Finite Group Databasebeta

← #101479

Subgroups — #101479

Order 512.

386 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 82 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#29392C211yesmaximal, characteristic
2#21267C211yesmaximal, characteristic
3#9877C211yesmaximal, characteristic, complemented
4#7642C211yesmaximal, characteristic
5#7624C211yesmaximal, characteristic, complemented
6#7041C211yesmaximal, characteristic, complemented
7#7019C211yesmaximal, characteristic, complemented
8C2×#433—21no—
9C4×C4⋊Q8C2211yescharacteristic
10#2293C2211yescharacteristic
11#2084C411yescharacteristic
12#2071C411yescharacteristic
13#2058C411yescharacteristic
14#2023C411yescharacteristic
15#1955—21no—
16#1940—21no—
17#1926—21no—
18#1924—22no—
19#1920—21no—
20#1919—21no—
21#1864—21no—
22#1839C2211yescharacteristic
23#1839C2211yescharacteristic
24#1831C2211yescharacteristic
25#1767C2211yescharacteristic
26#1750—21no—
27C4×#97C2211yescharacteristic
28#1279—21no—
29#1279—21no—
30#1272—22no—
31#1271—21no—
32#1270—21no—
33C2×C4⋊Q8C2×C411yescharacteristic
34C2×C4×Q8—21no—
35C2×C4×D4D411yescharacteristic
36C2×#117—21no—
37#433—42no—
38#433—42no—
39C2×#106—21no—
40C2×#106—21no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×#102—21no—
45C2×#102—21no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×#97—21no—
49C2×#97—21no—
50C2×#97—21no—
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53#391—21no—
54#390C2×C411yescharacteristic
55#390D411yescharacteristic
56#389—21no—
57#389—21no—
58#389C2×C411yescharacteristic
59#389D411yescharacteristic
60#388—21no—
61#388—21no—
62#384C2×C411yescharacteristic
63C4×C4⋊C4—21no—
64C4×C4⋊C4—21no—
65C4×C4⋊C4D411yescharacteristic
66C4×C22⋊C4—21no—
67C43C2311yescharacteristic
68#340—21no—
69#340—21no—
70#340—21no—
71#340—21no—
72#336—22no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336C2×C411yescharacteristic
81#336C2×C411yescharacteristic
82#336D412yes—
83C22×Q8—21no—
84C22×D4C22⋊C411yescharacteristic
85C23×C4—21no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—22no—
91C22×C8—41no—
92C22×C8—41no—
93C4⋊Q8—42no—
94C4×Q8—82no—
95C4×D4—41no—
96C4×D4—41no—
97#117—41no—
98#117—41no—
99#117—41no—
100#117—41no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4C22⋊C411yescharacteristic
110C2×C4⋊C4C22⋊C411yescharacteristic
111C2×C4⋊C4C22⋊C411yescharacteristic
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C42—21no—
118C2×C42—41no—
119C2×C42C22×C411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42#4111yescharacteristic
125C2×C42#4111yescharacteristic
126#106—42no—
127#106—42no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—44no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133#102—42no—
134#102—42no—
135C22⋊C8—44no—
136C22⋊C8—44no—
137#97—41no—
138#97—41no—
139#97—42no—
140#97—41no—
141#97—41no—
142C4×C8—21no—
143C4×C8—21no—
144C4×C8—21no—
145C4×C8SD1612yes—
146#95—21no—
147#95—21no—
148#95—21no—
149#95—41no—
150#95—41no—
151C24—21no—
152C2×Q8—22no—
153C2×Q8—41no—
154C2×Q8—81no—
155C2×Q8—81no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—41no—
159C2×D4—41no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4C2×C22⋊C411yescharacteristic
170C22×C4C4×D411yescharacteristic
171C22×C4C4×D411yescharacteristic
172C22×C4#12011yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12311yescharacteristic
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—42no—
183C2×C8—41no—
184C2×C8—42no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—42no—
191C2×C8—81no—
192C2×C8—81no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8—81no—
196C2×C8—81no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—81no—
213C4⋊C4—82no—
214C22⋊C4—42no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—81no—
220C22⋊C4—81no—
221C42—21no—
222C42—21no—
223C42—41no—
224C42—41no—
225C42—41no—
226C42—41no—
227C42—41no—
228C42—41no—
229C42—41no—
230C42—82no—
231C42C2×SD1611yescharacteristic
232C42#13511yescharacteristic
233C42AGL(1,ℤ/8)11yescharacteristic
234C42#13711yescharacteristic
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23#38611yescharacteristic
243Q8—82no—
244Q8—82no—
245D4—42no—
246D4—81no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
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—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4#42111yescharacteristic
282C2×C4#42111yescharacteristic
283C2×C4C4×SD1611yescharacteristic
284C2×C4#44011yescharacteristic
285C2×C4#45011yescharacteristic
286C2×C4#45211yescharacteristic
287C2×C4#46111yescharacteristic
288C2×C4#46311yescharacteristic
289C8—42no—
290C8—42no—
291C8—82no—
292C8—81no—
293C8—81no—
294C8—161no—
295C8—161no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22#187711yescharacteristic
304C22#188111yescharacteristic
305C22#189111yescharacteristic
306C22#189711yescharacteristic
307C22#190311yescharacteristic
308C22#190611yescharacteristic
309C22#190911yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—81no—
322C4—82no—
323C4—81no—
324C4—81no—
325C4—161no—
326C2—161no—
327C2#904711yescharacteristic
328C2#923511yescharacteristic
329C2#928711yescharacteristic
330C2#931311yescharacteristic
331C2#936911yescharacteristic
332C2#949311yescharacteristic
333C2#953211yescharacteristic
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. #9877C2semidirect
5. #7624C2semidirect
6. #7041C2semidirect
7. #7019C2semidirect

Maximal subgroups

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

subgroupindex
#293922
#212672
#98772
#76422
#76242
#70412
#70192