Finite Group Databasebeta

← #120818

Subgroups — #120818

Order 512.

591 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21931C211yesmaximal, characteristic
2C2×#1298C211yesmaximal, characteristic
3#8051C211yesmaximal, characteristic
4#8012C211yesmaximal, characteristic
5#7982C211yesmaximal, characteristic
6#7925C211yesmaximal, characteristic
7#7907C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10C22×C22⋊C8C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2280C2211yescharacteristic
13#2280C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1850—22no—
16#1850—22no—
17#1798—22no—
18#1298—22no—
19#1297—22no—
20#1294—22no—
21#1289C412yescomplemented
22#1289C412yescomplemented
23#1282—22no—
24#1280—22no—
25C24×C4C2311yescharacteristic
26C22×M16Q812yes—
27C23×C8—21no—
28C23×C8—21no—
29C23×C8—21no—
30C2×#117—21no—
31C2×#117—21no—
32C22×C22⋊C4—21no—
33C22×C42—21no—
34C2×C4⋊C8—24no—
35#407—21no—
36#407—21no—
37#407—21no—
38#407—21no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
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⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8C2×C412yes—
57C2×C22⋊C8D412yes—
58C2×C22⋊C8D412yes—
59#381—21no—
60#381Q812yes—
61#380—21no—
62#380—21no—
63#380C2×C411yescharacteristic
64#380C2×C411yescharacteristic
65#380C2×C412yes—
66#379—21no—
67#379—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4—21no—
70C4×C22⋊C4—21no—
71C4×C22⋊C4—21no—
72C4×C22⋊C4—21no—
73C4×C22⋊C4—21no—
74C4×C22⋊C4—21no—
75C4×C22⋊C4D412yes—
76C2×#95D412yes—
77#336—24no—
78C25#4111yescharacteristic
79C23×C4—22no—
80C23×C4—82no—
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×Q811yescharacteristic
87C23×C4C2×Q811yescharacteristic
88C23×C4#4111yescharacteristic
89C23×C4#4111yescharacteristic
90C23×C4#4111yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4#4111yescharacteristic
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C2×M16—22no—
96C2×M16—22no—
97C2×M16—82no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8—41no—
107C22×C8—41no—
108C22×C8—82no—
109C22×C8—82no—
110C22×C8—82no—
111#117—82no—
112#117—82no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C42—22no—
128C2×C42—41no—
129C2×C42—41no—
130C2×C42—41no—
131C2×C42—82no—
132C4⋊C8—84no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138C22⋊C8—22no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149#95—22no—
150#95—21no—
151#95—21no—
152#95—22no—
153#95—22no—
154#95—41no—
155#95C4⋊C414yes—
156C24—22no—
157C24—22no—
158C24—42no—
159C24—81no—
160C24C4×D411yescharacteristic
161C24C22⋊Q811yescharacteristic
162C24#12311yescharacteristic
163C24#12311yescharacteristic
164C24#12511yescharacteristic
165C24#12611yescharacteristic
166C24#12611yescharacteristic
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4C2×C4⋊C411yescharacteristic
190C22×C4#11711yescharacteristic
191C22×C4#11711yescharacteristic
192C22×C4C4×D411yescharacteristic
193C22×C4C4×D411yescharacteristic
194C22×C4C4×Q811yescharacteristic
195C22×C4#12011yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4C22⋊Q811yescharacteristic
200C22×C4C22⋊Q811yescharacteristic
201C22×C4C22⋊Q811yescharacteristic
202C22×C4C22⋊Q811yescharacteristic
203C22×C4#12311yescharacteristic
204C22×C4#12311yescharacteristic
205C22×C4#12311yescharacteristic
206C22×C4#12411yescharacteristic
207C22×C4#12411yescharacteristic
208C22×C4#12411yescharacteristic
209C22×C4#12411yescharacteristic
210C22×C4#12511yescharacteristic
211C22×C4#12511yescharacteristic
212C22×C4#12511yescharacteristic
213C22×C4#12511yescharacteristic
214C22×C4#12611yescharacteristic
215C22×C4#12611yescharacteristic
216C22×C4C4⋊Q811yescharacteristic
217M16—82no—
218M16—82no—
219C2×C8—44no—
220C2×C8—44no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—82no—
226C2×C8—82no—
227C2×C8—82no—
228C2×C8—82no—
229C2×C8—82no—
230C2×C8—82no—
231C2×C8—82no—
232C2×C8—82no—
233C4⋊C4—82no—
234C4⋊C4—81no—
235C4⋊C4—81no—
236C22⋊C4—44no—
237C22⋊C4—44no—
238C22⋊C4—44no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—82no—
244C22⋊C4—82no—
245C22⋊C4—82no—
246C42—44no—
247C42—82no—
248C42—82no—
249C42—82no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—21no—
256C23—21no—
257C23—22no—
258C23—22no—
259C23—21no—
260C23—21no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—84no—
273C23—84no—
274C23#38511yescharacteristic
275C23#38711yescharacteristic
276C23#38811yescharacteristic
277C23#39711yescharacteristic
278C23#39811yescharacteristic
279C23#40011yescharacteristic
280C23#40011yescharacteristic
281C23#44312yes—
282C23#44412yes—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
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—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—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—82no—
316C2×C4—84no—
317C2×C4—82no—
318C2×C4—84no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4#38211yescharacteristic
326C2×C4#38711yescharacteristic
327C2×C4#38811yescharacteristic
328C2×C4#38911yescharacteristic
329C2×C4#39311yescharacteristic
330C2×C4#39411yescharacteristic
331C2×C4#39811yescharacteristic
332C2×C4#40011yescharacteristic
333C8—82no—
334C8—82no—
335C8—82no—
336C8—82no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—44no—
359C22—44no—
360C22—82no—
361C22—81no—
362C22—81no—
363C22—82no—
364C22#144211yescharacteristic
365C22#186612yes—
366C22#192212yes—
367C22#193711yescharacteristic
368C22#193811yescharacteristic
369C22#208012yes—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—84no—
378C4—84no—
379C4—161no—
380C4—161no—
381C4#203212yes—
382C2—22no—
383C2—22no—
384C2—21no—
385C2—21no—
386C2—22no—
387C2—41no—
388C2—41no—
389C2—41no—
390C2#1127911yescharacteristic
391C2#1128111yescharacteristic
392C2#1129111yescharacteristic
subgroupG/H
centerC2×C4#393
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
socleC22#1442
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1289C4semidirect
22. #1289C4semidirect

Maximal subgroups

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

subgroupindex
#219312
C2×#12982
#80512
#80122
#79822
#79252
#79072