q8—81no—
331Q8—81no—
332C2×C4—34no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—62no—
347C2×C4—62no—
348C2×C4—62no—
349C2×C4—62no—
350C2×C4—62no—
351C2×C4—62no—
352C2×C4—62no—
353C2×C4—122no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—241no—
362C2×C4#67⋊C411yescharacteristic
363C2×C4#88411yescharacteristic
364C2×C4C4×#6711yescharacteristic
365C2×C4#93811yescharacteristic
366C8—81no—
367C8—241no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3#187211yescharacteristic
375C2×C3#191211yescharacteristic
376C2×C3#191311yescharacteristic
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22D3×#10311yescharacteristic
388C22#480311yescharacteristic
389C22#480511yescharacteristic
390C22#481111yescharacteristic
391C22#495611yescharacteristic
392C22#516911yescharacteristic
393C22#518611yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—64no—
399C4—82no—
400C4—81no—
401C4—81no—
402C4—122no—
403C4—242no—
404C4—242no—
405C3#920011yescharacteristic, complemented, Sylow
406C2—21no—
407C2—21no—
408C2—21no—
409C2—21no—
410C2—21no—
411C2—21no—
412C2#7003411yescharacteristic
413C2#7026811yescharacteristic
414C2#7041111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1520C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1520D3
hypocenterC3#9200
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
14. #5230C4semidirect
18. #4985C4semidirect
405. C3#9200semidirect

Maximal subgroups

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

subgroupindex
#827882
#738692
C3×#15202
#716902
C2×#44692
#676762
#676252
#92003