Finite Group Databasebeta

← #137757

Subgroups — #137757

Order 512.

361 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 361 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33422C211yesmaximal, characteristic
2C2×#1922C211yesmaximal, characteristic
3#12848C211yesmaximal, characteristic
4#11444C211yesmaximal, characteristic
5#11441C211yesmaximal, characteristic
6#7184C211yesmaximal, characteristic
7#7183C211yesmaximal, characteristic
8C2×#427C411yescharacteristic
9C2×#424C2211yescharacteristic
10C2×#405C2211yescharacteristic
11#2164—22no—
12#2163—21no—
13#2162—21no—
14#2151—21no—
15#2112C2211yescharacteristic
16#2112C2211yescharacteristic
17C2×C22⋊C16C2211yescharacteristic
18C2×C22⋊C16C2211yescharacteristic
19#2108—21no—
20#1922—21no—
21#1922—21no—
22#1922—21no—
23#1922—21no—
24C2×#341C411yescharacteristic
25C2×#339C411yescharacteristic
26C2×#339C411yescharacteristic
27C2×#336C2211yescharacteristic
28#1379—21no—
29#1379—21no—
30#1323—22no—
31#1322—21no—
32#1322—21no—
33C22×M16C2×C411yescharacteristic
34C22×M16C2×C411yescharacteristic
35C23×C8C2×C411yescharacteristic
36C23×C8C2311yescharacteristic
37C2×#117C2×C411yescharacteristic
38C2×#117C2×C411yescharacteristic
39C22×C42C2×C411yescharacteristic
40C2×M32—21no—
41C2×M32—21no—
42C22×C16—21no—
43C22×C16—21no—
44#427—21no—
45#427—21no—
46#427—21no—
47#427—21no—
48C2×#107C811yescharacteristic
49C2×#107C811yescharacteristic
50C2×#106C811yescharacteristic
51C2×#106C811yescharacteristic
52#424—21no—
53#424—21no—
54#424—21no—
55#424—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58#405—21no—
59#405—21no—
60#405D411yescharacteristic
61#405D411yescharacteristic
62#405D411yescharacteristic
63#405D411yescharacteristic
64C2×#97—21no—
65C2×C4×C8—21no—
66C4⋊C16—22no—
67C4⋊C16—22no—
68C22⋊C16—21no—
69C22⋊C16—21no—
70C22⋊C16—21no—
71C22⋊C16—21no—
72C22⋊C16—21no—
73C22⋊C16—21no—
74C22⋊C16D411yescharacteristic
75C22⋊C16D411yescharacteristic
76C22⋊C16D412yes—
77#346—22no—
78C4×C16—22no—
79#341—21no—
80#341—21no—
81#339—21no—
82#339—21no—
83#339—21no—
84#339—21no—
85#336—21no—
86#336—21no—
87C23×C4—21no—
88C23×C4C22×C411yescharacteristic
89C2×M16—21no—
90C2×M16—22no—
91C2×M16—21no—
92C2×M16—21no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16C22⋊C411yescharacteristic
97C2×M16C22⋊C411yescharacteristic
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8C2×C811yescharacteristic
107C22×C8C2×C811yescharacteristic
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D411yescharacteristic
110C22×C8C2×D411yescharacteristic
111C22×C8C2×D411yescharacteristic
112C22×C8#4111yescharacteristic
113C22×C8#4111yescharacteristic
114#117—21no—
115#117—21no—
116#117—21no—
117#117—21no—
118#117—21no—
119#117—21no—
120#117C22⋊C411yescharacteristic
121#117C22⋊C411yescharacteristic
122#117M1611yescharacteristic
123#117M1611yescharacteristic
124C2×C4⋊C4C2×C811yescharacteristic
125C2×C4⋊C4C2×C811yescharacteristic
126C2×C4⋊C4C2×C811yescharacteristic
127C2×C4⋊C4C2×C811yescharacteristic
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—41no—
135C2×C42—41no—
136M32—41no—
137M32—41no—
138M32—41no—
139M32—41no—
140C2×C16—21no—
141C2×C16—22no—
142C2×C16—21no—
143C2×C16—41no—
144C2×C16—41no—
145C2×C16—41no—
146C2×C16—41no—
147C2×C16—41no—
148C2×C16—41no—
149#107—21no—
150#107—21no—
151#107—21no—
152#107—21no—
153#106—21no—
154#106—21no—
155#106—21no—
156#106—21no—
157C4⋊C8—41no—
158C4⋊C8—41no—
159C4⋊C8—41no—
160C4⋊C8—41no—
161#97—21no—
162#97—21no—
163#97—21no—
164#97—21no—
165C4×C8—21no—
166C4×C8—21no—
167C4×C8—21no—
168C4×C8—21no—
169C24#13111yescharacteristic
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4C2×C22⋊C411yescharacteristic
184C22×C4C4×D411yescharacteristic
185C22×C4C4×D411yescharacteristic
186C22×C4C22×C811yescharacteristic
187C22×C4C2×M1611yescharacteristic
188C22×C4#13111yescharacteristic
189M16—21no—
190M16—21no—
191M16—21no—
192M16—42no—
193M16—41no—
194M16—41no—
195M16—81no—
196M16#10112yes—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—81no—
217C2×C8#12011yescharacteristic
218C2×C8#12111yescharacteristic
219C2×C8#12111yescharacteristic
220C2×C8#12311yescharacteristic
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4C22⋊C811yescharacteristic
228C4⋊C4C22⋊C811yescharacteristic
229C4⋊C4C22⋊C811yescharacteristic
230C4⋊C4C22⋊C811yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—21no—
233C22⋊C4—21no—
234C22⋊C4—21no—
235C22⋊C4—41no—
236C22⋊C4—41no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—41no—
242C42—41no—
243C42—41no—
244C42—41no—
245C42—81no—
246C16—41no—
247C16—41no—
248C16—42no—
249C16—81no—
250C16—81no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23#40811yescharacteristic
257C23C8×D411yescharacteristic
258C23#43511yescharacteristic
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—82no—
286C2×C4#38611yescharacteristic
287C2×C4C2×C22⋊C811yescharacteristic
288C2×C4#41011yescharacteristic
289C2×C4#41011yescharacteristic
290C2×C4C2×#10111yescharacteristic
291C2×C4#41311yescharacteristic
292C2×C4C8×D411yescharacteristic
293C2×C4#43511yescharacteristic
294C8—21no—
295C8—21no—
296C8—21no—
297C8—21no—
298C8—22no—
299C8—41no—
300C8—41no—
301C8—41no—
302C8—81no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22#147011yescharacteristic
315C22#147311yescharacteristic
316C22#186911yescharacteristic
317C22#217011yescharacteristic
318C22#217111yescharacteristic
319C4—21no—
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—82no—
326C4—81no—
327C4—81no—
328C4#145811yescharacteristic
329C4#146111yescharacteristic
330C4#189911yescharacteristic
331C4#190111yescharacteristic
332C2—21no—
333C2—21no—
334C2—21no—
335C2—21no—
336C2—41no—
337C2#908211yescharacteristic
338C2#1298011yescharacteristic
339C2#1298611yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C22×C8
Frattini, cosocleC23×C8C23
socleC22#1869
prime radicalC24#131
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#334222
C2×#19222
#128482
#114442
#114412
#71842
#71832