Finite Group Databasebeta

← #120784

Subgroups — #120784

Order 512.

500 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21981C211yesmaximal, characteristic
2#10857C211yesmaximal, characteristic
3C2×#1299C211yesmaximal, characteristic
4#8043C211yesmaximal, characteristic
5#8041C211yesmaximal, characteristic
6#7896C211yesmaximal, characteristic
7#7880C211yesmaximal, characteristic
8C2×#431C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#384C2211yescharacteristic
15#1948—21no—
16#1948—21no—
17#1853—22no—
18#1844—21no—
19#1834—21no—
20C2×#343—22no—
21#1299C414yescomplemented
22#1296—22no—
23#1296—21no—
24#1296—21no—
25#1279—22no—
26#1277—21no—
27#1277—21no—
28C22×M16—21no—
29C22×M16—21no—
30C22×M16—21no—
31C2×#117—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4D412yes—
35C22×C42C2311yescharacteristic
36#431—21no—
37#431—21no—
38#431—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42#431C2×C412yes—
43#431Q812yes—
44C2×#106—22no—
45C2×#106—22no—
46#423—21no—
47#423—21no—
48#423—21no—
49#423—21no—
50#423—21no—
51#423—21no—
52#423C2×C412yes—
53#423Q812yes—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8D412yes—
59C2×C4⋊C8D411yescharacteristic
60C2×C4⋊C8D411yescharacteristic
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C4×M16—21no—
64C4×M16—21no—
65C4×M16—21no—
66C4×M16C2×C412yes—
67C2×#97—21no—
68C2×C4×C8—21no—
69C2×C4×C8—21no—
70C2×C4×C8—21no—
71#384—21no—
72#384—21no—
73#384—21no—
74#384—21no—
75#384D411yescharacteristic
76#384D411yescharacteristic
77#382—21no—
78#380—21no—
79#380—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82#343—44no—
83#340—22no—
84#340—22no—
85C23×C4—22no—
86C23×C4C2×D411yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4#4111yescharacteristic
89C2×M16—22no—
90C2×M16—22no—
91C2×M16—22no—
92C2×M16—22no—
93C2×M16—22no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—41no—
99C2×M16—42no—
100C2×M16—42no—
101C2×M16—81no—
102C2×M16—81no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—22no—
106C22×C8—41no—
107C22×C8—41no—
108C22×C8—41no—
109#117—82no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—81no—
125C2×C4⋊C4—82no—
126C2×C4⋊C4—81no—
127C2×C22⋊C4—41no—
128C2×C42—41no—
129C2×C42C22×C411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×Q811yescharacteristic
134C2×C42C2×Q811yescharacteristic
135C2×C42#4111yescharacteristic
136C2×C42#4111yescharacteristic
137C2×C42#4111yescharacteristic
138C2×C42#4111yescharacteristic
139C2×C42#4111yescharacteristic
140C2×C42#4111yescharacteristic
141#106—82no—
142#106—82no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—41no—
152C4⋊C8—41no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155C4⋊C8C4⋊C412yes—
156C4⋊C8C4⋊C412yes—
157C22⋊C8—22no—
158C22⋊C8—22no—
159C22⋊C8—41no—
160C22⋊C8—41no—
161#97—22no—
162#97—41no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—41no—
167C4×C8—41no—
168C4×C8—41no—
169#95—41no—
170#95—41no—
171C24#12311yescharacteristic
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—44no—
186C22×C4—82no—
187C22×C4—82no—
188C22×C4#11711yescharacteristic
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×C4#12311yescharacteristic
198C22×C4#12411yescharacteristic
199C22×C4#12411yescharacteristic
200C22×C4#12511yescharacteristic
201C22×C4#12511yescharacteristic
202C22×C4#12511yescharacteristic
203C22×C4#12611yescharacteristic
204C22×C4#12611yescharacteristic
205C22×C4#12611yescharacteristic
206M16—24no—
207M16—42no—
208M16—42no—
209M16—42no—
210M16—44no—
211M16—82no—
212M16—82no—
213M16—82no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—81no—
229C2×C8—81no—
230C2×C8—81no—
231C2×C8—81no—
232C4⋊C4—41no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—82no—
243C4⋊C4—81no—
244C4⋊C4—81no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—81no—
248C4⋊C4—82no—
249C4⋊C4—81no—
250C22⋊C4—82no—
251C42—82no—
252C42C2×C4⋊C411yescharacteristic
253C42#11711yescharacteristic
254C42C4×D411yescharacteristic
255C42C4×D411yescharacteristic
256C42C22⋊Q811yescharacteristic
257C42C22⋊Q811yescharacteristic
258C42C22⋊Q811yescharacteristic
259C42#12311yescharacteristic
260C42#12311yescharacteristic
261C42#12311yescharacteristic
262C42#12411yescharacteristic
263C42#12411yescharacteristic
264C42#12511yescharacteristic
265C42#12511yescharacteristic
266C42#12611yescharacteristic
267C42C4⋊Q811yescharacteristic
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23#38711yescharacteristic
275C23#39411yescharacteristic
276C23#40011yescharacteristic
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4—164no—
313C2×C4#38211yescharacteristic
314C2×C4#38511yescharacteristic
315C2×C4#38711yescharacteristic
316C2×C4#38811yescharacteristic
317C2×C4#38811yescharacteristic
318C2×C4#38911yescharacteristic
319C2×C4#39311yescharacteristic
320C2×C4#39711yescharacteristic
321C2×C4#39811yescharacteristic
322C2×C4#39811yescharacteristic
323C2×C4#40011yescharacteristic
324C2×C4#40011yescharacteristic
325C2×C4#45311yescharacteristic
326C2×C4#45411yescharacteristic
327C2×C4#45511yescharacteristic
328C2×C4#45611yescharacteristic
329C2×C4#47112yes—
330C2×C4#47112yes—
331C2×C4#47111yescharacteristic
332C2×C4#47111yescharacteristic
333C2×C4#47211yescharacteristic
334C2×C4#47311yescharacteristic
335C8—44no—
336C8—42no—
337C8—42no—
338C8—82no—
339C8—82no—
340C8—82no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22#144211yescharacteristic
352C22#203711yescharacteristic
353C22#203811yescharacteristic
354C22#204611yescharacteristic
355C22#204611yescharacteristic
356C22#205111yescharacteristic
357C22#205211yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—161no—
365C4—161no—
366C4—162no—
367C4#185911yescharacteristic
368C4#186011yescharacteristic
369C4#193011yescharacteristic
370C4#193111yescharacteristic
371C4#195112yes—
372C4#202711yescharacteristic
373C4#202811yescharacteristic
374C4#207912yes—
375C4#208911yescharacteristic
376C4#209011yescharacteristic
377C2—41no—
378C2—41no—
379C2#1048911yescharacteristic
380C2#1049111yescharacteristic
381C2#1049211yescharacteristic
382C2#1049411yescharacteristic
383C2#1127111yescharacteristic
384C2#1127611yescharacteristic
385C2#1129011yescharacteristic
subgroupG/H
center, socleC23#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1299C4semidirect

Maximal subgroups

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

subgroupindex
#219812
#108572
C2×#12992
#80432
#80412
#78962
#78802