Finite Group Databasebeta

← #100702

Subgroups — #100702

Order 512.

357 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 80 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#29982C211yesmaximal, characteristic
2#22838C211yesmaximal, characteristic
3#7722C211yesmaximal, characteristic, complemented
4#7682C211yesmaximal, characteristic, complemented
5#7580C211yesmaximal, characteristic
6#7041C211yesmaximal, characteristic, complemented
7#7018C211yesmaximal, characteristic, complemented
8#2862C2211yescharacteristic
9#2293C2211yescharacteristic
10#2101—21no—
11#2092—21no—
12#2058C411yescharacteristic
13#2057C411yescharacteristic
14#2023C411yescharacteristic
15#2023C411yescharacteristic
16#1955—21no—
17#1949—21no—
18#1924—21no—
19#1923—21no—
20#1839C2211yescharacteristic
21#1831C2211yescharacteristic
22#1831C2211yescharacteristic
23#1818—21no—
24#1804—21no—
25#1771—21no—
26#1767C2211yescharacteristic
27#1750—21no—
28#1744C2211yescharacteristic
29#1272—21no—
30#1272—21no—
31#1272—21no—
32#1272—21no—
33#1271—21no—
34#1270—21no—
35C2×C4⋊Q8C2×C411yescharacteristic
36C2×C4⋊Q8C2×C411yescharacteristic
37C2×C4×D4D411yescharacteristic
38C2×#117—21no—
39C2×#107—21no—
40C2×#106—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×#102—21no—
45C2×#102—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×#97—21no—
51C2×#97—21no—
52C2×C4×C8—21no—
53C2×C4×C8—21no—
54#391—21no—
55#391—21no—
56#391—21no—
57#391—21no—
58#390C2×C411yescharacteristic
59#390D411yescharacteristic
60#389C2×C411yescharacteristic
61#389D411yescharacteristic
62#388—21no—
63#388—21no—
64#382—21no—
65#382—21no—
66C4×C4⋊C4D411yescharacteristic
67C4×C22⋊C4—21no—
68C43C2311yescharacteristic
69#340—21no—
70#340—21no—
71#340—21no—
72#340—21no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336C2×C411yescharacteristic
83#336C2×C411yescharacteristic
84#336D411yescharacteristic
85#336D411yescharacteristic
86#336D411yescharacteristic
87#336D411yescharacteristic
88C22×Q8—21no—
89C22×Q8—21no—
90C22×D4C22⋊C411yescharacteristic
91C23×C4—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—41no—
99C22×C8—41no—
100C4⋊Q8—81no—
101C4⋊Q8—81no—
102C4×D4—21no—
103C4×D4—21no—
104C4×D4—21no—
105C4×D4—21no—
106#117—81no—
107#117—81no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4C22⋊C411yescharacteristic
115C2×C4⋊C4C22⋊C411yescharacteristic
116C2×C4⋊C4C22⋊C411yescharacteristic
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C42—21no—
123C2×C42C22×C411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42#4111yescharacteristic
129C2×C42#4111yescharacteristic
130#107—81no—
131#106—81no—
132C4⋊C8—42no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137#102—81no—
138#102—81no—
139C22⋊C8—42no—
140C22⋊C8—42no—
141C22⋊C8—81no—
142C22⋊C8—81no—
143#97—41no—
144#97—41no—
145#97—81no—
146C4×C8—22no—
147C4×C8—41no—
148C4×C8—81no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—41no—
153#95—41no—
154#95—41no—
155#95—41no—
156C24—21no—
157C2×Q8—42no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—81no—
161C2×Q8—81no—
162C2×Q8—81no—
163C2×Q8—81no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—41no—
167C2×D4—41no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4C2×C22⋊C411yescharacteristic
180C22×C4C4×D411yescharacteristic
181C22×C4C4×D411yescharacteristic
182C22×C4#12011yescharacteristic
183C22×C4#12111yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4#12311yescharacteristic
186C2×C8—22no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—42no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—81no—
206C2×C8—81no—
207C2×C8—81no—
208C2×C8—81no—
209C2×C8—81no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—81no—
227C22⋊C4—81no—
228C22⋊C4—81no—
229C42—21no—
230C42—21no—
231C42—41no—
232C42—41no—
233C42—41no—
234C42—41no—
235C42—41no—
236C42—41no—
237C42—41no—
238C42#13511yescharacteristic
239C42#13511yescharacteristic
240C42#13711yescharacteristic
241C42#13711yescharacteristic
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23#38611yescharacteristic
250Q8—161no—
251Q8—161no—
252Q8—162no—
253D4—42no—
254D4—81no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
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—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4#41611yescharacteristic
292C2×C4#41911yescharacteristic
293C2×C4#44311yescharacteristic
294C2×C4#44411yescharacteristic
295C2×C4#45211yescharacteristic
296C2×C4#45211yescharacteristic
297C2×C4#48511yescharacteristic
298C2×C4#48511yescharacteristic
299C8—42no—
300C8—81no—
301C8—81no—
302C8—81no—
303C8—81no—
304C8—81no—
305C8—82no—
306C8—161no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22#187311yescharacteristic
315C22#187611yescharacteristic
316C22#188111yescharacteristic
317C22#190411yescharacteristic
318C22#190611yescharacteristic
319C22#190911yescharacteristic
320C22#191411yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—81no—
333C4—81no—
334C4—81no—
335C4—161no—
336C4—161no—
337C2—161no—
338C2#899411yescharacteristic
339C2#917711yescharacteristic
340C2#919211yescharacteristic
341C2#926511yescharacteristic
342C2#927911yescharacteristic
343C2#933311yescharacteristic
344C2#936911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7722C2semidirect
4. #7682C2semidirect
6. #7041C2semidirect
7. #7018C2semidirect

Maximal subgroups

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

subgroupindex
#299822
#228382
#77222
#76822
#75802
#70412
#70182