Finite Group Databasebeta

← #109727

Subgroups — #109727

Order 512.

620 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21944C211yesmaximal, characteristic
2#20529C211yesmaximal, characteristic
3C2×#1306C211yesmaximal, characteristic
4#8024C211yesmaximal, characteristic
5#8015C211yesmaximal, characteristic
6C2×#1293C211yesmaximal, characteristic
7#7915C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2280C2211yescharacteristic
13C2×#381C2211yescharacteristic
14C2×#380C2211yescharacteristic
15#1854—22no—
16#1853C412yescomplemented
17#1852C412yescomplemented
18#1809—22no—
19#1753—22no—
20#1753—22no—
21#1750—22no—
22#1306C414yescomplemented
23#1294—22no—
24#1294—22no—
25#1293C414yescomplemented
26#1280—22no—
27C24×C4C2311yescharacteristic
28C22×M16—21no—
29C22×M16—21no—
30C22×M16—21no—
31C23×C8—21no—
32C2×#117D412yes—
33C22×C4⋊C4C2×C412yes—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C2×#107—22no—
37C2×#106—22no—
38#407—21no—
39#407—21no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407C2×C412yes—
45#407C2×C412yes—
46#407C2×C412yes—
47#407C2×C412yes—
48#407Q812yes—
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⋊C8—21no—
58C2×C22⋊C8C2×C412yes—
59C2×C22⋊C8C2×C412yes—
60C2×C22⋊C8D412yes—
61C2×C22⋊C8D412yes—
62C2×#97—22no—
63C2×C4×C8—22no—
64#381—21no—
65#381—21no—
66#381—21no—
67#381—21no—
68#381C2×C412yes—
69#380—21no—
70#380—21no—
71#380—21no—
72#380—21no—
73#380C2×C411yescharacteristic
74#380C2×C411yescharacteristic
75#380D412yes—
76#379—21no—
77C4×C22⋊C4—21no—
78C4×C22⋊C4—21no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—21no—
82#340—22no—
83#340—22no—
84#336—24no—
85C25#4111yescharacteristic
86C23×C4—22no—
87C23×C4—41no—
88C23×C4—82no—
89C23×C4C22×C411yescharacteristic
90C23×C4C22×C411yescharacteristic
91C23×C4C22×C411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4C2×Q811yescharacteristic
97C23×C4C2×Q811yescharacteristic
98C23×C4#4111yescharacteristic
99C23×C4#4111yescharacteristic
100C23×C4#4111yescharacteristic
101C23×C4#4111yescharacteristic
102C23×C4#4111yescharacteristic
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—82no—
110C2×M16—82no—
111C2×M16—82no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8—82no—
119#117—82no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—82no—
123C2×C4⋊C4C22⋊C412yes—
124C2×C4⋊C4C22⋊C412yes—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—82no—
134C2×C22⋊C4—82no—
135C2×C42—22no—
136#107—82no—
137#106—82no—
138C22⋊C8—22no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—22no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8C4214yes—
152C22⋊C8C4⋊C414yes—
153C22⋊C8C4⋊C414yes—
154#97—82no—
155C4×C8—82no—
156#95—21no—
157#95—21no—
158#95—22no—
159#95—22no—
160#95—22no—
161#95—41no—
162#95—41no—
163#95—41no—
164#95—41no—
165C24—41no—
166C24—42no—
167C24—42no—
168C24—41no—
169C24#11711yescharacteristic
170C24C4×D411yescharacteristic
171C24C4×Q811yescharacteristic
172C24#12111yescharacteristic
173C24#12311yescharacteristic
174C24#12511yescharacteristic
175C24#12611yescharacteristic
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4—82no—
196C22×C4—82no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4—82no—
200C22×C4—82no—
201C22×C4C2×C4211yescharacteristic
202C22×C4C2×C22⋊C411yescharacteristic
203C22×C4C2×C4⋊C411yescharacteristic
204C22×C4C2×C4⋊C411yescharacteristic
205C22×C4#11711yescharacteristic
206C22×C4#11711yescharacteristic
207C22×C4C4×D411yescharacteristic
208C22×C4C4×D411yescharacteristic
209C22×C4C4×D411yescharacteristic
210C22×C4C4×D411yescharacteristic
211C22×C4C4×D411yescharacteristic
212C22×C4C4×D411yescharacteristic
213C22×C4C4×D411yescharacteristic
214C22×C4C4×Q811yescharacteristic
215C22×C4C4×Q811yescharacteristic
216C22×C4C4×Q811yescharacteristic
217C22×C4#12011yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4C22⋊Q811yescharacteristic
220C22×C4C22⋊Q811yescharacteristic
221C22×C4C22⋊Q811yescharacteristic
222C22×C4C22⋊Q811yescharacteristic
223C22×C4#12311yescharacteristic
224C22×C4#12311yescharacteristic
225C22×C4#12411yescharacteristic
226C22×C4#12511yescharacteristic
227C22×C4#12611yescharacteristic
228C22×C4C4⋊Q811yescharacteristic
229M16—82no—
230M16—82no—
231M16—82no—
232M16—82no—
233M16—82no—
234M16—82no—
235C2×C8—24no—
236C2×C8—24no—
237C2×C8—24no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—82no—
246C2×C8—82no—
247C2×C8—82no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—82no—
253C22⋊C4—82no—
254C22⋊C4—81no—
255C22⋊C4—81no—
256C22⋊C4—82no—
257C22⋊C4—82no—
258C22⋊C4—82no—
259C22⋊C4—82no—
260C22⋊C4—82no—
261C42—42no—
262C42—42no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—84no—
288C23—84no—
289C23C4×C4⋊C411yescharacteristic
290C23#38611yescharacteristic
291C23#38711yescharacteristic
292C23#38711yescharacteristic
293C23#38711yescharacteristic
294C23#38811yescharacteristic
295C23#38911yescharacteristic
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
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—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—162no—
337C2×C4—162no—
338C2×C4—164no—
339C2×C4C4×C22⋊C411yescharacteristic
340C2×C4C4×C4⋊C411yescharacteristic
341C2×C4#38511yescharacteristic
342C2×C4#38511yescharacteristic
343C2×C4#38711yescharacteristic
344C2×C4#38811yescharacteristic
345C2×C4#38911yescharacteristic
346C2×C4#39111yescharacteristic
347C8—44no—
348C8—44no—
349C8—44no—
350C8—82no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—82no—
376C22—82no—
377C22—82no—
378C22—82no—
379C22#143611yescharacteristic
380C22#188112yes—
381C22#190211yescharacteristic
382C22#190311yescharacteristic
383C22#191312yes—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—82no—
393C4—82no—
394C4—162no—
395C4—162no—
396C2—41no—
397C2—41no—
398C2—41no—
399C2—41no—
400C2—41no—
401C2—41no—
402C2#933112yes—
403C2#933212yes—
404C2#1090011yescharacteristic
405C2#1090411yescharacteristic
406C2#1091511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1853C4semidirect
17. #1852C4semidirect
22. #1306C4semidirect
25. #1293C4semidirect

Maximal subgroups

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

subgroupindex
#219442
#205292
C2×#13062
#80242
#80152
C2×#12932
#79152