Finite Group Databasebeta

← #101692

Subgroups — #101692

Order 512.

437 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 98 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#29160C211yesmaximal, characteristic
2#23241C211yesmaximal, characteristic
3#9885C211yesmaximal, characteristic
4#8578C211yesmaximal, characteristic
5#7667C211yesmaximal, characteristic
6#7029C211yesmaximal, characteristic
7#7025C211yesmaximal, characteristic
8#2857C2211yescharacteristic
9C4×C22⋊Q8C2211yescharacteristic
10#2100C411yescharacteristic, complemented
11#2095C411yescharacteristic, complemented
12#2057C411yescharacteristic, complemented
13#2024C411yescharacteristic, complemented
14#1985—21no—
15#1921—21no—
16#1917—21no—
17#1916—21no—
18#1892—21no—
19#1863—21no—
20#1850—21no—
21#1808—21no—
22#1797C2211yescharacteristic
23#1771—21no—
24#1752—21no—
25C4×C22⋊C8C2211yescharacteristic
26C2×#336C2211yescharacteristic
27C2×#336C2211yescharacteristic
28C2×#328—21no—
29C4⋊#95C2211yescharacteristic
30#1294—21no—
31#1294—21no—
32#1294—21no—
33#1272—21no—
34#1272—21no—
35C23×C8—21no—
36C23×C8—21no—
37C2×C4⋊Q8C2×C411yescharacteristic
38C2×C22⋊Q8—21no—
39C2×C22⋊Q8D411yescharacteristic
40C2×C4×Q8D411yescharacteristic
41C22×C4⋊C4—21no—
42C22×C42C2311yescharacteristic
43C2×#107—21no—
44C2×#107—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×#103—21no—
49C2×#103—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8D411yescharacteristic
52C2×C22⋊C8D411yescharacteristic
53C2×#97—21no—
54C2×#97—21no—
55C2×C4×C8—21no—
56#391—21no—
57#391C2×C411yescharacteristic
58#389—21no—
59#389—21no—
60#389C2×C411yescharacteristic
61#387—21no—
62#385—21no—
63#384C2×C411yescharacteristic
64#380—21no—
65#380—21no—
66#380D411yescharacteristic
67C4×C4⋊C4—21no—
68C4×C4⋊C4D411yescharacteristic
69C4×C22⋊C4—21no—
70C2×#95—21no—
71C2×#95—21no—
72#340—21no—
73#340—21no—
74#340—21no—
75#340—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336C2×C411yescharacteristic
85#336C2×C411yescharacteristic
86#336D411yescharacteristic
87#336D411yescharacteristic
88#328—42no—
89C22×Q8—21no—
90C22×Q8C22⋊C411yescharacteristic
91C23×C4—41no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4#4111yescharacteristic
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—42no—
102C22×C8—42no—
103C22×C8—41no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8—41no—
107C22×C8—41no—
108C22×C8—81no—
109C22×C8—81no—
110C4⋊Q8—81no—
111C22⋊Q8—21no—
112C22⋊Q8—21no—
113C22⋊Q8—21no—
114C22⋊Q8—21no—
115C22⋊Q8—81no—
116C22⋊Q8—81no—
117C4×Q8—81no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—81no—
130C2×C4⋊C4—81no—
131C2×C4⋊C4C22⋊C411yescharacteristic
132C2×C4⋊C4C22⋊C411yescharacteristic
133C2×C4⋊C4C22⋊C411yescharacteristic
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—41no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—81no—
140C2×C42C22×C411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42#4111yescharacteristic
144#107—42no—
145#107—81no—
146C4⋊C8—42no—
147C4⋊C8—81no—
148C4⋊C8—81no—
149#103—42no—
150#103—81no—
151C22⋊C8—21no—
152C22⋊C8—21no—
153C22⋊C8—21no—
154C22⋊C8—41no—
155C22⋊C8—41no—
156C22⋊C8Q1612yes—
157#97—42no—
158#97—81no—
159C4×C8—81no—
160#95—21no—
161#95—21no—
162#95—21no—
163#95—21no—
164#95—21no—
165#95—41no—
166#95—41no—
167#95—41no—
168#95—41no—
169#95—41no—
170#95—41no—
171C24#12111yescharacteristic
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—81no—
179C2×Q8—81no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4—81no—
201C22×C4—81no—
202C22×C4C2×C22⋊C411yescharacteristic
203C22×C4C4×D411yescharacteristic
204C22×C4C4×D411yescharacteristic
205C22×C4#12011yescharacteristic
206C22×C4#12111yescharacteristic
207C22×C4#12311yescharacteristic
208C22×C4C2×Q1611yescharacteristic
209C22×C4#13511yescharacteristic
210C22×C4AGL(1,ℤ/8)11yescharacteristic
211C22×C4#13711yescharacteristic
212C22×C4#13711yescharacteristic
213C22×C4#13711yescharacteristic
214C22×C4#13711yescharacteristic
215C22×C4#13711yescharacteristic
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—82no—
235C2×C8—82no—
236C2×C8—82no—
237C2×C8—82no—
238C2×C8—81no—
239C2×C8—81no—
240C2×C8—81no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—81no—
254C4⋊C4—81no—
255C4⋊C4—81no—
256C4⋊C4—81no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—81no—
265C22⋊C4—81no—
266C42—41no—
267C42—41no—
268C42—41no—
269C42—41no—
270C42—41no—
271C42—41no—
272C42—41no—
273C42—41no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23#38611yescharacteristic
279C23#46211yescharacteristic
280C23#46411yescharacteristic
281C23#46811yescharacteristic
282C23#47011yescharacteristic
283C23#47111yescharacteristic
284C23#47211yescharacteristic
285Q8—42no—
286Q8—81no—
287Q8—161no—
288Q8—161no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—161no—
336C2×C4—161no—
337C2×C4—161no—
338C2×C4—161no—
339C2×C4#41711yescharacteristic
340C2×C4#41911yescharacteristic
341C2×C4C4×Q1611yescharacteristic
342C2×C4#44111yescharacteristic
343C2×C4#44311yescharacteristic
344C2×C4#44411yescharacteristic
345C2×C4#45111yescharacteristic
346C2×C4#45211yescharacteristic
347C2×C4#46811yescharacteristic
348C2×C4#47011yescharacteristic
349C8—42no—
350C8—42no—
351C8—42no—
352C8—81no—
353C8—81no—
354C8—81no—
355C8—161no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22#169211yescharacteristic
369C22#169611yescharacteristic
370C22#169811yescharacteristic
371C22#170411yescharacteristic
372C22#187611yescharacteristic
373C22#187811yescharacteristic
374C22#188411yescharacteristic
375C22#189411yescharacteristic
376C22#190611yescharacteristic
377C22#190711yescharacteristic
378C22#191311yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—41no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4—162no—
395C4—161no—
396C4—161no—
397C2—21no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2#905611yescharacteristic
402C2#920911yescharacteristic
403C2#922411yescharacteristic
404C2#926411yescharacteristic
405C2#927611yescharacteristic
406C2#938511yescharacteristic
407C2#943211yescharacteristic
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
10. #2100C4semidirect
11. #2095C4semidirect
12. #2057C4semidirect
13. #2024C4semidirect

Maximal subgroups

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

subgroupindex
#291602
#232412
#98852
#85782
#76672
#70292
#70252