Finite Group Databasebeta

← #100509

Subgroups — #100509

Order 512.

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

normal subgroups only

This group has 425 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#33201C211yesmaximal, characteristic
2#22839C211yesmaximal, characteristic
3#8565C211yesmaximal, characteristic
4#7662C211yesmaximal, characteristic, complemented
5#7662C211yesmaximal, characteristic, complemented
6#7040C211yesmaximal, characteristic, complemented
7#7031C211yesmaximal, characteristic, complemented
8C2×#476C411yescharacteristic
9C2×#474C411yescharacteristic
10#2862C2211yescharacteristic
11C42×D4C2211yescharacteristic
12#1988C411yescharacteristic
13#1987C411yescharacteristic
14#1954—21no—
15#1949—21no—
16#1949—21no—
17#1924—22no—
18#1842C2211yescharacteristic
19#1831C2211yescharacteristic
20#1831C2211yescharacteristic
21#1817—21no—
22#1804—21no—
23#1804—21no—
24#1770—21no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27#1749—21no—
28C2×D4⋊C8—21no—
29#1279—21no—
30#1272—21no—
31#1272—21no—
32#1272—22no—
33C2×C4⋊Q8C2×C411yescharacteristic
34C2×C4⋊Q8C2×C411yescharacteristic
35C2×C4×D4C2×C411yescharacteristic
36C2×C4×D4D411yescharacteristic
37C2×C4×D4D411yescharacteristic
38C22×C42—21no—
39#476—42no—
40#474—42no—
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×C4⋊C8—21no—
49C2×C4⋊C8C2×C411yescharacteristic
50C2×C4⋊C8C2×C411yescharacteristic
51C2×#102—21no—
52C2×#102—21no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×#97—22no—
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58#391—21no—
59#391—21no—
60#391—21no—
61#391—21no—
62#382—21no—
63#382—21no—
64C4×C4⋊C4C2×C411yescharacteristic
65C4×C4⋊C4D411yescharacteristic
66C4×C4⋊C4D411yescharacteristic
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4—21no—
70C43C2311yescharacteristic
71#340—21no—
72#340—21no—
73#340—21no—
74#340—21no—
75#336—22no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336D411yescharacteristic
81#336D412yes—
82#336D411yescharacteristic
83D4⋊C8—42no—
84C22×Q8—21no—
85C22×Q8—21no—
86C22×D4C22⋊C411yescharacteristic
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—22no—
95C22×C8—41no—
96C22×C8—41no—
97C4⋊Q8—41no—
98C4⋊Q8—41no—
99C4⋊Q8—41no—
100C4⋊Q8—41no—
101C4×D4—22no—
102C4×D4—21no—
103C4×D4—21no—
104C4×D4—21no—
105C4×D4—21no—
106C4×D4—21no—
107C4×D4—21no—
108C4×D4—21no—
109C4×D4—21no—
110C4×D4—41no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4C22⋊C411yescharacteristic
116C2×C4⋊C4C22⋊C411yescharacteristic
117C2×C4⋊C4C22⋊C411yescharacteristic
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—82no—
125C2×C42C22×C411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42#4111yescharacteristic
131C2×C42#4111yescharacteristic
132#107—42no—
133#106—42no—
134C4⋊C8—22no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—42no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147#102—42no—
148#102—42no—
149C22⋊C8—44no—
150C22⋊C8—82no—
151#97—82no—
152C4×C8—41no—
153C4×C8—41no—
154C4×C8—41no—
155C4×C8—41no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—41no—
160#95—41no—
161#95—41no—
162#95—41no—
163C24—21no—
164C2×Q8—42no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—81no—
168C2×Q8—81no—
169C2×Q8—81no—
170C2×Q8—81no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4#10012yes—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4C2×C22⋊C411yescharacteristic
195C22×C4C4×D411yescharacteristic
196C22×C4C4×D411yescharacteristic
197C22×C4#12011yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4#12311yescharacteristic
201C2×C8—22no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—41no—
207C2×C8—42no—
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—42no—
217C2×C8—81no—
218C2×C8—81no—
219C2×C8—81no—
220C2×C8—81no—
221C2×C8—81no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—81no—
233C4⋊C4—81no—
234C22⋊C4—42no—
235C22⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—41no—
244C42—41no—
245C42—41no—
246C42—42no—
247C42—81no—
248C42—81no—
249C42#13511yescharacteristic
250C42#13511yescharacteristic
251C42AGL(1,ℤ/8)11yescharacteristic
252C42#13711yescharacteristic
253C42#13711yescharacteristic
254C42#13711yescharacteristic
255C42#13711yescharacteristic
256C42#13711yescharacteristic
257C23—22no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23#38611yescharacteristic
265Q8—161no—
266Q8—161no—
267Q8—162no—
268D4—22no—
269D4—22no—
270D4—41no—
271D4—41no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4C2×#10011yescharacteristic
307C2×C4#41311yescharacteristic
308C2×C4#41611yescharacteristic
309C2×C4#41711yescharacteristic
310C2×C4#41911yescharacteristic
311C2×C4#41911yescharacteristic
312C2×C4#44311yescharacteristic
313C2×C4#44411yescharacteristic
314C2×C4#45211yescharacteristic
315C2×C4#45211yescharacteristic
316C2×C4#46811yescharacteristic
317C2×C4#46811yescharacteristic
318C2×C4#47011yescharacteristic
319C2×C4#47011yescharacteristic
320C2×C4#48511yescharacteristic
321C2×C4#48511yescharacteristic
322C8—42no—
323C8—81no—
324C8—82no—
325C8—82no—
326C8—81no—
327C8—81no—
328C8—161no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22#187311yescharacteristic
337C22#187611yescharacteristic
338C22#188111yescharacteristic
339C22#189411yescharacteristic
340C22#190011yescharacteristic
341C22#190411yescharacteristic
342C22#190611yescharacteristic
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—41no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4—161no—
358C4—161no—
359C4#155111yescharacteristic
360C4#155411yescharacteristic
361C4#155811yescharacteristic
362C4#155911yescharacteristic
363C2—82no—
364C2#899411yescharacteristic
365C2#913611yescharacteristic
366C2#917711yescharacteristic
367C2#920911yescharacteristic
368C2#924311yescharacteristic
369C2#930711yescharacteristic
370C2#934911yescharacteristic
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
4. #7662C2semidirect
5. #7662C2semidirect
6. #7040C2semidirect
7. #7031C2semidirect

Maximal subgroups

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

subgroupindex
#332012
#228392
#85652
#76622
#70402
#70312