Finite Group Databasebeta

← #120896

Subgroups — #120896

Order 512.

542 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21949C211yesmaximal, characteristic
2#8115C211yesmaximal, characteristic
3#8114C211yesmaximal, characteristic
4#8107C211yesmaximal, characteristic
5#8016C211yesmaximal, characteristic
6#8015C211yesmaximal, characteristic
7#7943C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12#2280C2211yescharacteristic
13C2×#381C2211yescharacteristic
14C2×#380C2211yescharacteristic
15#1855—22no—
16#1855—22no—
17#1811—22no—
18#1306—22no—
19#1306C414yescomplemented
20#1305—22no—
21#1294—22no—
22#1294—22no—
23#1284—22no—
24C24×C4C2311yescharacteristic
25C22×M16—21no—
26C22×M16—21no—
27C22×M16—21no—
28C22×M16—21no—
29C2×#117Q812yes—
30C22×C4⋊C4—21no—
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
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#407C2×C412yes—
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#380—21no—
66#380—21no—
67#380—21no—
68#380D412yes—
69#379—21no—
70C4×C22⋊C4—21no—
71C4×C22⋊C4—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75#341—24no—
76C25#4111yescharacteristic
77C23×C4—41no—
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—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—82no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—82no—
123C2×C22⋊C4—82no—
124C2×C42—22no—
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—41no—
151C24—42no—
152C24—41no—
153C24—81no—
154C24#11711yescharacteristic
155C24C22⋊Q811yescharacteristic
156C24#12311yescharacteristic
157C24#12411yescharacteristic
158C24#12411yescharacteristic
159C24#12511yescharacteristic
160C24#12611yescharacteristic
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—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—82no—
183C22×C4—82no—
184C22×C4—82no—
185C22×C4C2×C4⋊C411yescharacteristic
186C22×C4#11711yescharacteristic
187C22×C4C4×D411yescharacteristic
188C22×C4C4×D411yescharacteristic
189C22×C4C4×D411yescharacteristic
190C22×C4C4×Q811yescharacteristic
191C22×C4#12011yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4C22⋊Q811yescharacteristic
196C22×C4C22⋊Q811yescharacteristic
197C22×C4C22⋊Q811yescharacteristic
198C22×C4C22⋊Q811yescharacteristic
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×C4#12611yescharacteristic
212C22×C4C4⋊Q811yescharacteristic
213M16—44no—
214M16—44no—
215M16—82no—
216M16—82no—
217M16—82no—
218M16—82no—
219M16—82no—
220M16—82no—
221C2×C8—24no—
222C2×C8—44no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—82no—
227C2×C8—82no—
228C2×C8—82no—
229C4⋊C4—82no—
230C4⋊C4—82no—
231C4⋊C4—82no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C22⋊C4—82no—
235C22⋊C4—82no—
236C22⋊C4—82no—
237C22⋊C4—82no—
238C22⋊C4—82no—
239C22⋊C4—82no—
240C42—82no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—22no—
246C23—21no—
247C23—21no—
248C23—22no—
249C23—22no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—84no—
266C23—84no—
267C23#38211yescharacteristic
268C23#38711yescharacteristic
269C23#38811yescharacteristic
270C23#39311yescharacteristic
271C23#39711yescharacteristic
272C23#39811yescharacteristic
273C23#40011yescharacteristic
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
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—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—164no—
312C2×C4—162no—
313C2×C4—162no—
314C2×C4#38511yescharacteristic
315C2×C4#38711yescharacteristic
316C2×C4#38811yescharacteristic
317C2×C4#38911yescharacteristic
318C2×C4#39411yescharacteristic
319C2×C4#39811yescharacteristic
320C2×C4#40011yescharacteristic
321C2×C4#40011yescharacteristic
322C8—44no—
323C8—82no—
324C8—82no—
325C8—82no—
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—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—42no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—81no—
350C22—82no—
351C22—82no—
352C22—81no—
353C22—82no—
354C22—82no—
355C22#144211yescharacteristic
356C22#203212yes—
357C22#208111yescharacteristic
358C22#208211yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—162no—
368C4—162no—
369C4—162no—
370C2—21no—
371C2—22no—
372C2—21no—
373C2—41no—
374C2—41no—
375C2—41no—
376C2—41no—
377C2—41no—
378C2#1127311yescharacteristic
379C2#1127911yescharacteristic
380C2#1128311yescharacteristic
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
19. #1306C4semidirect

Maximal subgroups

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

subgroupindex
#219492
#81152
#81142
#81072
#80162
#80152
#79432