Finite Group Databasebeta

← #101813

Subgroups — #101813

Order 512.

490 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 88 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#25674C211yesmaximal, characteristic
2#22511C211yesmaximal, characteristic
3#9888C211yesmaximal, characteristic
4#8333C211yesmaximal, characteristic
5#7727C211yesmaximal, characteristic
6#7024C211yesmaximal, characteristic
7#7021C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#500C411yescharacteristic, complemented
10C2×#487C411yescharacteristic, complemented
11C2×#446—21no—
12#2857C2211yescharacteristic
13#2036—21no—
14#2036—21no—
15#1990C411yescharacteristic, complemented
16#1985—21no—
17#1985—21no—
18#1984—21no—
19#1956C411yescharacteristic, complemented
20#1863—21no—
21#1863—21no—
22#1817C2211yescharacteristic
23#1798—21no—
24#1797C2211yescharacteristic
25#1750—21no—
26#1749C2211yescharacteristic
27C4×C22⋊C8C2211yescharacteristic
28C2×#328—21no—
29C2×#328—21no—
30#1434C2211yescharacteristic
31#1294—21no—
32#1294—21no—
33#1272—21no—
34#1272—21no—
35#533—42no—
36#533—42no—
37C2×C4⋊Q8C2×C411yescharacteristic
38C2×C4⋊Q8C2×C411yescharacteristic
39C2×C4⋊Q8C2×C411yescharacteristic
40C2×#125—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8D411yescharacteristic
44C2×C22⋊Q8D411yescharacteristic
45C2×C4×Q8—21no—
46C2×C4×Q8D411yescharacteristic
47C2×C4×Q8D411yescharacteristic
48C2×#117—21no—
49C22×C42C2311yescharacteristic
50#500—42no—
51#487—42no—
52#446—42no—
53#446—42no—
54C2×#107—21no—
55C2×#107—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×#103—21no—
60C2×#103—21no—
61C2×#103—21no—
62C2×#103—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8D411yescharacteristic
67C2×C22⋊C8D411yescharacteristic
68C2×#97—21no—
69C2×#97—21no—
70C2×C4×C8—21no—
71C2×C4×C8C2×C411yescharacteristic
72C2×C4×C8C2×C411yescharacteristic
73#391—21no—
74#389—21no—
75#389—21no—
76#384C2×C411yescharacteristic
77#380—21no—
78#380—21no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—21no—
82#340—21no—
83#340—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336—21no—
89#336—21no—
90#336—21no—
91#336—21no—
92#336D411yescharacteristic
93#336D411yescharacteristic
94#328—42no—
95#328—42no—
96C22×Q8—21no—
97C22×Q8—21no—
98C22×Q8C22⋊C411yescharacteristic
99C22×Q8C22⋊C411yescharacteristic
100C23×C4—41no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4#4111yescharacteristic
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—41no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—41no—
114C4⋊Q8—41no—
115C4⋊Q8—41no—
116C4⋊Q8—41no—
117#125—44no—
118C22⋊Q8—22no—
119C22⋊Q8—22no—
120C22⋊Q8—41no—
121C22⋊Q8—41no—
122C22⋊Q8—44no—
123C22⋊Q8—82no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—41no—
127C4×Q8—41no—
128C4×Q8—42no—
129C4×Q8—42no—
130#117—42no—
131#117—42no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4C22⋊C411yescharacteristic
146C2×C4⋊C4C22⋊C411yescharacteristic
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—41no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—41no—
153C2×C42—41no—
154C2×C42C22×C411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42#4111yescharacteristic
158#107—42no—
159#107—42no—
160C4⋊C8—42no—
161C4⋊C8—42no—
162C4⋊C8—41no—
163C4⋊C8—41no—
164#103—42no—
165#103—42no—
166#103—42no—
167#103—42no—
168C22⋊C8—21no—
169C22⋊C8—21no—
170C22⋊C8—21no—
171C22⋊C8—42no—
172C22⋊C8—41no—
173C22⋊C8—41no—
174C22⋊C8—41no—
175C22⋊C8—41no—
176C22⋊C8Q1612yes—
177#97—42no—
178#97—41no—
179#97—41no—
180C4×C8—41no—
181C4×C8—41no—
182C4×C8—42no—
183#95—22no—
184#95—41no—
185#95—41no—
186#95—41no—
187#95—41no—
188#95—41no—
189#95—41no—
190#95—41no—
191C24#12111yescharacteristic
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8—21no—
195C2×Q8—42no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—81no—
205C2×Q8—81no—
206C2×Q8—81no—
207C2×Q8#9912yes—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4#12011yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4C2×Q1611yescharacteristic
238C22×C4#13511yescharacteristic
239C22×C4#13711yescharacteristic
240C22×C4#13711yescharacteristic
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—81no—
260C2×C8—81no—
261C2×C8—81no—
262C4⋊C4—24no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—81no—
277C4⋊C4—81no—
278C4⋊C4—81no—
279C4⋊C4—82no—
280C4⋊C4#9912yes—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—82no—
284C42—22no—
285C42—22no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—41no—
291C42—41no—
292C42—41no—
293C42—41no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23#38611yescharacteristic
301C23#46211yescharacteristic
302C23#46411yescharacteristic
303Q8—42no—
304Q8—42no—
305Q8—81no—
306Q8—82no—
307Q8—82no—
308Q8—81no—
309Q8—161no—
310Q8—161no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—161no—
355C2×C4—161no—
356C2×C4—161no—
357C2×C4—161no—
358C2×C4C2×#9911yescharacteristic
359C2×C4C2×#9911yescharacteristic
360C2×C4C4×Q1611yescharacteristic
361C2×C4#44111yescharacteristic
362C2×C4#44311yescharacteristic
363C2×C4#44411yescharacteristic
364C2×C4#45111yescharacteristic
365C2×C4#45211yescharacteristic
366C8—42no—
367C8—42no—
368C8—81no—
369C8—81no—
370C8—81no—
371C8—81no—
372C8—161no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22#187611yescharacteristic
384C22#187811yescharacteristic
385C22#188411yescharacteristic
386C22#189811yescharacteristic
387C22#190311yescharacteristic
388C22#190511yescharacteristic
389C22#190911yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—41no—
399C4—42no—
400C4—41no—
401C4—42no—
402C4—81no—
403C4—81no—
404C4—162no—
405C4—161no—
406C4—161no—
407C2—41no—
408C2—41no—
409C2#905611yescharacteristic
410C2#924111yescharacteristic
411C2#925811yescharacteristic
412C2#928011yescharacteristic
413C2#928811yescharacteristic
414C2#941811yescharacteristic
415C2#944311yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C2×#500C4semidirect
10. C2×#487C4semidirect
15. #1990C4semidirect
19. #1956C4semidirect

Maximal subgroups

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

subgroupindex
#256742
#225112
#98882
#83332
#77272
#70242
#70212