Finite Group Databasebeta

← #101470

Subgroups — #101470

Order 512.

429 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 96 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#28202C211yesmaximal, characteristic
2#22311C211yesmaximal, characteristic, complemented
3#9644C211yesmaximal, characteristic
4#8591C211yesmaximal, characteristic, complemented
5#7651C211yesmaximal, characteristic
6#7040C211yesmaximal, characteristic, complemented
7#7018C211yesmaximal, characteristic, complemented
8C2×#476—21no—
9C4×C4⋊Q8C2211yescharacteristic
10C4×#127C2211yescharacteristic
11#2084C412yes—
12#2041C412yes—
13#1985—22no—
14#1939—21no—
15#1926—22no—
16#1917—22no—
17#1842C2211yescharacteristic
18#1831C2211yescharacteristic
19#1831C2211yescharacteristic
20#1831C2211yescharacteristic
21#1802—21no—
22#1770—21no—
23#1744C2211yescharacteristic
24C2×#332—21no—
25C2×D4⋊C8—21no—
26C2×D4⋊C8—21no—
27#1296—21no—
28#1270—22no—
29C2×C4⋊Q8D411yescharacteristic
30C2×#127C2×C411yescharacteristic
31C2×C4×Q8—21no—
32C2×C4×D4—21no—
33C2×C4×D4—21no—
34C2×C4×D4—21no—
35#476—42no—
36#476—42no—
37C2×#106—22no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8—21no—
45C2×#102—22no—
46C2×#102—22no—
47C2×#97—22no—
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50#391—21no—
51#390—21no—
52#390—21no—
53#390—21no—
54#389—21no—
55#389C2×C412yes—
56#384C2×C411yescharacteristic
57#384D411yescharacteristic
58C4×C4⋊C4—21no—
59C4×C4⋊C4D412yes—
60C43C2311yescharacteristic
61#340—22no—
62#336—22no—
63#336—22no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69#336—21no—
70#336C2×C412yes—
71#336D412yes—
72#332—42no—
73D4⋊C8—42no—
74D4⋊C8—42no—
75C22×Q8—21no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—21no—
79C23×C4—41no—
80C22×C8—21no—
81C22×C8—22no—
82C22×C8—21no—
83C22×C8—22no—
84C22×C8—41no—
85C22×C8—41no—
86C4⋊Q8—21no—
87C4⋊Q8—21no—
88C4⋊Q8—21no—
89C4⋊Q8—21no—
90#127—42no—
91C4×Q8—82no—
92C4×D4—41no—
93C4×D4—41no—
94C4×D4—41no—
95C4×D4—41no—
96C4×D4—41no—
97C4×D4—41no—
98C4×D4—41no—
99C4×D4—41no—
100C4×D4—81no—
101C4×D4—81no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4C22⋊C412yes—
110C2×C4⋊C4C22⋊C412yes—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C42—22no—
115C2×C42C22×C411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42#4111yescharacteristic
121C2×C42#4111yescharacteristic
122#106—44no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—81no—
134C4⋊C8—82no—
135#102—44no—
136#102—44no—
137#97—82no—
138C4×C8—21no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8SD1612yes—
142#95—22no—
143#95—22no—
144C24—41no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×Q8—81no—
148C2×Q8—81no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—41no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—81no—
164C2×D4—81no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—41no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4C2×C22⋊C411yescharacteristic
175C22×C4C4×D411yescharacteristic
176C22×C4C4×D411yescharacteristic
177C22×C4#12011yescharacteristic
178C22×C4#12111yescharacteristic
179C22×C4#12111yescharacteristic
180C22×C4#12311yescharacteristic
181C2×C8—22no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—41no—
192C2×C8—42no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8—81no—
196C2×C8—81no—
197C2×C8—81no—
198C2×C8—81no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—42no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—81no—
211C4⋊C4—82no—
212C4⋊C4—81no—
213C22⋊C4—82no—
214C22⋊C4—81no—
215C22⋊C4—81no—
216C22⋊C4—81no—
217C22⋊C4—81no—
218C42—21no—
219C42—21no—
220C42—41no—
221C42—41no—
222C42—41no—
223C42—41no—
224C42—42no—
225C42—42no—
226C42C2×SD1611yescharacteristic
227C42#13511yescharacteristic
228C42AGL(1,ℤ/8)11yescharacteristic
229C42AGL(1,ℤ/8)11yescharacteristic
230C42AGL(1,ℤ/8)11yescharacteristic
231C42AGL(1,ℤ/8)11yescharacteristic
232C42#13711yescharacteristic
233C42#13711yescharacteristic
234C23—42no—
235C23—81no—
236C23—81no—
237C23—81no—
238C23—81no—
239C23—81no—
240C23—81no—
241C23#38611yescharacteristic
242Q8—82no—
243Q8—82no—
244D4—42no—
245D4—42no—
246D4—42no—
247D4—42no—
248D4—81no—
249D4—81no—
250D4—81no—
251D4—81no—
252D4—82no—
253D4—82no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—161no—
278C2×C4—161no—
279C2×C4—161no—
280C2×C4—161no—
281C2×C4#41711yescharacteristic
282C2×C4#41811yescharacteristic
283C2×C4C4×SD1611yescharacteristic
284C2×C4#44011yescharacteristic
285C2×C4#44811yescharacteristic
286C2×C4#44911yescharacteristic
287C2×C4#46011yescharacteristic
288C2×C4#46411yescharacteristic
289C2×C4#46811yescharacteristic
290C2×C4#46811yescharacteristic
291C2×C4#46911yescharacteristic
292C2×C4#46911yescharacteristic
293C2×C4#47111yescharacteristic
294C2×C4#47211yescharacteristic
295C8—42no—
296C8—82no—
297C8—82no—
298C8—81no—
299C8—161no—
300C8—161no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—161no—
305C22—161no—
306C22—161no—
307C22—161no—
308C22#188011yescharacteristic
309C22#188111yescharacteristic
310C22#189111yescharacteristic
311C22#189511yescharacteristic
312C22#190411yescharacteristic
313C22#190711yescharacteristic
314C22#191211yescharacteristic
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—82no—
326C4—82no—
327C4—161no—
328C4#168611yescharacteristic
329C4#168811yescharacteristic
330C4#169811yescharacteristic
331C4#170211yescharacteristic
332C2—162no—
333C2#905011yescharacteristic
334C2#923011yescharacteristic
335C2#930211yescharacteristic
336C2#930911yescharacteristic
337C2#937211yescharacteristic
338C2#949011yescharacteristic
339C2#952911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC2×#127C2×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
2. #22311C2semidirect
4. #8591C2semidirect
6. #7040C2semidirect
7. #7018C2semidirect

Maximal subgroups

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

subgroupindex
#282022
#223112
#96442
#85912
#76512
#70402
#70182