Finite Group Databasebeta

← #120875

Subgroups — #120875

Order 512.

484 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20765C211yesmaximal, characteristic
2#10855C211yesmaximal, characteristic
3#8026C211yesmaximal, characteristic
4#7981C211yesmaximal, characteristic
5#7957C211yesmaximal, characteristic
6C2×#1286C211yesmaximal, characteristic
7#7882C211yesmaximal, characteristic
8C2×#431C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12#2285C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#382C2211yescharacteristic
15#1836—22no—
16#1835—22no—
17C2×#344—21no—
18C2×#344—21no—
19#1736—21no—
20#1736—21no—
21#1294—22no—
22#1289—22no—
23#1286—22no—
24#1286—22no—
25#1278C414yescomplemented
26C22×M16—21no—
27C22×M16D411yescharacteristic
28C22×M16D411yescharacteristic
29C2×#117—21no—
30C2×#117—21no—
31C22×C4⋊C4—21no—
32C22×C42C2311yescharacteristic
33#431—21no—
34#431—21no—
35#431—21no—
36#431—21no—
37#431—21no—
38#431—21no—
39#431—21no—
40#431C2×C412yes—
41#423—21no—
42#423—21no—
43#423—21no—
44#423—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8Q812yes—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C4×M16—21no—
52C4×M16—21no—
53C4×M16C2×C412yes—
54C4×M16Q812yes—
55C2×#97—21no—
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58C2×C4×C8C2×C412yes—
59#384—21no—
60#384—21no—
61#382—21no—
62#382—21no—
63#382—21no—
64#382—21no—
65#382D412yes—
66#382D412yes—
67#380—21no—
68#380—21no—
69#380—21no—
70#380D412yes—
71C4×C4⋊C4—21no—
72C4×C4⋊C4—21no—
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75C2×#95—21no—
76C2×#95—21no—
77#344—42no—
78#344—42no—
79#340—24no—
80#340—24no—
81C23×C4—41no—
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4#4111yescharacteristic
85C2×M16—21no—
86C2×M16—21no—
87C2×M16—21no—
88C2×M16—21no—
89C2×M16—21no—
90C2×M16—21no—
91C2×M16—21no—
92C2×M16—21no—
93C2×M16—21no—
94C2×M16—21no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—41no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16C4⋊C412yes—
102C2×M16C4⋊C412yes—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—41no—
108#117—41no—
109#117—41no—
110#117—41no—
111#117—41no—
112#117—41no—
113#117—41no—
114#117—41no—
115#117—41no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—41no—
130C2×C42—41no—
131C2×C42—41no—
132C2×C42C22×C411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×Q811yescharacteristic
136C2×C42C2×Q811yescharacteristic
137C2×C42#4111yescharacteristic
138C2×C42#4111yescharacteristic
139C2×C42#4111yescharacteristic
140C2×C42#4111yescharacteristic
141C2×C42#4111yescharacteristic
142C2×C42#4111yescharacteristic
143C2×C42#4111yescharacteristic
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8—41no—
152C22⋊C8—22no—
153C22⋊C8—41no—
154C22⋊C8—41no—
155C22⋊C8—41no—
156#97—22no—
157#97—22no—
158C4×C8—22no—
159C4×C8—22no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—22no—
163C4×C8—41no—
164#95—22no—
165#95—22no—
166#95—22no—
167#95—22no—
168#95—22no—
169#95—41no—
170#95—41no—
171#95—41no—
172C24#12111yescharacteristic
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4C2×C4⋊C411yescharacteristic
183C22×C4C4×D411yescharacteristic
184C22×C4C4×Q811yescharacteristic
185C22×C4#12011yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4C22⋊Q811yescharacteristic
189C22×C4#12311yescharacteristic
190C22×C4#12311yescharacteristic
191C22×C4#12311yescharacteristic
192C22×C4#12311yescharacteristic
193C22×C4#12311yescharacteristic
194C22×C4#12411yescharacteristic
195C22×C4#12411yescharacteristic
196C22×C4#12511yescharacteristic
197C22×C4#12611yescharacteristic
198C22×C4#12611yescharacteristic
199C22×C4C4⋊Q811yescharacteristic
200C22×C4#13511yescharacteristic
201C22×C4#13511yescharacteristic
202C22×C4#13511yescharacteristic
203C22×C4#13511yescharacteristic
