Finite Group Databasebeta

← #190286

Subgroups — #190286

Order 512.

870 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34189C211yesmaximal, characteristic
2#20728C211yesmaximal, characteristic
3#17366C211yesmaximal, characteristic
4#8771C214yesmaximal, complemented
5#8768C214yesmaximal, complemented
6#8094C212yesmaximal, complemented
7#7959C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9#2920C2212yes—
10#2920C2212yes—
11#2357—21no—
12#2357C2212yes—
13#2356—21no—
14#2356C2212yes—
15#2353—22no—
16#2353C414yes—
17#2314—22no—
18#2311—21no—
19#2285C2211yescharacteristic
20#2283C412yes—
21#2283C412yes—
22C2×#382C2211yescharacteristic
23#2274—21no—
24#1822—21no—
25#1801C2212yes—
26#1800C2212yes—
27#1785—21no—
28C4×#101—21no—
29C4×#100—21no—
30#1465—22no—
31#1464—22no—
32#1461C2212yes—
33#1460C2212yes—
34#1303C2218yes—
35#1286C2218yes—
36C22×#41D411yescharacteristic
37C4×#41—22no—
38C4×#41—22no—
39C4×#41—22no—
40C4×#41—22no—
41C2×C4×Q8D411yescharacteristic
42C2×C4×Q8C2311yescharacteristic
43C2×C4×D4C2×C412yes—
44C2×C4×D4D412yes—
45C2×C4×D4D411yescharacteristic
46C2×C4×D4C2311yescharacteristic
47C2×#117—21no—
48C2×#117D411yescharacteristic
49C2×#117D412yes—
50C22×C4⋊C4—21no—
51C22×C42C2×C412yes—
52C22×C42C2311yescharacteristic
53#432—22no—
54#432—22no—
55#432—22no—
56#432—22no—
57#431—22no—
58#431—22no—
59#431—22no—
60#431C2314yes—
61#423—22no—
62#423C2314yes—
63C2×#101—22no—
64C2×#100—22no—
65#407—22no—
66#407—22no—
67C4×M16—22no—
68C4×M16—22no—
69#391—22no—
70#390—22no—
71#388—24no—
72#388—24no—
73#387—24no—
74#387—24no—
75#386—24no—
76#384—21no—
77#383—22no—
78#383C2×C414yes—
79#382—21no—
80#382—21no—
81#382C2×C412yes—
82#382C2×C411yescharacteristic
83#382C2×C411yescharacteristic
84#382C2312yes—
85#381—22no—
86#381C2×C414yes—
87#380—22no—
88#380—22no—
89#380—21no—
90#380C2×C412yes—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4C2312yes—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—24no—
99C4×C22⋊C4C2×C414yes—
100#376—22no—
101#376C2×C414yes—
102C2×#95—21no—
103C2×#95C2×C412yes—
104#343—22no—
105#343—22no—
106#324—24no—
107#323—24no—
108C2×#41—22no—
109C2×#41—22no—
110C2×#41—42no—
111C2×#41—42no—
112C22×Q8C2×D411yescharacteristic
113C22×D4C22⋊C412yes—
114C22×D4C2×D411yescharacteristic
115C23×C4—22no—
116C23×C4C22⋊C412yes—
117C23×C4C22⋊C412yes—
118C23×C4C22×C412yes—
119C23×C4C22×C411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C2×M16—24no—
123C2×M16—42no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×D4—24no—
129C4×D4—24no—
130C4×D4—24no—
131C4×D4—24no—
132C4×D4—22no—
133C4×D4—22no—
134C4×D4—22no—
135C4×D4—22no—
136#117—22no—
137#117—22no—
138#117—24no—
139#117—24no—
140#117—44no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C2×D412yes—
150C2×C4⋊C4C2×D411yescharacteristic
151C2×C4⋊C4C2×D411yescharacteristic
152C2×C22⋊C4—22no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4C22⋊C412yes—
156C2×C22⋊C4C22⋊C412yes—
157C2×C22⋊C4C2×D412yes—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—24no—
164C2×C42—24no—
165C2×C42C22⋊C412yes—
166C2×C42C22⋊C412yes—
167C2×C42C22×C412yes—
168C2×C42C22×C411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42#4112yes—
172C2×C42#4112yes—
173C2×C42#4112yes—
174C2×C42#4112yes—
175C2×C42C2411yescharacteristic
176C4⋊C8—24no—
177C4⋊C8—24no—
178C4⋊C8—24no—
179C4⋊C8—42no—
180#101—44no—
181#100—44no—
182C22⋊C8—24no—
183C22⋊C8—42no—
184C22⋊C8—42no—
185C22⋊C8—42no—
186#97—42no—
187#97—42no—
188C4×C8—24no—
189C4×C8—42no—
190#95—22no—
191#95—22no—
192#95—21no—
193#95—24no—
194#95—24no—
195#95—21no—
196#95—24no—
197#95—24no—
198#95—22no—
199#95C22×C414yes—
200#95C22×C412yes—
201#95C22×C412yes—
202C24C2×C22⋊C412yes—
203C24C2×C22⋊C411yescharacteristic
204#41—48no—
205#41—48no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×D4—24no—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—42no—
214C2×D4—42no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4#11712yes—
242C22×C4#11712yes—
243C22×C4#12312yes—
244C22×C4#12312yes—
245C22×C4#12312yes—
246C22×C4#12312yes—
247C22×C4C23×C411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4C22×D411yescharacteristic
250M16—44no—
251M16—44no—
252M16—84no—
253C2×C8—44no—
254C2×C8—44no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—44no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—44no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—44no—
269C22⋊C4—84no—
270C22⋊C4—84no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—44no—
278C42—84no—
279C42—84no—
280C42#11711yescharacteristic
281C42#11711yescharacteristic
282C42#11711yescharacteristic
283C42#11711yescharacteristic
284C42#12312yes—
285C42#12312yes—
286C42#12312yes—
287C42#12312yes—
288C42C2×#4112yes—
289C42C2×#4112yes—
290C23—21no—
291C23—21no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—41no—
296C23—41no—
297C23—42no—
298C23—42no—
299C23#38112yes—
300C23C22×C22⋊C411yescharacteristic
301Q8—44no—
302D4—48no—
303D4—44no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—164no—
339C2×C4#38111yescharacteristic
340C2×C4#38111yescharacteristic
341C2×C4#38111yescharacteristic
342C2×C4#38111yescharacteristic
343C2×C4#38112yes—
344C2×C4C2×#11711yescharacteristic
345C2×C4C2×#11711yescharacteristic
346C2×C4C2×#12312yes—
347C2×C4C2×#12312yes—
348C2×C4#57512yes—
349C2×C4#57512yes—
350C8—84no—
351C8—84no—
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—82no—
363C22—82no—
364C22C2×#38111yescharacteristic
365C22#289611yescharacteristic
366C22#289611yescharacteristic
367C4—21no—
368C4—21no—
369C4—22no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—22no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—82no—
379C4—82no—
380C4—84no—
381C4#309314yes—
382C2—21no—
383C2—21no—
384C2—42no—
385C2—82no—
386C2#2070111yescharacteristic
387C2#2070811yescharacteristic
388C2#2071311yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8771C2semidirect
5. #8768C2semidirect
6. #8094C2semidirect
7. #7959C2semidirect

Maximal subgroups

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

subgroupindex
#341892
#207282
#173662
#87712
#87682
#80942
#79592