Finite Group Databasebeta

← #10693218

Subgroups — #10693218

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#69502C211yesmaximal, characteristic, complemented
6#67687C211yesmaximal, characteristic, complemented
7#67626C211yesmaximal, characteristic, complemented
8#9521—31nomaximal, Sylow
9C3×#525C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×#380C2211yescharacteristic
13#5234—21no—
14#5232—21no—
15#5207C2211yescharacteristic
16#5207C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5044—21no—
19#5043C412yescomplemented
20#5034—21no—
21#5033C412yescomplemented
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#1824—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—61no—
91#483—32no—
92#481—61no—
93#480—32no—
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—82no—
135C3×M16—41no—
136C3×M16—41no—
137C2×C3×C8—41no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4C22⋊C411yescharacteristic
143C3×C4⋊C4C22⋊C411yescharacteristic
144C3×C22⋊C4—21no—
145C3×C22⋊C4—21no—
146C3×C22⋊C4—21no—
147C3×C22⋊C4—21no—
148C3×C22⋊C4—21no—
149C3×C22⋊C4—81no—
150C3×C42—21no—
151C4⋊Dic3—22no—
152C4⋊Dic3—22no—
153C4⋊Dic3—41no—
154C4⋊Dic3—41no—
155#208—41no—
156#208—41no—
157#208—41no—
158#208—41no—
159#208—81no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—41no—
166C2×C3⋊C8—41no—
167C2×C3⋊C8—81no—
168C2×#41D1211yescharacteristic
169C23×C4—61no—
170C23×C4C22×D311yescharacteristic
171C2×M16—21no—
172C2×M16—31no—
173C2×M16—31no—
174C2×M16—61no—
175C2×M16—61no—
176C22×C8—31no—
177C22×C8—31no—
178C22×C8—61no—
179#123—21no—
180C22⋊Q8C4×D311yescharacteristic
181C22⋊Q8#6711yescharacteristic
182#121C4×D311yescharacteristic
183#121#6711yescharacteristic
184#120—21no—
185C4×D4—21no—
186C4×D4—21no—
187#117D1211yescharacteristic
188C2×C4⋊C4—21no—
189C2×C4⋊C4—32no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—61no—
195C2×C22⋊C4—21no—
196#107—62no—
197#106—62no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200#103—62no—
201#102—62no—
202C22⋊C8—21no—
203C22⋊C8—21no—
204C22⋊C8—32no—
205C22⋊C8—61no—
206C22⋊C8—61no—
207C22⋊C8—61no—
208C22⋊C8—61no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212#97—61no—
213C4×C8—61no—
214#95—21no—
215C23×C3—21no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3C4×D411yescharacteristic
221C23×C3#12111yescharacteristic
222C23×C3#12311yescharacteristic
223C3×Q8—41no—
224C3×D4—41no—
225C3×D4—41no—
226C3×D4—41no—
227C3×D4—81no—
228C3×D4—81no—
229C2×C3×C4—21no—
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—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—81no—
244C2×C3×C4—81no—
245C2×C3×C4—81no—
246C2×C3×C4C2×C22⋊C411yescharacteristic
247C2×C3×C4C4×D411yescharacteristic
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×Dic3—42no—
251C2×Dic3—82no—
252C2×Dic3—82no—
253C2×Dic3—82no—
254C3×C8—81no—
255C3⋊C8—42no—
256C3⋊C8—41no—
257C3⋊C8—41no—
258C3⋊C8—41no—
259C3⋊C8—41no—
260C24#23711yescharacteristic
261#41—41no—
262#41—41no—
263C2×Q8(C2×D3)⋊C411yescharacteristic
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4(C2×D3)⋊C411yescharacteristic
268C22×C4—21no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—62no—
273C22×C4—121no—
274C22×C4—121no—
275C22×C4—242no—
276C22×C4C2×C4×D311yescharacteristic
277C22×C4C2×D1211yescharacteristic
278C22×C4D3×D411yescharacteristic
279C22×C4D3×D411yescharacteristic
280C22×C4#23711yescharacteristic
281C22×C4C2×#6711yescharacteristic
282M16—41no—
283M16—41no—
284M16—121no—
285M16—121no—
286M16—121no—
287M16—121no—
288M16—241no—
289C2×C8—41no—
290C2×C8—61no—
291C2×C8—61no—
292C2×C8—61no—
293C2×C8—61no—
294C2×C8—62no—
295C2×C8—121no—
296C2×C8—121no—
297C2×C8—241no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—62no—
303C4⋊C4—62no—
304C4⋊C4—121no—
305C4⋊C4—121no—
306C4⋊C4(C2×D3)⋊C411yescharacteristic
307C4⋊C4(C2×D3)⋊C411yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—81no—
314C42—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3#38611yescharacteristic
329C22×C3#47111yescharacteristic
330C22×C3#47311yescharacteristic
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—41no—
337C3×C4—161no—
338C3×C4#45811yescharacteristic
339C3×C4#45811yescharacteristic
340Dic3—82no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—41no—
345C23—41no—
346C23#67⋊C411yescharacteristic
347C23#88611yescharacteristic
348C23#93911yescharacteristic
349Q8—41no—
350D4—41no—
351D4—41no—
352D4—41no—
353D4—81no—
354D4—81no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—122no—
373C2×C4—242no—
374C2×C4—242no—
375C2×C4—242no—
376C2×C4#67⋊C411yescharacteristic
377C2×C4#88211yescharacteristic
378C2×C4C2×(C2×D3)⋊C411yescharacteristic
379C2×C4#93911yescharacteristic
380C8—81no—
381C8—122no—
382C8—121no—
383C8—121no—
384C8—121no—
385C8—121no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—41no—
389C2×C3—41no—
390C2×C3—81no—
391C2×C3#189011yescharacteristic
392C2×C3#190311yescharacteristic
393C2×C3#190811yescharacteristic
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—42no—
403C22—42no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22#495811yescharacteristic
408C22#520011yescharacteristic
409C22#520411yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—161no—
417C4—242no—
418C4#474211yescharacteristic
419C4#474211yescharacteristic
420C3#952111yescharacteristic, complemented, Sylow
421C2—21no—
422C2—21no—
423C2—41no—
424C2—41no—
425C2—81no—
426C2#7002311yescharacteristic
427C2#7030411yescharacteristic
428C2#7040011yescharacteristic
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#9521
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. #69502C2semidirect
6. #67687C2semidirect
7. #67626C2semidirect
19. #5043C4semidirect
21. #5033C4semidirect
420. C3#9521semidirect

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
#695022
#676872
#676262
#95213