Finite Group Databasebeta

← #99626

Subgroups — #99626

Order 512.

456 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 94 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#23757C211yesmaximal, characteristic
2#22289C211yesmaximal, characteristic
3#11370C211yesmaximal, characteristic
4#8710C211yesmaximal, characteristic
5C4×#328C211yesmaximal, characteristic
6#7028C211yesmaximal, characteristic
7#7026C211yesmaximal, characteristic
8C4×C22⋊Q8C2211yescharacteristic
9#2285C2211yescharacteristic
10#2100C411yescharacteristic, complemented
11#2095C411yescharacteristic, complemented
12#2083C411yescharacteristic, complemented
13#2076—21no—
14#2072C411yescharacteristic, complemented
15#2036—21no—
16#1984—21no—
17#1940—21no—
18#1917—21no—
19#1916—21no—
20#1851—21no—
21#1797C2211yescharacteristic
22C4×#107—21no—
23C4×#103—21no—
24#1753—21no—
25C4×C22⋊C8C2211yescharacteristic
26C4×C22⋊C8C2211yescharacteristic
27C2×#339—21no—
28C2×#339—21no—
29#1728C2211yescharacteristic
30C2×#328—21no—
31C4×#95C2211yescharacteristic
32#1294—21no—
33#1279—21no—
34#1279—21no—
35C22×M16D411yescharacteristic
36C22×M16D411yescharacteristic
37C2×C22⋊Q8D411yescharacteristic
38C2×C4×Q8D411yescharacteristic
39C2×#117—21no—
40C22×C42—21no—
41C22×C42C2311yescharacteristic
42C2×#107—21no—
43C2×#107—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8D411yescharacteristic
47C2×C4⋊C8D411yescharacteristic
48C2×#103—21no—
49C2×#103—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×#97—21no—
54C2×C4×C8—21no—
55C2×C4×C8—21no—
56C2×C4×C8—21no—
57#391C2×C411yescharacteristic
58#389C2×C411yescharacteristic
59#387—21no—
60#385—21no—
61#384C2×C411yescharacteristic
62#384C2×C411yescharacteristic
63#382—21no—
64#380—21no—
65#380—21no—
66#380D411yescharacteristic
67C4×C4⋊C4—21no—
68C4×C4⋊C4—21no—
69C4×C4⋊C4—21no—
70C4×C4⋊C4D411yescharacteristic
71C4×C22⋊C4—21no—
72C2×#95—21no—
73C2×#95—21no—
74#340—21no—
75#340—21no—
76#340—21no—
77#340—21no—
78#339—42no—
79#339—42no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336C2×C411yescharacteristic
89#336C2×C411yescharacteristic
90#328—24no—
91C22×Q8C22⋊C411yescharacteristic
92C23×C4—41no—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4#4111yescharacteristic
96C2×M16—21no—
97C2×M16—21no—
98C2×M16—21no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—41no—
103C2×M16Q1612yes—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—41no—
111C22⋊Q8—22no—
112C22⋊Q8—41no—
113C4×Q8—41no—
114C4×Q8—41no—
115#117—41no—
116#117—41no—
117#117—41no—
118#117—41no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4C22⋊C411yescharacteristic
129C2×C4⋊C4C22⋊C411yescharacteristic
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—41no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—22no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42—41no—
140C2×C42—41no—
141C2×C42—41no—
142C2×C42—41no—
143C2×C42—41no—
144C2×C42—41no—
145C2×C42C22×C411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42#4111yescharacteristic
149#107—42no—
150#107—42no—
151C4⋊C8—41no—
152C4⋊C8—41no—
153C4⋊C8—81no—
154C4⋊C8—81no—
155#103—42no—
156#103—42no—
157C22⋊C8—22no—
158C22⋊C8—42no—
159C22⋊C8—41no—
160C22⋊C8—41no—
161C22⋊C8—41no—
162#97—81no—
163C4×C8—41no—
164C4×C8—41no—
165C4×C8—41no—
166C4×C8—41no—
167C4×C8—81no—
168#95—21no—
169#95—21no—
170#95—21no—
171#95—21no—
172#95—21no—
173#95—22no—
174#95—22no—
175#95—41no—
176#95—41no—
177#95—41no—
178#95—41no—
179C24#12111yescharacteristic
180C2×Q8—21no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8#10412yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4C2×C22⋊C411yescharacteristic
210C22×C4C4×D411yescharacteristic
211C22×C4C4×D411yescharacteristic
212C22×C4#12011yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12311yescharacteristic
215C22×C4C2×Q1611yescharacteristic
216C22×C4#13511yescharacteristic
217C22×C4#13711yescharacteristic
218C22×C4#13711yescharacteristic
219M16—42no—
220M16—42no—
221M16—81no—
222M16—81no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—42no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—81no—
243C2×C8—81no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—81no—
263C22⋊C4—42no—
264C22⋊C4—81no—
265C22⋊C4—81no—
266C22⋊C4—81no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—41no—
272C42—41no—
273C42—42no—
274C42—42no—
275C42—41no—
276C42—41no—
277C42—41no—
278C42—41no—
279C42—81no—
280C42—81no—
281C42—81no—
282C42—81no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23#38611yescharacteristic
290C23#46211yescharacteristic
291C23#46411yescharacteristic
292Q8—42no—
293Q8—81no—
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—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—44no—
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—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4C2×#10411yescharacteristic
344C2×C4#42111yescharacteristic
345C2×C4C4×Q1611yescharacteristic
346C2×C4#44111yescharacteristic
347C2×C4#44311yescharacteristic
348C2×C4#44411yescharacteristic
349C2×C4#45111yescharacteristic
350C2×C4#45211yescharacteristic
351C2×C4#45311yescharacteristic
352C2×C4#45411yescharacteristic
353C2×C4#45811yescharacteristic
354C2×C4#45811yescharacteristic
355C8—42no—
356C8—42no—
357C8—81no—
358C8—82no—
359C8—81no—
360C8—81no—
361C8—81no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22#187111yescharacteristic
373C22#187611yescharacteristic
374C22#187811yescharacteristic
375C22#188111yescharacteristic
376C22#188311yescharacteristic
377C22#188411yescharacteristic
378C22#189011yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—41no—
386C4—42no—
387C4—41no—
388C4—84no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—161no—
393C4—161no—
394C4#161411yescharacteristic
395C4#161511yescharacteristic
396C4#161711yescharacteristic
397C4#161811yescharacteristic
398C2—41no—
399C2—41no—
400C2#897011yescharacteristic
401C2#899011yescharacteristic
402C2#899711yescharacteristic
403C2#901111yescharacteristic
404C2#903811yescharacteristic
405C2#904611yescharacteristic
406C2#905611yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ 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. #2083C4semidirect
14. #2072C4semidirect

Maximal subgroups

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

subgroupindex
#237572
#222892
#113702
#87102
C4×#3282
#70282
#70262