Finite Group Databasebeta

← #10693225

Subgroups — #10693225

Order 768.

472 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 99 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#82919C211yesmaximal, characteristic
2#82914C211yesmaximal, characteristic
3C3×#1513C211yesmaximal, characteristic
4#69973C211yesmaximal, characteristic, complemented
5#69501C211yesmaximal, characteristic, complemented
6#67687C211yesmaximal, characteristic, complemented
7#67626C211yesmaximal, characteristic, complemented
8#9522—31nomaximal, Sylow
9C3×#525C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×#380C2211yescharacteristic
13#5234—21no—
14#5232—21no—
15#5207C2211yescharacteristic
16#5207C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5044C412yescomplemented
19#5043—21no—
20#5034C412yescomplemented
21#5033—21no—
22#5012—21no—
23#5011—21no—
24#5011—21no—
25D4×C3⋊C8—21no—
26#5000—21no—
27#4995—21no—
28C3×#328—21no—
29C3×#327—21no—
30#4554—21no—
31#4554—21no—
32#4529—21no—
33#4529—21no—
34#3092—31no—
35#2976—31no—
36#1823—31no—
37#1821—31no—
38#1513D311yescharacteristic
39#1305—31no—
40#1293—31no—
41C2×C3×#41D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×Dic3—21no—
44C2×C3×M16—21no—
45C3×#123—21no—
46C3×C22⋊Q8C2×C411yescharacteristic
47C3×C22⋊Q8D411yescharacteristic
48C3×#121C2×C411yescharacteristic
49C3×#121D411yescharacteristic
50C3×#120—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×#117D411yescharacteristic
54C2×C3×C4⋊C4—21no—
55C2×C3×C22⋊C4—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3C2×C412yes—
62C2×#208—21no—
63C2×#208—21no—
64C2×#208D411yescharacteristic
65C2×#208D411yescharacteristic
66C22×C3⋊C8—21no—
67C22×C3⋊C8D411yescharacteristic
68C22×C3⋊C8D411yescharacteristic
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#95—21no—
72#835—22no—
73#832—22no—
74#830—21no—
75#830—21no—
76#830—21no—
77#830—21no—
78#830—21no—
79#830—21no—
80#830—21no—
81#830C2×C412yes—
82#808—22no—
83#807—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86#803—21no—
87C4×C3⋊C8—21no—
88#525C2×D311yescharacteristic
89C22×C4⋊C4—31no—
90#484—32no—
91#483—61no—
92#481—32no—
93#480—61no—
94#436—61no—
95#435—61no—
96#435—61no—
97C8×D4—61no—
98#433—61no—
99#428—61no—
100#417—61no—
101#408—61no—
102#407—31no—
103#407—31no—
104#407C2×D311yescharacteristic
105C2×C22⋊C8—31no—
106#380C2×D311yescharacteristic
107#343—61no—
108#343—61no—
109#340—61no—
110#340—61no—
111#328—21no—
112#327—21no—
113C24×C3#4111yescharacteristic
114C3×#41—41no—
115C3×#41—41no—
116C2×C3×Q8C22⋊C411yescharacteristic
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4C22⋊C411yescharacteristic
121C22×C3×C4—21no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4C22×C411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C22×Dic3—22no—
132C22×Dic3—41no—
133C22×Dic3—41no—
134C22×Dic3—81no—
135C22×Dic3—81no—
136C3×M16—41no—
137C3×M16—41no—
138C2×C3×C8—41no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4C22⋊C411yescharacteristic
144C3×C4⋊C4C22⋊C411yescharacteristic
145C3×C22⋊C4—21no—
146C3×C22⋊C4—21no—
147C3×C22⋊C4—21no—
148C3×C22⋊C4—21no—
149C3×C22⋊C4—21no—
150C3×C22⋊C4—81no—
151C3×C42—21no—
152C4⋊Dic3—22no—
153C4⋊Dic3—22no—
154C4⋊Dic3—41no—
155C4⋊Dic3—41no—
156#208—41no—
157#208—41no—
158#208—41no—
159#208—41no—
160#208—81no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—41no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—81no—
169C2×#41D1211yescharacteristic
170C23×C4—61no—
171C23×C4C22×D311yescharacteristic
172C2×M16—21no—
173C2×M16—31no—
174C2×M16—31no—
175C2×M16—61no—
176C2×M16—61no—
177C22×C8—31no—
178C22×C8—31no—
179C22×C8—61no—
180#123—21no—
181C22⋊Q8C4×D311yescharacteristic
182C22⋊Q8#6711yescharacteristic
183#121C4×D311yescharacteristic
184#121#6711yescharacteristic
185#120—21no—
186C4×D4—21no—
187C4×D4—21no—
188#117D1211yescharacteristic
189C2×C4⋊C4—21no—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—61no—
196C2×C22⋊C4—21no—
197#107—62no—
198#106—62no—
199C4⋊C8—61no—
200C4⋊C8—61no—
201#103—62no—
202#102—62no—
203C22⋊C8—21no—
204C22⋊C8—21no—
205C22⋊C8—32no—
