Finite Group Databasebeta

← #122124

Subgroups — #122124

Order 512.

560 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37609C211yesmaximal, characteristic
2#8113C211yesmaximal, characteristic
3#8110C211yesmaximal, characteristic
4#8070C211yesmaximal, characteristic
5#7976C211yesmaximal, characteristic
6#7959C211yesmaximal, characteristic
7#7959C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2845—21no—
13#2844—21no—
14#2666—22no—
15#2666—22no—
16#2627—22no—
17#2612—22no—
18#2569—21no—
19#2569C412yes—
20#2549—21no—
21#2549C412yes—
22#2283C2211yescharacteristic
23#2283C2211yescharacteristic
24C2×#382C2211yescharacteristic
25#1306—22no—
26#1305—21no—
27#1305—21no—
28#1300—22no—
29#1288—22no—
30#1286—21no—
31#1286—21no—
32#1286—22no—
33C2×C22⋊Q8—22no—
34C22×C4⋊C4—21no—
35C22×C42C2311yescharacteristic
36#432—21no—
37#432—21no—
38#432—21no—
39#432—21no—
40#432—21no—
41#432—21no—
42#432—21no—
43#432Q812yescomplemented
44#431—21no—
45#431—21no—
46#431—21no—
47#431—21no—
48#431—21no—
49#431—21no—
50#431Q812yescomplemented
51#431Q812yescomplemented
52#423—21no—
53#423—21no—
54#423—21no—
55#423—21no—
56#407—21no—
57#407—21no—
58#407—21no—
59#407—21no—
60C4×M16—21no—
61C4×M16—21no—
62C4×M16—21no—
63C4×M16Q812yescomplemented
64#400—22no—
65#400—22no—
66#400—22no—
67#400—22no—
68#398—22no—
69#398—22no—
70#398—22no—
71#398—22no—
72#398—22no—
73#398—22no—
74#398—22no—
75#398—22no—
76#395—22no—
77#395—22no—
78#395—22no—
79#395—22no—
80#391—22no—
81#391—22no—
82#389—22no—
83#389—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#385—22no—
89#385—22no—
90#383—21no—
91#383D412yes—
92#382—21no—
93#382—21no—
94#382—21no—
95#382—21no—
96#382D411yescharacteristic
97#382D411yescharacteristic
98#382D412yes—
99#381—21no—
100#381D412yes—
101#380—22no—
102#380C2×C411yescharacteristic
103#380C2×C411yescharacteristic
104#380D411yescharacteristic
105#380D411yescharacteristic
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4C2×C412yes—
110#376—21no—
111#376D412yes—
112C2×#95—21no—
113C2×#95C2×C412yes—
114C22×Q8—22no—
115C23×C4—22no—
116C23×C4C22×C411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C2×M16—22no—
120C2×M16—41no—
121C2×M16—41no—
122C2×M16—41no—
123C22⋊Q8—84no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—82no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4C22⋊C412yes—
144C2×C22⋊C4C22⋊C412yes—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×Q811yescharacteristic
152C2×C42C2×Q811yescharacteristic
153C2×C42C2×Q811yescharacteristic
154C2×C42C2×Q811yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42#4111yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42#4111yescharacteristic
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—41no—
163C4⋊C8—41no—
164C4⋊C8—41no—
165C4⋊C8—41no—
166C4⋊C8—41no—
167C22⋊C8—22no—
168C22⋊C8—22no—
169C22⋊C8—22no—
170C22⋊C8—41no—
171C22⋊C8—41no—
172C22⋊C8—41no—
173C22⋊C8—41no—
174C22⋊C8—41no—
175#97—22no—
176#97—22no—
177#97—41no—
178#97—41no—
179C4×C8—22no—
180C4×C8—22no—
181C4×C8—22no—
182C4×C8—41no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95C22⋊C412yes—
202#95C22⋊C412yes—
203#95C22⋊C412yes—
204#95C22⋊C412yes—
205C24C2×C22⋊C411yescharacteristic
206C2×Q8—82no—
207C2×Q8—82no—
208C2×Q8—82no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—44no—
218C22×C4—81no—
219C22×C4—81no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C4×Q811yescharacteristic
225C22×C4C4×Q811yescharacteristic
226C22×C4C4×Q811yescharacteristic
227C22×C4C4×Q811yescharacteristic
228C22×C4#12011yescharacteristic
229C22×C4#12011yescharacteristic
230C22×C4#12011yescharacteristic
231C22×C4#12011yescharacteristic
232C22×C4C22⋊Q811yescharacteristic
233C22×C4C22⋊Q811yescharacteristic
234C22×C4C22⋊Q811yescharacteristic
235C22×C4C22⋊Q811yescharacteristic
236C22×C4#12411yescharacteristic
237C22×C4#12411yescharacteristic
238C22×C4C4⋊Q811yescharacteristic
239C22×C4C4⋊Q811yescharacteristic
240M16—81no—
241M16—81no—
242M16—81no—
243M16—81no—
244M16—82no—
245M16—82no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C4⋊C4—81no—
251C4⋊C4—82no—
252C4⋊C4—81no—
253C4⋊C4—81no—
254C4⋊C4—81no—
255C4⋊C4—81no—
256C4⋊C4—81no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C42—82no—
271C42—82no—
272C42C22⋊Q811yescharacteristic
273C42C22⋊Q811yescharacteristic
274C42C22⋊Q811yescharacteristic
275C42C22⋊Q811yescharacteristic
276C42C22⋊Q811yescharacteristic
277C42C22⋊Q811yescharacteristic
278C42C22⋊Q811yescharacteristic
279C42C22⋊Q811yescharacteristic
280C42#12411yescharacteristic
281C42#12411yescharacteristic
282C42#12411yescharacteristic
283C42#12411yescharacteristic
284C42C4⋊Q811yescharacteristic
285C42C4⋊Q811yescharacteristic
286C42C4⋊Q811yescharacteristic
287C42C4⋊Q811yescharacteristic
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23#37911yescharacteristic
293C23#39111yescharacteristic
294C23#39111yescharacteristic
295Q8—164no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
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—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—164no—
335C2×C4#39111yescharacteristic
336C2×C4#39111yescharacteristic
337C2×C4#39111yescharacteristic
338C2×C4#39111yescharacteristic
339C2×C4#39311yescharacteristic
340C2×C4#39311yescharacteristic
341C2×C4#39311yescharacteristic
342C2×C4#39311yescharacteristic
343C2×C4#39511yescharacteristic
344C2×C4#39511yescharacteristic
345C2×C4#39511yescharacteristic
346C2×C4#39511yescharacteristic
347C8—82no—
348C8—82no—
349C8—82no—
350C8—82no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—42no—
360C22—42no—
361C22#144611yescharacteristic
362C22#179511yescharacteristic
363C22#179611yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—82no—
377C4—82no—
378C4—82no—
379C4—82no—
380C4—82no—
381C4—82no—
382C2—21no—
383C2—21no—
384C2—41no—
385C2—41no—
386C2#1138811yescharacteristic
387C2#1138911yescharacteristic
388C2#1139511yescharacteristic
subgroupG/H
center, socleC22#1446
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ 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
43. #432Q8semidirect
50. #431Q8semidirect
51. #431Q8semidirect
63. C4×M16Q8semidirect

Maximal subgroups

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

subgroupindex
#376092
#81132
#81102
#80702
#79762
#79592