Finite Group Databasebeta

← #129545

Subgroups — #129545

Order 512.

637 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2969C211yesmaximal, characteristic
2#12785C211yesmaximal, characteristic, complemented
3C4×#349C211yesmaximal, characteristic, complemented
4#11607C212yesmaximal, complemented
5#11587C212yesmaximal, complemented
6C2×#529C411yescharacteristic
7#2969—24no—
8#2969—24no—
9C2×#446C411yescharacteristic
10C2×C8×Q8C2211yescharacteristic
11C2×#436—21no—
12C2×#436C2211yescharacteristic
13C2×#435—21no—
14C2×C8×D4C411yescharacteristic
15C2×#433—21no—
16C2×#431C411yescharacteristic
17C2×#405—21no—
18C2×C4×M16C2211yescharacteristic
19#2150C2212yes—
20C2×#349—21no—
21C2×#349—21no—
22C4×M32C2212yes—
23#1377—21no—
24#1377—21no—
25Q8⋊C16—24no—
26#1330—24no—
27C22×M16—21no—
28C22×M16D411yescharacteristic
29C22×M16D411yescharacteristic
30C23×C8—21no—
31#529—22no—
32#529—22no—
33#529—22no—
34#529—22no—
35C2×#126—21no—
36C2×#125C811yescharacteristic
37C2×#124C811yescharacteristic
38C2×#123—21no—
39C2×C22⋊Q8C811yescharacteristic
40C2×#121C811yescharacteristic
41C2×C4×Q8C2×C411yescharacteristic
42C2×C4×D4—21no—
43C2×C4×D4C2×C411yescharacteristic
44C2×#117—21no—
45C22×C42C2×C411yescharacteristic
46C2×M32—22no—
47C2×M32—22no—
48#446—22no—
49#446—22no—
50C8×Q8—22no—
51C8×Q8D414yes—
52#436—28no—
53#436—22no—
54#436D414yes—
55#435—28no—
56C8×D4—22no—
57C8×D4—22no—
58#433—28no—
59#431—22no—
60#431—22no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8C2×C411yescharacteristic
64C2×C4⋊C8C2×C411yescharacteristic
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68#405—28no—
69C4×M16—22no—
70C4×M16—22no—
71C2×#97—21no—
72C2×#97D411yescharacteristic
73C2×#97D411yescharacteristic
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8C2×C411yescharacteristic
77C2×C4×C8C2311yescharacteristic
78C4⋊C16—22no—
79C4⋊C16—22no—
80#349—24no—
81#349—42no—
82#346—22no—
83C4×C16—22no—
84C22×Q8C2×C811yescharacteristic
85C22×D4—21no—
86C22×D4C2×C811yescharacteristic
87C23×C4—21no—
88C23×C4C22⋊C411yescharacteristic
89C23×C4C2×C811yescharacteristic
90C23×C4M1611yescharacteristic
91C2×M16—22no—
92C2×M16—24no—
93C2×M16—22no—
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—21no—
97C2×M16—21no—
98C2×M16—41no—
99C2×M16—42no—
100C2×M16—41no—
101C2×M16—41no—
102C22×C8—24no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—42no—
106C22×C8—42no—
107C22×C8C22⋊C411yescharacteristic
108C22×C8C22⋊C411yescharacteristic
109C22×C8C2×D411yescharacteristic
110C22×C8C2×D411yescharacteristic
111#126—24no—
112#126—24no—
113#125—22no—
114#125—22no—
115#124—22no—
116#124—22no—
117#123—24no—
118#123—24no—
119C22⋊Q8—22no—
120C22⋊Q8—22no—
121#121—22no—
122#121—22no—
123C4×Q8—21no—
124C4×Q8—21no—
125C4×Q8C22⋊C412yes—
126C4×Q8M1612yes—
127C4×D4—24no—
128C4×D4—24no—
129C4×D4—21no—
130C4×D4—21no—
131C4×D4C22⋊C412yes—
132C4×D4M1612yes—
133#117—24no—
134#117—24no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4C2×C811yescharacteristic
140C2×C4⋊C4C2×C811yescharacteristic
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42C22⋊C411yescharacteristic
152C2×C42C2×C811yescharacteristic
153C2×C42M1611yescharacteristic
154C2×C42C22×C411yescharacteristic
155M32—24no—
156M32—42no—
