Finite Group Databasebeta

← #223175

Subgroups — #223175

Order 512.

919 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36211C211yesmaximal, characteristic
2#22748C212yesmaximal, complemented
3#9901C212yesmaximal, complemented
4#9134C212yesmaximal, complemented
5#9133C212yesmaximal, complemented
6#8727C212yesmaximal, complemented
7#8592C211yesmaximal, characteristic, complemented
8C4×#331C211yesmaximal, characteristic, complemented
9#8039C212yesmaximal, complemented
10C2×#431C2212yes—
11C22×C4⋊C8C2212yes—
12#2859C2211yescharacteristic
13#2856C2212yes—
14#2535C412yes—
15#2513—22no—
16#2490C414yes—
17#2420—22no—
18#2414—22no—
19#2363—22no—
20#2357C412yes—
21C4×#127C2211yescharacteristic
22#2285C2212yes—
23C42×D4C2211yescharacteristic
24#1923—22no—
25#1916—22no—
26#1869—22no—
27#1831C2212yes—
28#1797C2212yes—
29#1797C2212yes—
30C4×C4⋊C8C2212yes—
31C8×C22⋊C4—22no—
32C2×#331C2212yes—
33C2×#331C2211yescharacteristic
34C2×#331C2211yescharacteristic
35#1296C2211yescharacteristic
36#1296C2212yes—
37#1296C2212yes—
38#1296C2212yes—
39#1296C2211yescharacteristic
40#1296C2212yes—
41#1296C2212yes—
42C23×C8—22no—
43C2×#127C2311yescharacteristic
44C2×#125C2×C412yes—
45C2×#121—22no—
46C2×#121—22no—
47C2×#121C2×C414yes—
48C2×C4×D4—22no—
49C2×C4×D4C2×C412yes—
50C2×C4×D4C2×C411yescharacteristic
51C2×C4×D4D412yes—
52C2×#117—22no—
53C22×C42C2312yes—
54#431—24no—
55#431—42no—
56C2×C4⋊C8—24no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—42no—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C4×C8—22no—
67C2×C4×C8—22no—
68#397—22no—
69#397—24no—
70#394—24no—
71#392—22no—
72#390—24no—
73#390—22no—
74#390—22no—
75#390—22no—
76#390C2×C412yes—
77#390C2×C414yes—
78#389—22no—
79#389—22no—
80#388—24no—
81#388—24no—
82#386—24no—
83#384C2311yescharacteristic
84#384C2312yes—
85#382—22no—
86#382C2×C412yes—
87#381—22no—
88#380—22no—
89#380C2×C414yes—
90C4×C4⋊C4—22no—
91C4×C4⋊C4C2×C414yes—
92C4×C4⋊C4C2×C411yescharacteristic
93C4×C4⋊C4D412yes—
94C4×C22⋊C4—22no—
95C4×C22⋊C4C2×C412yes—
96C4×C22⋊C4D412yes—
97C4×C22⋊C4D412yes—
98C43C2311yescharacteristic
99#336—22no—
100#336—22no—
101#336—22no—
102#336—22no—
103#336—22no—
104#331—24no—
105#331—41no—
106#331—41no—
107C22×D4—24no—
108C22×D4—22no—
109C22×D4C22⋊C411yescharacteristic
110C22×D4C22×C412yes—
111C23×C4—22no—
112C23×C4C22×C412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—44no—
120C22×C8—44no—
121C22×C8—82no—
122#127—22no—
123#127—21no—
124#127—21no—
125#125—82no—
126#121—44no—
127#121—44no—
128#121—82no—
129#121—82no—
130#121—84no—
131C4×D4—24no—
132C4×D4—24no—
133C4×D4—42no—
134C4×D4—44no—
135C4×D4—82no—
136C4×D4—82no—
137#117—82no—
138#117—82no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C411yescharacteristic
145C2×C4⋊C4C22×C412yes—
146C2×C4⋊C4C22×C414yes—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4C22⋊C412yes—
154C2×C22⋊C4C22⋊C412yes—
155C2×C22⋊C4C22⋊C412yes—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—24no—
161C2×C42C22×C411yescharacteristic
162C2×C42C22×C412yes—
163C2×C42C22×C411yescharacteristic
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C2×C42C2×D412yes—
167C2×C42C2×D412yes—
168C2×C42C2411yescharacteristic
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—44no—
177C4⋊C8#4114yes—
178C22⋊C8—44no—
179C22⋊C8—44no—
180C22⋊C8—82no—
181C4×C8—44no—
182C4×C8—82no—
183#95—22no—
184#95—24no—
185#95—24no—
186#95—22no—
187#95C22⋊C411yescharacteristic
188#95C22⋊C412yes—
189#95C22⋊C412yes—
190#95C22⋊C411yescharacteristic
191C24—22no—
192C24C2×C22⋊C412yes—
193C2×D4—24no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—41no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—48no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—84no—
207C2×D4—84no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—84no—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C412yes—
232C22×C4#13514yes—
233C22×C4#13514yes—
234C22×C4C23×C411yescharacteristic
235C22×C4C22×D412yes—
236C2×C8—24no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—88no—
249C4⋊C4—24no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—44no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—84no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—44no—
264C22⋊C4—48no—
265C22⋊C4—84no—
266C22⋊C4—84no—
267C22⋊C4—84no—
268C22⋊C4—84no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—41no—
281C42—84no—
282C42C2×#4112yes—
283C42H(5,𝔽2)11yescharacteristic
284C422−1+411yescharacteristic
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—44no—
295C23—44no—
296C23#41614yes—
297C23C22×C22⋊C411yescharacteristic
298D4—44no—
299D4—82no—
300D4—82no—
301D4—82no—
302D4—82no—
303D4—82no—
304D4—82no—
305D4—168no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—164no—
338C2×C4#41312yes—
339C2×C4#41614yes—
340C2×C4C4×#4111yescharacteristic
341C2×C4#52011yescharacteristic
342C2×C4#54612yes—
343C2×C4#54712yes—
344C2×C4C2×#13512yes—
345C2×C4C2×#13512yes—
346C2×C4#57612yes—
347C2×C4#57712yes—
348C8—44no—
349C8—82no—
350C8—82no—
351C8—82no—
352C22—24no—
353C22—24no—
354C22—24no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—82no—
360C22—84no—
361C22—82no—
362C22—82no—
363C22—84no—
364C22#233911yescharacteristic
365C22C2×#41311yescharacteristic
366C22#288811yescharacteristic
367C22C2×#41612yes—
368C22#289712yes—
369C4—22no—
370C4—21no—
371C4—21no—
372C4—24no—
373C4—41no—
374C4—42no—
375C4—42no—
376C4—42no—
377C4—84no—
378C4—82no—
379C4#329712yes—
380C4#330112yes—
381C4#333712yes—
382C4#333912yes—
383C2—44no—
384C2—162no—
385C2#1770011yescharacteristic
386C2#1772412yes—
387C2#1777111yescharacteristic
388C2#2248311yescharacteristic
389C2#2248812yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36211C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22748C2semidirect
3. #9901C2semidirect
4. #9134C2semidirect
5. #9133C2semidirect
6. #8727C2semidirect
7. #8592C2semidirect
8. C4×#331C2semidirect
9. #8039C2semidirect

Maximal subgroups

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

subgroupindex
#362112
#227482
#99012
#91342
#91332
#87272
#85922
C4×#3312
#80392