Finite Group Databasebeta

← #100045

Subgroups — #100045

Order 512.

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

normal subgroups only

This group has 433 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#22385C211yesmaximal, characteristic
2#22273C211yesmaximal, characteristic
3#9864C211yesmaximal, characteristic, complemented
4#7604C211yesmaximal, characteristic, complemented
5#7584C211yesmaximal, characteristic
6#7023C211yesmaximal, characteristic, complemented
7#7023C211yesmaximal, characteristic, complemented
8C2×#497C411yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10C4×#124C2211yescharacteristic
11#1984—21no—
12#1960C411yescharacteristic
13#1959C411yescharacteristic
14#1923—21no—
15#1923—21no—
16#1880—21no—
17#1878—21no—
18#1878—21no—
19#1877—21no—
20#1869—21no—
21#1850—21no—
22#1839C2211yescharacteristic
23#1839C2211yescharacteristic
24#1836C411yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C8×C22⋊C4—21no—
28#1745C2211yescharacteristic
29#1279—21no—
30#1279—21no—
31#1272—21no—
32#1271—21no—
33#1271—21no—
34#1270—21no—
35C22×Q16—21no—
36C22×SD16—21no—
37C23×C8—21no—
38C2×C4⋊Q8C2×C411yescharacteristic
39C2×#124C2×C411yescharacteristic
40C2×C4×Q8—21no—
41C2×C4×Q8D411yescharacteristic
42C2×C4×D4D411yescharacteristic
43#497—22no—
44#497C814yescomplemented
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×#103—21no—
49C2×#103—21no—
50C2×#103—21no—
51C2×#102—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×#97C2×C411yescharacteristic
56C2×#97C2×C411yescharacteristic
57C2×C4×C8—21no—
58C2×C4×C8—21no—
59C2×C4×C8D411yescharacteristic
60C2×C4×C8D411yescharacteristic
61#391—21no—
62#391D411yescharacteristic
63#390D411yescharacteristic
64#389—21no—
65#389—21no—
66#388—21no—
67#388—21no—
68#384C2×C411yescharacteristic
69#382C2×C411yescharacteristic
70C4×C4⋊C4—21no—
71C4×C4⋊C4—21no—
72C4×C22⋊C4—21no—
73C4×C22⋊C4—21no—
74C43C2311yescharacteristic
75#340—21no—
76#340—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336D411yescharacteristic
89#336D411yescharacteristic
90C22×Q8—21no—
91C22×Q8C22⋊C411yescharacteristic
92C22×D4C22⋊C411yescharacteristic
93C23×C4—21no—
94C2×Q16—24no—
95C2×Q16—81no—
96C2×Q16—81no—
97C2×SD16—24no—
98C2×SD16—81no—
99C2×SD16—81no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—42no—
107C22×C8—41no—
108C22×C8—41no—
109C22×C8—81no—
110C22×C8—81no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8C2×C812yescomplemented
115#124—21no—
116#124—21no—
117#124C2×C812yes—
118#124M1612yes—
119C4×Q8—81no—
120C4×Q8—81no—
121C4×Q8—81no—
122C4×D4—81no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C4⋊C4C22⋊C411yescharacteristic
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C42—21no—
138C2×C42—41no—
139C2×C42C22×C411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42#4111yescharacteristic
145C2×C42#4111yescharacteristic
146C4⋊C8—22no—
147C4⋊C8—41no—
148C4⋊C8—42no—
149C4⋊C8—81no—
150#103—81no—
151#103—81no—
152#103—81no—
153#102—81no—
154C22⋊C8—42no—
155C22⋊C8—81no—
156C22⋊C8—81no—
157#97—21no—
158#97—21no—
159#97—21no—
160#97C2×C812yescomplemented
161C4×C8—41no—
162C4×C8—41no—
163C4×C8—81no—
164C4×C8—81no—
165#95—21no—
166#95—21no—
167#95—21no—
168#95—21no—
169#95—41no—
170#95—41no—
171C24—21no—
172C2×Q8—21no—
173C2×Q8—22no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8C22⋊C812yes—
182C2×D4—21no—
183C2×D4—41no—
184C2×D4—41no—
185C2×D4C22⋊C812yes—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4C2×C22⋊C411yescharacteristic
198C22×C4C4×D411yescharacteristic
199C22×C4C4×D411yescharacteristic
200C22×C4#12011yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4#12311yescharacteristic
204Q16—84no—
205SD16—84no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—42no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—84no—
226C2×C8—81no—
227C2×C8—81no—
228C2×C8—81no—
229C2×C8—81no—
230C2×C8—81no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—81no—
241C4⋊C4—81no—
242C4⋊C4—82no—
243C22⋊C4—22no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—81no—
250C22⋊C4—81no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—41no—
256C42—41no—
257C42—41no—
258C42—41no—
259C42—41no—
260C42—41no—
261C42—82no—
262C42C22×C811yescharacteristic
263C42C2×M1611yescharacteristic
264C42#13111yescharacteristic
265C42#13111yescharacteristic
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—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
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×C22⋊C811yescharacteristic
318C2×C4#40811yescharacteristic
319C2×C4C8×D411yescharacteristic
320C2×C4C8×D411yescharacteristic
321C2×C4#43511yescharacteristic
322C2×C4#43511yescharacteristic
323C2×C4#45511yescharacteristic
324C2×C4#45611yescharacteristic
325C2×C4#47111yescharacteristic
326C2×C4#47311yescharacteristic
327C8—42no—
328C8—42no—
329C8—81no—
330C8—81no—
331C8—81no—
332C8—81no—
333C8—82no—
334C8—161no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22#186911yescharacteristic
343C22#188111yescharacteristic
344C22#188211yescharacteristic
345C22#190611yescharacteristic
346C22#190811yescharacteristic
347C22#191011yescharacteristic
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—82no—
363C4—81no—
364C4—81no—
365C4—161no—
366C2—82no—
367C2#895311yescharacteristic
368C2#909611yescharacteristic
369C2#910111yescharacteristic
370C2#937011yescharacteristic
371C2#937311yescharacteristic
372C2#937611yescharacteristic
373C2#937911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C22×C8
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9864C2semidirect
4. #7604C2semidirect
6. #7023C2semidirect
7. #7023C2semidirect
44. #497C8semidirect
114. C4⋊Q8C2×C8semidirect
160. #97C2×C8semidirect

Maximal subgroups

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

subgroupindex
#223852
#222732
#98642
#76042
#75842
#70232