Finite Group Databasebeta

← #121122

Subgroups — #121122

Order 512.

555 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 555 conjugacy classes of subgroups — too many to draw. Only its 110 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#8115C211yesmaximal, characteristic
2#8051C211yesmaximal, characteristic
3C2×#1297C211yesmaximal, characteristic
4#8009C211yesmaximal, characteristic
5#7942C211yesmaximal, characteristic
6#7929C211yesmaximal, characteristic
7#7911C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12#2280C2211yescharacteristic
13C2×C4×C22⋊C4C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1306—22no—
16#1297—22no—
17#1297—22no—
18#1293—22no—
19#1284—22no—
20#1282—22no—
21#1280C414yescomplemented
22C24×C4C2311yescharacteristic
23C22×M16—21no—
24C22×M16—21no—
25C22×M16—21no—
26C22×M16—21no—
27C2×#117—21no—
28C22×C22⋊C4—21no—
29C22×C22⋊C4—21no—
30C22×C42—21no—
31C22×C42—21no—
32#407—21no—
33#407—21no—
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#407—21no—
46#407D412yes—
47#407Q812yes—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8C2×C412yes—
55C2×C22⋊C8C2×C412yes—
56#381D412yes—
57#380—21no—
58#380—21no—
59#379—21no—
60C4×C22⋊C4—21no—
61C4×C22⋊C4—21no—
62C4×C22⋊C4—21no—
63C4×C22⋊C4—21no—
64C4×C22⋊C4—21no—
65C4×C22⋊C4—21no—
66C4×C22⋊C4—21no—
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4D412yes—
70C2×#95—21no—
71C2×#95C2×C412yes—
72C25#4111yescharacteristic
73C23×C4—22no—
74C23×C4—41no—
75C23×C4—82no—
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C23×C4C2×Q811yescharacteristic
81C23×C4#4111yescharacteristic
82C23×C4#4111yescharacteristic
83C23×C4#4111yescharacteristic
84C23×C4#4111yescharacteristic
85C23×C4#4111yescharacteristic
86C23×C4#4111yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4#4111yescharacteristic
89C23×C4#4111yescharacteristic
90C2×M16—22no—
91C2×M16—22no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—82no—
99C2×M16—82no—
100C2×M16—82no—
101C2×M16—82no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—41no—
105C22×C8—41no—
106#117—82no—
107C2×C4⋊C4—41no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—44no—
111C2×C22⋊C4—44no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C42—22no—
118C2×C42—41no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—82no—
123C2×C42—82no—
124C22⋊C8—22no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140#95—22no—
141#95—22no—
142#95—22no—
143#95—22no—
144#95—22no—
145#95—41no—
146C24—42no—
147C24—42no—
148C24—81no—
149C24#11711yescharacteristic
150C24#12311yescharacteristic
151C24#12311yescharacteristic
152C24#12411yescharacteristic
153C24#12511yescharacteristic
154C24#12611yescharacteristic
155C24#12611yescharacteristic
156C22×C4—24no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—44no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—82no—
177C22×C4—82no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4#11711yescharacteristic
181C22×C4#11711yescharacteristic
182C22×C4C4×D411yescharacteristic
183C22×C4C4×D411yescharacteristic
184C22×C4C4×D411yescharacteristic
185C22×C4C4×Q811yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4C22⋊Q811yescharacteristic
190C22×C4C22⋊Q811yescharacteristic
191C22×C4C22⋊Q811yescharacteristic
192C22×C4#12311yescharacteristic
193C22×C4#12311yescharacteristic
194C22×C4#12311yescharacteristic
195C22×C4#12311yescharacteristic
196C22×C4#12411yescharacteristic
197C22×C4#12411yescharacteristic
198C22×C4#12511yescharacteristic
199C22×C4#12511yescharacteristic
200C22×C4#12611yescharacteristic
201C22×C4#12611yescharacteristic
202C22×C4#12611yescharacteristic
203C22×C4#12611yescharacteristic
204C22×C4#12611yescharacteristic
205C22×C4#12611yescharacteristic
206C22×C4#12611yescharacteristic
207C22×C4#12611yescharacteristic
208M16—82no—
209M16—82no—
210M16—82no—
211M16—82no—
212M16—82no—
213M16—82no—
214M16—82no—
215M16—82no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—82no—
221C2×C8—82no—
222C2×C8—82no—
223C2×C8—82no—
224C4⋊C4—82no—
225C22⋊C4—24no—
226C22⋊C4—42no—
227C22⋊C4—44no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—82no—
231C22⋊C4—82no—
232C22⋊C4—82no—
233C42—82no—
234C42—82no—
235C42—82no—
236C42—82no—
237C42—82no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—41no—
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—84no—
260C23—84no—
261C23#38311yescharacteristic
262C23#38711yescharacteristic
263C23#38811yescharacteristic
264C23#39711yescharacteristic
265C23#39711yescharacteristic
266C23#40011yescharacteristic
267C23#40011yescharacteristic
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
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—44no—
297C2×C4—44no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—84no—
302C2×C4—88no—
303C2×C4—82no—
304C2×C4—84no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4#38711yescharacteristic
309C2×C4#38711yescharacteristic
310C2×C4#38811yescharacteristic
311C2×C4#38811yescharacteristic
312C2×C4#39611yescharacteristic
313C2×C4#39711yescharacteristic
314C2×C4#40011yescharacteristic
315C2×C4#40111yescharacteristic
316C8—82no—
317C8—82no—
318C8—82no—
319C8—82no—
320C22—21no—
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—42no—
340C22—42no—
341C22—44no—
342C22—82no—
343C22—82no—
344C22—82no—
345C22—82no—
346C22—82no—
347C22#144311yescharacteristic
348C22#184112yes—
349C22#193212yes—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—84no—
358C4—84no—
359C4—162no—
360C4#210412yes—
361C2—22no—
362C2—22no—
363C2—41no—
364C2—41no—
365C2—41no—
366C2—41no—
367C2—41no—
368C2#1130011yescharacteristic
369C2#1130411yescharacteristic
370C2#1131811yescharacteristic
subgroupG/H
center, socleC22#1443
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
21. #1280C4semidirect

Maximal subgroups

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

subgroupindex
#81152
#80512
C2×#12972
#80092
#79422
#79292
#79112