206C22⋊C8—61no—
207C22⋊C8—61no—
208C22⋊C8—61no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213#97—61no—
214C4×C8—61no—
215#95—21no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—21no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3C4×D411yescharacteristic
222C23×C3#12111yescharacteristic
223C23×C3#12311yescharacteristic
224C3×Q8—41no—
225C3×D4—41no—
226C3×D4—41no—
227C3×D4—41no—
228C3×D4—81no—
229C3×D4—81no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—81no—
245C2×C3×C4—81no—
246C2×C3×C4—81no—
247C2×C3×C4C2×C22⋊C411yescharacteristic
248C2×C3×C4C4×D411yescharacteristic
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×Dic3—42no—
252C2×Dic3—82no—
253C2×Dic3—81no—
254C2×Dic3—81no—
255C2×Dic3—82no—
256C3×C8—81no—
257C3⋊C8—42no—
258C3⋊C8—41no—
259C3⋊C8—41no—
260C3⋊C8—41no—
261C3⋊C8—41no—
262C24#23711yescharacteristic
263#41—41no—
264#41—41no—
265C2×Q8(C2×D3)⋊C411yescharacteristic
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4(C2×D3)⋊C411yescharacteristic
270C22×C4—21no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—62no—
275C22×C4—121no—
276C22×C4—121no—
277C22×C4—241no—
278C22×C4—241no—
279C22×C4C2×C4×D311yescharacteristic
280C22×C4C2×D1211yescharacteristic
281C22×C4D3×D411yescharacteristic
282C22×C4D3×D411yescharacteristic
283C22×C4#23711yescharacteristic
284C22×C4C2×#6711yescharacteristic
285M16—41no—
286M16—41no—
287M16—121no—
288M16—121no—
289M16—121no—
290M16—121no—
291M16—241no—
292C2×C8—41no—
293C2×C8—61no—
294C2×C8—61no—
295C2×C8—61no—
296C2×C8—61no—
297C2×C8—62no—
298C2×C8—121no—
299C2×C8—121no—
300C2×C8—241no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—62no—
306C4⋊C4—62no—
307C4⋊C4—121no—
308C4⋊C4—121no—
309C4⋊C4(C2×D3)⋊C411yescharacteristic
310C4⋊C4(C2×D3)⋊C411yescharacteristic
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—81no—
317C42—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3#38611yescharacteristic
332C22×C3#47111yescharacteristic
333C22×C3#47211yescharacteristic
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—41no—
338C3×C4—41no—
339C3×C4—41no—
340C3×C4—161no—
341C3×C4#45811yescharacteristic
342C3×C4#45811yescharacteristic
343Dic3—82no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—41no—
348C23—41no—
349C23#67⋊C411yescharacteristic
350C23#88611yescharacteristic
351C23#93911yescharacteristic
352Q8—41no—
353D4—41no—
354D4—41no—
355D4—41no—
356D4—81no—
357D4—81no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—122no—
376C2×C4—242no—
377C2×C4—241no—
378C2×C4—241no—
379C2×C4—242no—
380C2×C4#67⋊C411yescharacteristic
381C2×C4#88211yescharacteristic
382C2×C4C2×(C2×D3)⋊C411yescharacteristic
383C2×C4#93911yescharacteristic
384C8—81no—
385C8—122no—
386C8—121no—
387C8—121no—
388C8—121no—
389C8—121no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—41no—
393C2×C3—41no—
394C2×C3—81no—
395C2×C3#189011yescharacteristic
396C2×C3#190211yescharacteristic
397C2×C3#190711yescharacteristic
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—42no—
407C22—42no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22#495811yescharacteristic
412C22#519911yescharacteristic
413C22#520311yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—161no—
421C4—242no—
422C4#474211yescharacteristic
423C4#474211yescharacteristic
424C3#952211yescharacteristic, complemented, Sylow
425C2—21no—
426C2—21no—
427C2—41no—
428C2—41no—
429C2—81no—
430C2#7002311yescharacteristic
431C2#7030311yescharacteristic
432C2#7039911yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1513C2
socleC22×C3#386
prime radicalC3×#525C22
cosocleC23×C3×C4C23
hypercenter#1513D3
hypocenterC3#9522
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1513

Complements

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

NH ≅ G/Ntype
4. #69973C2semidirect
5. #69501C2semidirect
6. #67687C2semidirect
7. #67626C2semidirect
18. #5044C4semidirect
20. #5034C4semidirect
424. C3#9522semidirect

Maximal subgroups

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

subgroupindex
#829192
#829142
C3×#15132
#699732
#695012
#676872
#676262
#95223