Finite Group Databasebeta

← #101491

Subgroups — #101491

Order 512.

419 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 90 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#29260C211yesmaximal, characteristic
2#22385C211yesmaximal, characteristic
3#9677C211yesmaximal, characteristic, complemented
4#8581C211yesmaximal, characteristic
5#7622C211yesmaximal, characteristic, complemented
6#7041C211yesmaximal, characteristic, complemented
7#7022C211yesmaximal, characteristic, complemented
8C4×C4⋊Q8C2211yescharacteristic
9C4×#124C2211yescharacteristic
10#2095C411yescharacteristic
11#2094C411yescharacteristic
12#2024C411yescharacteristic
13#2023C411yescharacteristic
14#1984—21no—
15#1983—21no—
16#1949—21no—
17#1923—21no—
18#1923—21no—
19#1869—21no—
20#1865—21no—
21#1862—21no—
22#1850—21no—
23#1839C2211yescharacteristic
24#1831C2211yescharacteristic
25#1767C2211yescharacteristic
26#1767C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28C8×C22⋊C4—21no—
29C2×#329—21no—
30C2×Q8⋊C8—21no—
31C2×Q8⋊C8—21no—
32#1295—21no—
33#1271—21no—
34#1271—21no—
35C23×C8—21no—
36C2×C4⋊Q8C2×C411yescharacteristic
37C2×#124D411yescharacteristic
38C2×C4×Q8—21no—
39C2×C4×Q8D411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×#107—21no—
42C2×#106—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×#103—21no—
49C2×#103—21no—
50C2×#102—21no—
51C2×#102—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×#97—21no—
56C2×#97—21no—
57C2×C4×C8—21no—
58C2×C4×C8D411yescharacteristic
59C2×C4×C8D411yescharacteristic
60#391—21no—
61#391C2×C411yescharacteristic
62#390C2×C411yescharacteristic
63#389—21no—
64#389—21no—
65#388—21no—
66#388—21no—
67#384C2×C411yescharacteristic
68#382D411yescharacteristic
69C4×C4⋊C4—21no—
70C4×C4⋊C4—21no—
71C4×C22⋊C4—21no—
72C4×C22⋊C4—21no—
73C43C2311yescharacteristic
74#340—21no—
75#340—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336C2×C411yescharacteristic
85#336C2×C411yescharacteristic
86#336D411yescharacteristic
87#336D411yescharacteristic
88#329—42no—
89Q8⋊C8—42no—
90Q8⋊C8—42no—
91C22×Q8—21no—
92C22×Q8C22⋊C411yescharacteristic
93C22×D4C22⋊C411yescharacteristic
94C23×C4—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—42no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—81no—
105C22×C8—81no—
106C4⋊Q8—42no—
107#124—21no—
108#124—21no—
109#124SD1612yes—
110#124Q1612yes—
111C4×Q8—41no—
112C4×Q8—41no—
113C4×Q8—41no—
114C4×Q8—41no—
115C4×Q8—81no—
116C4×D4—81no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4C22⋊C411yescharacteristic
125C2×C4⋊C4C22⋊C411yescharacteristic
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C42—21no—
132C2×C42—41no—
133C2×C42C22×C411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42#4111yescharacteristic
139C2×C42#4111yescharacteristic
140#107—81no—
141#106—81no—
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—42no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—81no—
150#103—42no—
151#103—81no—
152#102—42no—
153#102—81no—
154C22⋊C8—42no—
155C22⋊C8—81no—
156C22⋊C8—81no—
157#97—41no—
158#97—41no—
159#97—41no—
160#97—41no—
161C4×C8—41no—
162C4×C8—41no—
163C4×C8—81no—
164#95—21no—
165#95—21no—
166#95—21no—
167#95—21no—
168#95—41no—
169#95—41no—
170C24—21no—
171C2×Q8—21no—
172C2×Q8—22no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8#10312yes—
181C2×D4—21no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4#10312yes—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4C2×C22⋊C411yescharacteristic
197C22×C4C4×D411yescharacteristic
198C22×C4C4×D411yescharacteristic
199C22×C4#12011yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4#12311yescharacteristic
203C2×C8—22no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—84no—
224C2×C8—81no—
225C2×C8—81no—
226C2×C8—81no—
227C2×C8—81no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—81no—
239C4⋊C4—81no—
240C4⋊C4—81no—
241C4⋊C4—81no—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—81no—
249C22⋊C4—81no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—41no—
255C42—41no—
256C42—41no—
257C42—41no—
258C42—41no—
259C42—41no—
260C42—81no—
261C42—81no—
262C42C2×SD1611yescharacteristic
263C42C2×Q1611yescharacteristic
264C42#13511yescharacteristic
265C42#13511yescharacteristic
266C23—22no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23#38611yescharacteristic
274Q8—42no—
275Q8—42no—
276Q8—42no—
277Q8—81no—
278Q8—81no—
279Q8—81no—
280D4—42no—
281D4—81no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4C2×#10311yescharacteristic
318C2×C4#41611yescharacteristic
319C2×C4#44311yescharacteristic
320C2×C4#44411yescharacteristic
321C2×C4#45311yescharacteristic
322C2×C4#45411yescharacteristic
323C2×C4#46511yescharacteristic
324C2×C4#46511yescharacteristic
325C2×C4#46711yescharacteristic
326C2×C4#46711yescharacteristic
327C8—42no—
328C8—81no—
329C8—81no—
330C8—81no—
331C8—81no—
332C8—81no—
333C8—82no—
334C8—82no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22#187611yescharacteristic
343C22#188311yescharacteristic
344C22#189111yescharacteristic
345C22#189211yescharacteristic
346C22#190211yescharacteristic
347C22#190611yescharacteristic
348C22#191111yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—42no—
360C4—41no—
361C4—41no—
362C4—81no—
363C4—82no—
364C4—81no—
365C4—161no—
366C2—82no—
367C2#905211yescharacteristic
368C2#920711yescharacteristic
369C2#928511yescharacteristic
370C2#931711yescharacteristic
371C2#938111yescharacteristic
372C2#948311yescharacteristic
373C2#953111yescharacteristic
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. #9677C2semidirect
5. #7622C2semidirect
6. #7041C2semidirect
7. #7022C2semidirect

Maximal subgroups

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

subgroupindex
#292602
#223852
#96772
#85812
#76222
#70412
#70222