Finite Group Databasebeta

← #110967

Subgroups — #110967

Order 512.

597 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21836C211yesmaximal, characteristic
2C2×#1528C211yesmaximal, characteristic
3C2×#1470C211yesmaximal, characteristic
4#17314C211yesmaximal, characteristic
5#8073C211yesmaximal, characteristic
6#7986C211yesmaximal, characteristic
7#7926C211yesmaximal, characteristic
8C22×#125C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2283C2211yescharacteristic
15#1991—22no—
16#1989—22no—
17#1824C412yes—
18#1823C412yes—
19#1528—22no—
20#1528—22no—
21C2×#330C414yescomplemented
22#1470—22no—
23#1470C414yescomplemented
24C2×#324—22no—
25#1450—22no—
26C8×M16—22no—
27C8×M16—22no—
28#1300—22no—
29#1290—22no—
30#1282—22no—
31C22×M16—21no—
32C2×#125—21no—
33C2×#125—21no—
34C2×#125—21no—
35C2×#125—21no—
36C2×#125—21no—
37C2×#125—21no—
38C2×#125—21no—
39C2×#125—21no—
40C2×#125—21no—
41C2×#125—21no—
42C2×#125C2×C412yes—
43C2×#125C2×C412yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C42C2311yescharacteristic
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432C2×C412yes—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431C2×C412yes—
64#431D412yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8D412yes—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8C2×C412yes—
72C2×C22⋊C8D412yes—
73C4×M16—21no—
74C4×M16—21no—
75C4×M16D412yes—
76C4×M16D412yes—
77C2×#97—21no—
78C2×#97C2×C412yes—
79C2×#97C2×C412yes—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8D412yes—
83#383—21no—
84#382—21no—
85#381—21no—
86#380C2×C412yes—
87C4×C4⋊C4—21no—
88C4×C22⋊C4—21no—
89#376—21no—
90#340—24no—
91#340—24no—
92#330—28no—
93#324—28no—
94#322—24no—
95C82—24no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4C22×C411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4#4111yescharacteristic
101C2×M16—28no—
102C2×M16—22no—
103C2×M16—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—41no—
108C22×C8C22⋊C414yes—
109#125—22no—
110#125—22no—
111#125—22no—
112#125—22no—
113#125—22no—
114#125—22no—
115#125—22no—
116#125—22no—
117#125—41no—
118#125—41no—
119#125—41no—
120#125—41no—
121#125—41no—
122#125—41no—
123#125—41no—
124#125—41no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4C4212yes—
149C2×C4⋊C4C4212yes—
150C2×C22⋊C4—22no—
151C2×C42—41no—
152C2×C42C22×C411yescharacteristic
153C2×C42C22×C411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—41no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172C22⋊C8—22no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—41no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179C22⋊C8C22⋊C414yes—
180#97—22no—
181#97—22no—
182#97—22no—
183#97—22no—
184#97—41no—
185#97C22⋊C414yes—
186C4×C8—22no—
187C4×C8—22no—
188C4×C8—22no—
189C4×C8—22no—
190C4×C8—22no—
191C4×C8—22no—
192#95—22no—
193#95—41no—
194#95—41no—
195#95—41no—
196C24C4×D411yescharacteristic
197C22×C4—24no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—44no—
206C22×C4—42no—
207C22×C4—81no—
208C22×C4—82no—
209C22×C4C2×C4211yescharacteristic
210C22×C4C2×C22⋊C411yescharacteristic
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4#11711yescharacteristic
213C22×C4#11711yescharacteristic
214C22×C4C4×D411yescharacteristic
215C22×C4C4×D411yescharacteristic
216C22×C4C4×D411yescharacteristic
217C22×C4C4×D411yescharacteristic
218C22×C4C4×D411yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12311yescharacteristic
223C22×C4#12411yescharacteristic
224C22×C4#12411yescharacteristic
225C22×C4#12611yescharacteristic
226C22×C4#12711yescharacteristic
227M16—28no—
228M16—28no—
229C2×C8—28no—
230C2×C8—24no—
231C2×C8—24no—
232C2×C8—24no—
233C2×C8—44no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—82no—
237C2×C8—82no—
238C2×C8—82no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—44no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C22⋊C4—44no—
257C42—82no—
258C42C2×C22⋊C411yescharacteristic
259C42#11711yescharacteristic
260C42C4×D411yescharacteristic
261C42C4×D411yescharacteristic
262C42C4×D411yescharacteristic
263C42C4×D411yescharacteristic
264C42C4×D411yescharacteristic
265C42C4×D411yescharacteristic
266C42#12011yescharacteristic
267C42#12111yescharacteristic
268C42#12111yescharacteristic
269C42#12111yescharacteristic
270C42#12311yescharacteristic
271C42#12311yescharacteristic
272C42#12411yescharacteristic
273C42#12611yescharacteristic
274C23—41no—
275C23—41no—
276C23C4×C22⋊C411yescharacteristic
277C23#38811yescharacteristic
278C23#39011yescharacteristic
279C23#44314yes—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—41no—
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—42no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4C4×C22⋊C411yescharacteristic
310C2×C4C4×C22⋊C411yescharacteristic
311C2×C4#38611yescharacteristic
312C2×C4#38611yescharacteristic
313C2×C4#38611yescharacteristic
314C2×C4#38811yescharacteristic
315C2×C4#38811yescharacteristic
316C2×C4#38811yescharacteristic
317C2×C4#38811yescharacteristic
318C2×C4#38811yescharacteristic
319C2×C4#39011yescharacteristic
320C2×C4#39011yescharacteristic
321C8—28no—
322C8—44no—
323C8—44no—
324C8—82no—
325C8—82no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—44no—
333C22#143711yescharacteristic
334C22#176412yes—
335C22#193712yes—
336C22#197412yes—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—82no—
349C4—82no—
350C4—162no—
351C4#191512yes—
352C2—22no—
353C2—22no—
354C2—22no—
355C2#1092911yescharacteristic
356C2#1094011yescharacteristic
357C2#1095111yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×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
21. C2×#330C4semidirect
23. #1470C4semidirect

Maximal subgroups

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

subgroupindex
#218362
C2×#15282
C2×#14702
#173142
#80732
#79862
#79262