204C22×C4AGL(1,ℤ/8)11yescharacteristic
205C22×C4AGL(1,ℤ/8)11yescharacteristic
206C22×C4#13711yescharacteristic
207C22×C4#13711yescharacteristic
208M16—22no—
209M16—22no—
210M16—22no—
211M16—22no—
212M16—41no—
213M16—41no—
214M16—41no—
215M16—41no—
216M16—81no—
217M16—81no—
218M16—81no—
219M16—81no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—82no—
232C2×C8—82no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—82no—
238C4⋊C4—82no—
239C4⋊C4—82no—
240C4⋊C4—82no—
241C4⋊C4—81no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C4⋊C4—81no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—81no—
248C22⋊C4—81no—
249C42—81no—
250C42—81no—
251C42—81no—
252C42—81no—
253C42#11711yescharacteristic
254C42#11711yescharacteristic
255C42C4×D411yescharacteristic
256C42C4×D411yescharacteristic
257C42C22⋊Q811yescharacteristic
258C42C22⋊Q811yescharacteristic
259C42C22⋊Q811yescharacteristic
260C42C22⋊Q811yescharacteristic
261C42#12411yescharacteristic
262C42#12411yescharacteristic
263C42#12511yescharacteristic
264C42#12511yescharacteristic
265C42#12511yescharacteristic
266C42#12511yescharacteristic
267C42#12611yescharacteristic
268C42#12611yescharacteristic
269C23—41no—
270C23—41no—
271C23#38911yescharacteristic
272C23#39411yescharacteristic
273C23#39711yescharacteristic
274C23#46311yescharacteristic
275C23#46311yescharacteristic
276C23#46411yescharacteristic
277C23#46411yescharacteristic
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—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—162no—
300C2×C4—162no—
301C2×C4#38211yescharacteristic
302C2×C4#38511yescharacteristic
303C2×C4#38711yescharacteristic
304C2×C4#38711yescharacteristic
305C2×C4#38811yescharacteristic
306C2×C4#38811yescharacteristic
307C2×C4#39311yescharacteristic
308C2×C4#39811yescharacteristic
309C2×C4#39811yescharacteristic
310C2×C4#40011yescharacteristic
311C2×C4#40011yescharacteristic
312C2×C4#40011yescharacteristic
313C2×C4#42711yescharacteristic
314C2×C4#42711yescharacteristic
315C2×C4#42811yescharacteristic
316C2×C4#42811yescharacteristic
317C2×C4#44911yescharacteristic
318C2×C4#44911yescharacteristic
319C2×C4#45211yescharacteristic
320C2×C4#45211yescharacteristic
321C2×C4#47811yescharacteristic
322C2×C4#47811yescharacteristic
323C2×C4#47911yescharacteristic
324C2×C4#47911yescharacteristic
325C2×C4#48211yescharacteristic
326C2×C4#48211yescharacteristic
327C2×C4#48511yescharacteristic
328C2×C4#48511yescharacteristic
329C2×C4#48811yescharacteristic
330C2×C4#48811yescharacteristic
331C2×C4#48811yescharacteristic
332C2×C4#48811yescharacteristic
333C2×C4#48911yescharacteristic
334C2×C4#48911yescharacteristic
335C2×C4#49011yescharacteristic
336C2×C4#49111yescharacteristic
337C8—42no—
338C8—42no—
339C8—82no—
340C8—82no—
341C8—82no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22#144211yescharacteristic
353C22#194911yescharacteristic
354C22#194911yescharacteristic
355C22#204211yescharacteristic
356C22#204211yescharacteristic
357C22#207311yescharacteristic
358C22#207411yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—162no—
368C4—162no—
369C4—162no—
370C4#183511yescharacteristic
371C4#183611yescharacteristic
372C4#185411yescharacteristic
373C4#185411yescharacteristic
374C4#202511yescharacteristic
375C4#202611yescharacteristic
376C4#210111yescharacteristic
377C4#210111yescharacteristic
378C2—21no—
379C2—21no—
380C2—21no—
381C2—21no—
382C2—21no—
383C2—21no—
384C2#1126511yescharacteristic
385C2#1126811yescharacteristic
386C2#1127711yescharacteristic
subgroupG/H
center, socleC22#1442
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1278C4semidirect

Maximal subgroups

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

subgroupindex
#207652
#108552
#80262
#79812
#79572
C2×#12862
#78822