Finite Group Databasebeta

← #99991

Subgroups — #99991

Order 512.

504 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 104 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#21614C211yesmaximal, characteristic
2#20568C211yesmaximal, characteristic
3#9673C211yesmaximal, characteristic, complemented
4#8565C211yesmaximal, characteristic
5#7591C211yesmaximal, characteristic, complemented
6#7027C211yesmaximal, characteristic, complemented
7#7022C211yesmaximal, characteristic, complemented
8C2×#476C411yescharacteristic
9C2×#436—21no—
10C4×C4⋊Q8C2211yescharacteristic
11C42×D4C2211yescharacteristic
12#1987C411yescharacteristic
13#1947C411yescharacteristic
14#1919—21no—
15#1862—21no—
16#1831C2211yescharacteristic
17#1802C411yescharacteristic
18#1776—21no—
19#1770—21no—
20#1767C2211yescharacteristic
21C4×C4⋊C8C2211yescharacteristic
22C4×C4⋊C8C2211yescharacteristic
23#1762—21no—
24#1753—22no—
25#1749—21no—
26#1744C2211yescharacteristic
27#1728—21no—
28C2×Q8⋊C8—22no—
29C2×D4⋊C8—21no—
30#1279—21no—
31#1271—21no—
32#1271—21no—
33C22×M16—22no—
34C2×C4⋊Q8C2×C411yescharacteristic
35C2×C4×Q8—21no—
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4D411yescharacteristic
38C2×C4×D4D411yescharacteristic
39C22×C42—21no—
40#476—22no—
41#476—22no—
42#436—24no—
43#436—24no—
44C2×#107—21no—
45C2×#106—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8C2×C411yescharacteristic
50C2×C4⋊C8C2×C411yescharacteristic
51C2×#102—21no—
52C2×#102—21no—
53C2×C22⋊C8—22no—
54C2×#97—21no—
55C2×#97—22no—
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58C2×C4×C8D412yes—
59#391—21no—
60#389—21no—
61#389—21no—
62#384C2×C411yescharacteristic
63C4×C4⋊C4—21no—
64C4×C4⋊C4—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#336—21no—
83#336D411yescharacteristic
84#336D411yescharacteristic
85Q8⋊C8—44no—
86D4⋊C8—42no—
87C22×Q8—21no—
88C22×D4C22⋊C411yescharacteristic
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—21no—
92C2×M16—28no—
93C2×M16—82no—
94C2×M16—82no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—22no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—41no—
101C22×C8—41no—
102C4⋊Q8—21no—
103C4⋊Q8—21no—
104C4⋊Q8—21no—
105C4⋊Q8M1612yescomplemented
106C4×Q8—42no—
107C4×Q8—42no—
108C4×D4—22no—
109C4×D4—22no—
110C4×D4—21no—
111C4×D4—21no—
112C4×D4—21no—
113C4×D4—21no—
114C4×D4—21no—
115C4×D4—21no—
116C4×D4—21no—
117C4×D4—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×C4⋊C4C22⋊C411yescharacteristic
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—41no—
134C2×C42—82no—
135C2×C42C22×C411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42#4111yescharacteristic
141C2×C42#4111yescharacteristic
142#107—42no—
143#106—24no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—21no—
150C4⋊C8—41no—
151C4⋊C8—41no—
152C4⋊C8—41no—
153C4⋊C8M1612yescomplemented
154#102—24no—
155#102—42no—
156C22⋊C8—28no—
157#97—41no—
158#97—41no—
159#97—82no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—42no—
163C4×C8—41no—
164C4×C8—41no—
165#95—21no—
166#95—21no—
167#95—21no—
168#95—41no—
169#95—41no—
170C24—21no—
171C2×Q8—22no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—41no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4#10412yes—
181C22×C4—24no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4C2×C22⋊C411yescharacteristic
199C22×C4C4×D411yescharacteristic
200C22×C4C4×D411yescharacteristic
201C22×C4#12011yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12311yescharacteristic
205M16—88no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—24no—
209C2×C8—22no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—42no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—42no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—81no—
223C2×C8—81no—
224C2×C8—81no—
225C2×C8—81no—
226C2×C8—81no—
227C4⋊C4—21no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—22no—
234C4⋊C4—42no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—81no—
240C4⋊C4—81no—
241C4⋊C4—81no—
242C4⋊C4#10412yes—
243C22⋊C4—24no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—41no—
253C42—41no—
254C42—41no—
255C42—42no—
256C42—81no—
257C42—81no—
258C42—82no—
259C42C2×M1611yescharacteristic
260C42C2×M1611yescharacteristic
261C42#13111yescharacteristic
262C42#13111yescharacteristic
263C42AGL(1,ℤ/8)11yescharacteristic
264C42#13711yescharacteristic
265C42#13711yescharacteristic
266C42#13711yescharacteristic
267C23—22no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23#38611yescharacteristic
275Q8—44no—
276Q8—82no—
277D4—22no—
278D4—22no—
279D4—41no—
280D4—41no—
281C2×C4—21no—
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—22no—
294C2×C4—21no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4#40711yescharacteristic
316C2×C4#40811yescharacteristic
317C2×C4#41711yescharacteristic
318C2×C4#41911yescharacteristic
319C2×C4C2×#10411yescharacteristic
320C2×C4C2×#10411yescharacteristic
321C2×C4#43511yescharacteristic
322C2×C4#43511yescharacteristic
323C2×C4#43611yescharacteristic
324C2×C4#43611yescharacteristic
325C2×C4#45311yescharacteristic
326C2×C4#45411yescharacteristic
327C2×C4#46811yescharacteristic
328C2×C4#46811yescharacteristic
329C2×C4#47011yescharacteristic
330C2×C4#47011yescharacteristic
331C8—42no—
332C8—44no—
333C8—42no—
334C8—42no—
335C8—82no—
336C8—81no—
337C8—161no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22#187011yescharacteristic
346C22#188111yescharacteristic
347C22#188311yescharacteristic
348C22#189411yescharacteristic
349C22#189611yescharacteristic
350C22#190211yescharacteristic
351C22#190411yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—41no—
363C4—42no—
364C4—81no—
365C4—81no—
366C4—82no—
367C4—161no—
368C4#148311yescharacteristic
369C4#148511yescharacteristic
370C4#148911yescharacteristic
371C4#149111yescharacteristic
372C2—82no—
373C2#895611yescharacteristic
374C2#907711yescharacteristic
375C2#909111yescharacteristic
376C2#930711yescharacteristic
377C2#931211yescharacteristic
378C2#932111yescharacteristic
379C2#932511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×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. #9673C2semidirect
5. #7591C2semidirect
6. #7027C2semidirect
7. #7022C2semidirect
105. C4⋊Q8M16semidirect
153. C4⋊C8M16semidirect

Maximal subgroups

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

subgroupindex
#216142
#205682
#96732
#85652
#75912
#70272
#70222