157M32—42no—
158M32—42no—
159C2×C16—42no—
160C2×C16—42no—
161C4⋊C8—24no—
162C4⋊C8—24no—
163C4⋊C8—21no—
164C4⋊C8—21no—
165C4⋊C8C22⋊C412yes—
166C4⋊C8C22⋊C412yes—
167C22⋊C8—24no—
168C22⋊C8—24no—
169C22⋊C8—24no—
170#97—24no—
171#97—22no—
172#97—22no—
173C4×C8—24no—
174C4×C8—24no—
175C4×C8—21no—
176C4×C8—21no—
177C4×C8C2×D412yes—
178C4×C8C2×D412yes—
179C24—21no—
180C24C22⋊C811yescharacteristic
181C2×Q8—21no—
182C2×Q8—42no—
183C2×Q8C22⋊C812yes—
184C2×D4—24no—
185C2×D4—21no—
186C2×D4—42no—
187C2×D4—82no—
188C2×D4C22⋊C812yes—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—81no—
210C22×C4C22⋊C811yescharacteristic
211C22×C4C22⋊C811yescharacteristic
212C22×C4C22⋊C811yescharacteristic
213C22×C4C2×C22⋊C411yescharacteristic
214C22×C4C22×C811yescharacteristic
215C22×C4C2×M1611yescharacteristic
216M16—44no—
217M16—42no—
218M16—44no—
219M16—42no—
220M16—44no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—22no—
226C2×C8—21no—
227C2×C8—22no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—48no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8#10412yes—
236C2×C8#12012yes—
237C2×C8#12012yes—
238C2×C8#13512yes—
239C2×C8AGL(1,ℤ/8)11yescharacteristic
240C2×C8#13711yescharacteristic
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4C22⋊C812yes—
249C4⋊C4C22⋊C812yes—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C42—21no—
257C42—22no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—22no—
264C42C2×C22⋊C411yescharacteristic
265C42C2×C22⋊C411yescharacteristic
266C42C2×M1611yescharacteristic
267C42C2×M1611yescharacteristic
268C16—44no—
269C16—82no—
270C23—22no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—42no—
275C23—42no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×C22⋊C811yescharacteristic
281Q8—24no—
282Q8—42no—
283D4—42no—
284D4—42no—
285D4—84no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4#37911yescharacteristic
315C2×C4C2×C22⋊C811yescharacteristic
316C2×C4C2×C22⋊C811yescharacteristic
317C2×C4#40711yescharacteristic
318C2×C4#40711yescharacteristic
319C2×C4#40711yescharacteristic
320C2×C4#41611yescharacteristic
321C2×C4#41711yescharacteristic
322C2×C4C2×#10411yescharacteristic
323C2×C4#42111yescharacteristic
324C8—22no—
325C8—22no—
326C8—41no—
327C8—42no—
328C8—42no—
329C8#44912yes—
330C8#45212yes—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—42no—
335C22—41no—
336C22—42no—
337C22—41no—
338C22—42no—
339C22—81no—
340C22—81no—
341C22—82no—
342C22—82no—
343C22#147611yescharacteristic
344C22#147711yescharacteristic
345C22#177911yescharacteristic
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—42no—
351C4—41no—
352C4—42no—
353C4—42no—
354C4—41no—
355C4#148811yescharacteristic
356C4#148911yescharacteristic
357C4#178811yescharacteristic
358C4#178911yescharacteristic
359C4#211512yes—
360C2—21no—
361C2—21no—
362C2—81no—
363C2—82no—
364C2#831211yescharacteristic
365C2#1268211yescharacteristic
366C2#1268611yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1779
prime radicalC2×#121C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12785C2semidirect
3. C4×#349C2semidirect
4. #11607C2semidirect
5. #11587C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29692
#127852
C4×#3492
#116072
#115872