Finite Group Databasebeta

← #120887

Subgroups — #120887

Order 512.

560 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 116 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#21949C211yesmaximal, characteristic
2#8114C211yesmaximal, characteristic
3#8104C211yesmaximal, characteristic
4#7943C211yesmaximal, characteristic
5#7942C211yesmaximal, characteristic
6C2×#1284C211yesmaximal, characteristic
7#7911C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12#2280C2211yescharacteristic
13C2×#381C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1855—22no—
16#1855—22no—
17#1811—22no—
18#1306—22no—
19#1305—22no—
20#1284—22no—
21#1284—22no—
22#1284C414yescomplemented
23#1280—22no—
24C24×C4C2311yescharacteristic
25C22×M16—21no—
26C22×M16—21no—
27C22×M16—21no—
28C22×M16—21no—
29C2×#117—21no—
30C22×C22⋊C4—21no—
31C22×C22⋊C4—21no—
32C22×C42Q812yes—
33C2×#108—24no—
34#407—21no—
35#407—21no—
36#407—21no—
37#407—21no—
38#407—21no—
39#407—21no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407—21no—
45#407C2×C412yes—
46#407D412yes—
47#407D412yes—
48#407D412yes—
49#407Q812yes—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8D412yes—
58#381—21no—
59#381—21no—
60#381—21no—
61#381—21no—
62#381C2×C412yes—
63#380—21no—
64#380—21no—
65#379—21no—
66C4×C22⋊C4—21no—
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4—21no—
70C4×C22⋊C4—21no—
71C4×C22⋊C4C2×C412yes—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75#341—24no—
76C25#4111yescharacteristic
77C23×C4—22no—
78C23×C4—41no—
79C23×C4—82no—
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×Q811yescharacteristic
86C23×C4C2×Q811yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4#4111yescharacteristic
89C23×C4#4111yescharacteristic
90C23×C4#4111yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4#4111yescharacteristic
93C23×C4#4111yescharacteristic
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—22no—
97C2×M16—22no—
98C2×M16—22no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—82no—
103C2×M16—82no—
104C2×M16—82no—
105C2×M16—82no—
106C22×C8—22no—
107C22×C8—41no—
108C22×C8—41no—
109C22×C8—41no—
110#117—82no—
111C2×C4⋊C4—41no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—44no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—82no—
121C2×C42—22no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—82no—
125#108—84no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8C4⋊C414yes—
142#95—22no—
143#95—22no—
144#95—22no—
145#95—41no—
146#95—41no—
147#95—41no—
148#95—41no—
149#95—41no—
150C24—42no—
151C24—42no—
152C24—81no—
153C24C4×D411yescharacteristic
154C24C22⋊Q811yescharacteristic
155C24#12411yescharacteristic
156C24#12411yescharacteristic
157C24#12511yescharacteristic
158C24#12611yescharacteristic
159C24#12611yescharacteristic
160C22×C4—24no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—81no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4—82no—
183C22×C4—82no—
184C22×C4C2×C4⋊C411yescharacteristic
185C22×C4#11711yescharacteristic
186C22×C4#11711yescharacteristic
187C22×C4C4×D411yescharacteristic
188C22×C4C4×D411yescharacteristic
189C22×C4C4×Q811yescharacteristic
190C22×C4#12011yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4C22⋊Q811yescharacteristic
195C22×C4C22⋊Q811yescharacteristic
196C22×C4C22⋊Q811yescharacteristic
197C22×C4C22⋊Q811yescharacteristic
198C22×C4#12311yescharacteristic
199C22×C4#12311yescharacteristic
200C22×C4#12311yescharacteristic
201C22×C4#12311yescharacteristic
202C22×C4#12311yescharacteristic
203C22×C4#12411yescharacteristic
204C22×C4#12411yescharacteristic
205C22×C4#12511yescharacteristic
206C22×C4#12511yescharacteristic
207C22×C4#12511yescharacteristic
208C22×C4#12511yescharacteristic
209C22×C4#12611yescharacteristic
210C22×C4#12611yescharacteristic
211C22×C4C4⋊Q811yescharacteristic
212M16—44no—
213M16—44no—
214M16—82no—
215M16—82no—
216M16—82no—
217M16—82no—
218M16—82no—
219M16—82no—
220C2×C8—24no—
221C2×C8—44no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—82no—
226C2×C8—82no—
227C2×C8—82no—
228C4⋊C4—82no—
229C22⋊C4—44no—
230C22⋊C4—44no—
231C22⋊C4—82no—
232C22⋊C4—82no—
233C22⋊C4—82no—
234C22⋊C4—82no—
235C22⋊C4—82no—
236C42—82no—
237C42—82no—
238C42—82no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—84no—
261C23—84no—
262C23#38811yescharacteristic
263C23#38811yescharacteristic
264C23#38911yescharacteristic
265C23#39311yescharacteristic
266C23#40011yescharacteristic
267C23#40011yescharacteristic
268C23#40011yescharacteristic
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—44no—
298C2×C4—82no—
299C2×C4—84no—
300C2×C4—82no—
301C2×C4—84no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—164no—
309C2×C4#38211yescharacteristic
310C2×C4#38511yescharacteristic
311C2×C4#38711yescharacteristic
312C2×C4#38711yescharacteristic
313C2×C4#39411yescharacteristic
314C2×C4#39711yescharacteristic
315C2×C4#39811yescharacteristic
316C2×C4#39811yescharacteristic
317C8—44no—
318C8—82no—
319C8—82no—
320C8—82no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—42no—
341C22—42no—
342C22—44no—
343C22—82no—
344C22—82no—
345C22—82no—
346C22—82no—
347C22—82no—
348C22#144211yescharacteristic
349C22#203212yes—
350C22#210212yes—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—84no—
360C4—162no—
361C4—162no—
362C2—22no—
363C2—22no—
364C2—41no—
365C2—41no—
366C2—41no—
367C2—41no—
368C2—41no—
369C2#1127811yescharacteristic
370C2#1127911yescharacteristic
371C2#1128611yescharacteristic
subgroupG/H
center, socleC22#1442
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #1284C4semidirect

Maximal subgroups

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

subgroupindex
#219492
#81142
#81042
#79432
#79422
C2×#12842
#79112