Finite Group Databasebeta

← #190374

Subgroups — #190374

Order 512.

898 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34195C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3#17361C211yesmaximal, characteristic
4#8777C214yesmaximal, complemented
5#8776C214yesmaximal, complemented
6#8050C212yesmaximal, complemented
7#7945C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9#2920C2212yes—
10#2356C412yescomplemented
11#2353C2212yes—
12#2353C2212yes—
13#2349C412yescomplemented
14#2347—21no—
15#2343—22no—
16#2339—21no—
17#2320—21no—
18#2318—22no—
19#2283C412yescomplemented
20#2283C2211yescharacteristic
21#2283C2212yes—
22#2283C2211yescharacteristic
23C2×#383C412yescomplemented
24#2275—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×Q8C2×C411yescharacteristic
42C2×C4×Q8D411yescharacteristic
43C2×C4×D4—22no—
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#391—22no—
61#389—22no—
62#388—22no—
63#388—22no—
64#387—22no—
65#387—22no—
66#387—22no—
67#387—22no—
68#386—22no—
69#386—22no—
70#385—22no—
71#385—22no—
72#383—22no—
73#383—22no—
74#383—22no—
75#383C2×C412yes—
76#383C2×C412yes—
77#383C2×C411yescharacteristic
78#383C2×C412yes—
79#383C2×C411yescharacteristic
80#382—22no—
81#382—21no—
82#382—22no—
83#382C2×C412yes—
84#381—22no—
85#381—21no—
86#381—22no—
87#381C2312yes—
88#380—21no—
89#380—21no—
90#380—22no—
91#380C2×C412yes—
92#380C2×C412yes—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4C2×C412yes—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—21no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4C2314yes—
103C4×C22⋊C4C2311yescharacteristic
104C4×C22⋊C4C2311yescharacteristic
105#376—21no—
106#376—21no—
107#376—22no—
108#376C2×C412yes—
109#376C2×C412yes—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
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×C4—22no—
124C23×C4—22no—
125C23×C4C22⋊C412yes—
126C23×C4C22⋊C412yes—
127C23×C4C22×C412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2411yescharacteristic
131C2×M16—24no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×D4—24no—
137C4×D4—24no—
138C4×D4—24no—
139C4×D4—24no—
140C4×D4—22no—
141C4×D4—22no—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—48no—
145#117—22no—
146#117—22no—
147#117—24no—
148#117—24no—
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—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4C22⋊C412yes—
165C2×C22⋊C4C2×D412yes—
166C2×C22⋊C4C2×D412yes—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—24no—
177C2×C42C22⋊C412yes—
178C2×C42C22⋊C412yes—
179C2×C42C22×C412yes—
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42#4112yes—
185C2×C42#4112yes—
186C2×C42#4112yes—
187C2×C42#4112yes—
188C4⋊C8—42no—
189C4⋊C8—42no—
190#99—48no—
191C22⋊C8—24no—
192C22⋊C8—24no—
193#97—42no—
194C4×C8—42no—
195#95—21no—
196#95—21no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—21no—
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—42no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—22no—
229C2×D4—42no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
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—44no—
245C22×C4—44no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4#12312yes—
262C22×C4#12312yes—
263C22×C4#12312yes—
264C22×C4#12312yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C2×#4112yes—
268C22×C4C2×#4112yes—
269M16—84no—
270C2×C8—44no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—44no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—84no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—48no—
286C22⋊C4—81no—
287C22⋊C4—81no—
288C22⋊C4—82no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
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—41no—
315C23—41no—
316C23—42no—
317C23—42no—
318C23#42114yes—
319C23C22×C22⋊C411yescharacteristic
320C23C2×#12312yes—
321Q8—44no—
322D4—28no—
323D4—48no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—44no—
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—42no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—164no—
365C2×C4—164no—
366C2×C4#38111yescharacteristic
367C2×C4#38111yescharacteristic
368C2×C4#38111yescharacteristic
369C2×C4#38111yescharacteristic
370C2×C4#38112yes—
371C2×C4#38112yes—
372C2×C4C2×#11711yescharacteristic
373C2×C4C2×#11711yescharacteristic
374C2×C4C2×#12312yes—
375C8—84no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—24no—
380C22—22no—
381C22—22no—
382C22—44no—
383C22—44no—
384C22—42no—
385C22—42no—
386C22—82no—
387C22C2×#38111yescharacteristic
388C22C2×#42112yes—
389C4—21no—
390C4—21no—
391C4—22no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—42no—
400C4—42no—
401C4—82no—
402C4—82no—
403C4—82no—
404C4—82no—
405C4—164no—
406C2—22no—
407C2—42no—
408C2—44no—
409C2#2070511yescharacteristic
410C2#2070811yescharacteristic
411C2#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. #8777C2semidirect
5. #8776C2semidirect
6. #8050C2semidirect
7. #7945C2semidirect
10. #2356C4semidirect
13. #2349C4semidirect
19. #2283C4semidirect
23. C2×#383C4semidirect

Maximal subgroups

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

subgroupindex
#341952
#207232
#173612
#87772
#87762
#80502
#79452