Finite Group Databasebeta

← #100480

Subgroups — #100480

Order 512.

431 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 431 conjugacy classes of subgroups — too many to draw. Only its 94 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#33068C211yesmaximal, characteristic
2#21270C211yesmaximal, characteristic
3#9891C211yesmaximal, characteristic, complemented
4#8174C211yesmaximal, characteristic
5#7683C211yesmaximal, characteristic, complemented
6#7049C211yesmaximal, characteristic, complemented
7#7030C211yesmaximal, characteristic, complemented
8C2×#432—21no—
9#2859C2211yescharacteristic
10C42×D4C2211yescharacteristic
11#1954—21no—
12#1948C411yescharacteristic
13#1948C411yescharacteristic
14#1939—21no—
15#1924—22no—
16#1920—21no—
17#1920—21no—
18#1864—21no—
19#1864—21no—
20#1831C2211yescharacteristic
21#1802C411yescharacteristic
22#1802C411yescharacteristic
23#1767C2211yescharacteristic
24#1767C2211yescharacteristic
25#1749—21no—
26#1744C2211yescharacteristic
27C4×#97C2211yescharacteristic
28C2×D4⋊C8—21no—
29#1295—21no—
30#1295—21no—
31#1279—21no—
32#1271—22no—
33C2×#125—21no—
34C2×C4×D4C2×C411yescharacteristic
35C2×C4×D4D411yescharacteristic
36C2×C4×D4D411yescharacteristic
37C22×C42—21no—
38#432—42no—
39#432—42no—
40C2×#106—21no—
41C2×#106—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8C2×C411yescharacteristic
46C2×C4⋊C8C2×C411yescharacteristic
47C2×#102—21no—
48C2×#102—21no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×#97—21no—
52C2×#97—21no—
53C2×#97—21no—
54C2×#97D411yescharacteristic
55C2×#97D411yescharacteristic
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58#389—21no—
59#389—21no—
60#389—21no—
61#389—21no—
62#384C2×C411yescharacteristic
63#384C2×C411yescharacteristic
64#382—21no—
65C4×C4⋊C4C2×C411yescharacteristic
66C4×C4⋊C4D411yescharacteristic
67C4×C4⋊C4D411yescharacteristic
68C4×C22⋊C4—21no—
69C4×C22⋊C4—21no—
70C4×C22⋊C4—21no—
71C43C2311yescharacteristic
72#340—21no—
73#340—21no—
74#340—21no—
75#340—21no—
76#336—22no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336D412yes—
83D4⋊C8—24no—
84C22×D4C22⋊C411yescharacteristic
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—22no—
93C22×C8—41no—
94C22×C8—41no—
95#125—82no—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—41no—
99C4×D4—41no—
100C4×D4—41no—
101C4×D4—41no—
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⋊C4—41no—
110C2×C4⋊C4C22⋊C411yescharacteristic
111C2×C4⋊C4C22⋊C411yescharacteristic
112C2×C4⋊C4C22⋊C411yescharacteristic
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42C22×C411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42#4111yescharacteristic
129C2×C42#4111yescharacteristic
130#106—42no—
131#106—42no—
132C4⋊C8—22no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—44no—
139#102—42no—
140#102—42no—
141C22⋊C8—44no—
142C22⋊C8—44no—
143#97—21no—
144#97—21no—
145#97—21no—
146#97—42no—
147#97—41no—
148#97—41no—
149#97—41no—
150#97—41no—
151#97SD1612yes—
152C4×C8—22no—
153C4×C8—41no—
154C4×C8—41no—
155C4×C8—41no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—41no—
160#95—41no—
161C24—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4#10112yes—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4C2×C22⋊C411yescharacteristic
186C22×C4C4×D411yescharacteristic
187C22×C4C4×D411yescharacteristic
188C22×C4#12011yescharacteristic
189C22×C4#12111yescharacteristic
190C22×C4#12111yescharacteristic
191C22×C4#12311yescharacteristic
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—42no—
209C2×C8—81no—
210C2×C8—81no—
211C2×C8—81no—
212C2×C8—81no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—42no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4#10412yes—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—41no—
239C42—41no—
240C42—41no—
241C42—41no—
242C42—41no—
243C42—41no—
244C42—41no—
245C42—41no—
246C42—41no—
247C42C2×SD1611yescharacteristic
248C42#13511yescharacteristic
249C42AGL(1,ℤ/8)11yescharacteristic
250C42#13711yescharacteristic
251C23—22no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23#38611yescharacteristic
259D4—22no—
260D4—22no—
261D4—41no—
262D4—41no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4C2×#10111yescharacteristic
300C2×C4#41311yescharacteristic
301C2×C4C2×#10411yescharacteristic
302C2×C4#42111yescharacteristic
303C2×C4#42111yescharacteristic
304C2×C4#42111yescharacteristic
305C2×C4C4×SD1611yescharacteristic
306C2×C4#44011yescharacteristic
307C2×C4#45011yescharacteristic
308C2×C4#45211yescharacteristic
309C2×C4#46111yescharacteristic
310C2×C4#46311yescharacteristic
311C8—42no—
312C8—42no—
313C8—82no—
314C8—82no—
315C8—82no—
316C8—81no—
317C8—81no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22#187111yescharacteristic
326C22#187711yescharacteristic
327C22#188111yescharacteristic
328C22#189711yescharacteristic
329C22#190111yescharacteristic
330C22#190311yescharacteristic
331C22#190611yescharacteristic
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—42no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—81no—
343C4—81no—
344C4—161no—
345C4—161no—
346C4#152911yescharacteristic
347C4#153111yescharacteristic
348C4#153411yescharacteristic
349C4#153511yescharacteristic
350C2—82no—
351C2#898911yescharacteristic
352C2#912611yescharacteristic
353C2#916211yescharacteristic
354C2#923511yescharacteristic
355C2#924911yescharacteristic
356C2#931311yescharacteristic
357C2#935111yescharacteristic
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. #9891C2semidirect
5. #7683C2semidirect
6. #7049C2semidirect
7. #7030C2semidirect

Maximal subgroups

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

subgroupindex
#330682
#212702
#98912
#81742
#76832
#70492
#70302