Finite Group Databasebeta

← #190375

Subgroups — #190375

Order 512.

884 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34197C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3#17361C211yesmaximal, characteristic
4#8779C214yesmaximal, complemented
5#8778C214yesmaximal, complemented
6#8050C212yesmaximal, complemented
7#7945C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9#2920C2212yes—
10#2357C412yescomplemented
11#2357C2212yes—
12#2357C2212yes—
13#2351C412yescomplemented
14#2346—21no—
15#2346—22no—
16#2341—21no—
17#2319—22no—
18#2319—21no—
19#2283C412yescomplemented
20#2283C412yescomplemented
21#2283C2211yescharacteristic
22#2283C2212yes—
23#2283C2211yescharacteristic
24#2276—21no—
25#1820—21no—
26#1799—22no—
27#1782—21no—
28C4×#99C2214yes—
29#1467—21no—
30#1466—21no—
31#1465C2214yes—
32#1458—22no—
33#1297C2214yes—
34#1297C2214yes—
35#1284C2218yes—
36C22×#41D411yescharacteristic
37C4×#41—22no—
38C4×#41—22no—
39C4×#41—22no—
40C4×#41—22no—
41C2×C4×Q8—22no—
42C2×C4×Q8C2×C411yescharacteristic
43C2×C4×Q8D411yescharacteristic
44C2×C4×D4C2×C411yescharacteristic
45C2×C4×D4D412yes—
46C2×C4×D4D411yescharacteristic
47C2×C4×D4C2312yes—
48C2×#117D411yescharacteristic
49C2×#117D412yes—
50C22×C42C2×C412yes—
51C22×C42C2311yescharacteristic
52#432—22no—
53#432—22no—
54#431—22no—
55#431—22no—
56#423—22no—
57C2×#99—24no—
58#407C2314yes—
59C4×M16—22no—
60#390—22no—
61#390—22no—
62#390—22no—
63#389—22no—
64#389—22no—
65#389—22no—
66#388—22no—
67#388—22no—
68#388—22no—
69#388—22no—
70#388—22no—
71#388—22no—
72#383—22no—
73#383C2×C411yescharacteristic
74#383C2×C411yescharacteristic
75#383C2×C412yes—
76#382—22no—
77#382—21no—
78#382—22no—
79#382C2×C412yes—
80#382C2×C412yes—
81#382C2×C412yes—
82#381—22no—
83#381—21no—
84#381—22no—
85#381C2×C412yes—
86#381C2×C412yes—
87#381C2312yes—
88#380—22no—
89#380—22no—
90#380—21no—
91#380—21no—
92#380—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4C2×C412yes—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4C2×C412yes—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—21no—
104C4×C22⋊C4C2×C412yes—
105C4×C22⋊C4C2×C412yes—
106C4×C22⋊C4C2314yes—
107C4×C22⋊C4C2311yescharacteristic
108C4×C22⋊C4C2311yescharacteristic
109#376—22no—
110#376—22no—
111#376—21no—
112#376—21no—
113#376—22no—
114#342—24no—
115#324—24no—
116#323—24no—
117C2×#41—22no—
118C2×#41—22no—
119C2×#41—44no—
120C22×Q8C22⋊C411yescharacteristic
121C22×D4C22⋊C411yescharacteristic
122C22×D4C2×D412yes—
123C23×C4C22⋊C412yes—
124C23×C4C22⋊C412yes—
125C23×C4C22×C412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2411yescharacteristic
129C2×M16—24no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—48no—
135C4×D4—24no—
136C4×D4—24no—
137C4×D4—24no—
138C4×D4—24no—
139C4×D4—22no—
140C4×D4—22no—
141C4×D4—22no—
142C4×D4—22no—
143#117—22no—
144#117—22no—
145#117—24no—
146#117—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4C22⋊C411yescharacteristic
155C2×C4⋊C4C22⋊C411yescharacteristic
156C2×C4⋊C4C22⋊C412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4C22⋊C412yes—
167C2×C22⋊C4C2×D412yes—
168C2×C22⋊C4C2×D412yes—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—24no—
181C2×C42C22⋊C412yes—
182C2×C42C22⋊C412yes—
183C2×C42C22×C412yes—
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42#4112yes—
189C2×C42#4112yes—
190C2×C42#4112yes—
191C2×C42#4112yes—
192C4⋊C8—42no—
193C4⋊C8—42no—
194#99—48no—
195C22⋊C8—24no—
196C22⋊C8—24no—
197#97—42no—
198C4×C8—42no—
199#95—21no—
200#95—21no—
201#95—22no—
202#95—22no—
203#95—21no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—21no—
210#95—22no—
211#95—24no—
212#95—42no—
213#95—42no—
214#95—42no—
215#95C22×C412yes—
216#95C22×C412yes—
217#95C22×C412yes—
218#95C22×C412yes—
219C24C2×C22⋊C412yes—
220C24C22×D411yescharacteristic
221#41—48no—
222#41—48no—
223C2×Q8—22no—
224C2×Q8—24no—
225C2×D4—24no—
226C2×D4—22no—
227C2×D4—24no—
228C2×D4—42no—
229C2×D4—42no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4#12312yes—
256C22×C4#12312yes—
257C22×C4#12312yes—
258C22×C4#12312yes—
259C22×C4C23×C411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C2×#4112yes—
262C22×C4C2×#4112yes—
263M16—84no—
264C2×C8—44no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—84no—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—48no—
280C22⋊C4—82no—
281C22⋊C4—82no—
282C22⋊C4—82no—
283C22⋊C4—81no—
284C22⋊C4—81no—
285C22⋊C4—82no—
286C22⋊C4—82no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—44no—
294C42—44no—
295C42—44no—
296C42—81no—
297C42—81no—
298C42—82no—
299C42—82no—
300C42—84no—
301C42#11711yescharacteristic
302C42#11711yescharacteristic
303C42#11711yescharacteristic
304C42#11711yescharacteristic
305C42#11712yes—
306C42#11712yes—
307C42#12312yes—
308C42#12312yes—
309C42#12312yes—
310C42#12312yes—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—41no—
316C23—41no—
317C23—42no—
318C23—42no—
319C23C22×C22⋊C411yescharacteristic
320C23C2×#12312yes—
321Q8—28no—
322D4—48no—
323D4—44no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4#38111yescharacteristic
364C2×C4#38111yescharacteristic
365C2×C4#38111yescharacteristic
366C2×C4#38111yescharacteristic
367C2×C4#38112yes—
368C2×C4#38112yes—
369C2×C4#42114yes—
370C2×C4C2×#11711yescharacteristic
371C2×C4C2×#11711yescharacteristic
372C2×C4C2×#12312yes—
373C8—84no—
374C22—21no—
375C22—21no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—82no—
383C22—82no—
384C22—82no—
385C22C2×#38111yescharacteristic
386C22C2×#42112yes—
387C4—21no—
388C4—21no—
389C4—22no—
390C4—24no—
391C4—21no—
392C4—21no—
393C4—22no—
394C4—22no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4—42no—
399C4—82no—
400C4—82no—
401C4—82no—
402C4—82no—
403C4—164no—
404C2—22no—
405C2—42no—
406C2—82no—
407C2#2070511yescharacteristic
408C2#2070811yescharacteristic
409C2#2071011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8779C2semidirect
5. #8778C2semidirect
6. #8050C2semidirect
7. #7945C2semidirect
10. #2357C4semidirect
13. #2351C4semidirect
19. #2283C4semidirect
20. #2283C4semidirect

Maximal subgroups

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

subgroupindex
#341972
#207232
#173612
#87792
#87782
#80502
#79452