Finite Group Databasebeta

← #190370

Subgroups — #190370

Order 512.

955 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34123C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3#17361C211yesmaximal, characteristic
4#8777C214yesmaximal, complemented
5#8776C214yesmaximal, complemented
6#8052C212yesmaximal, complemented
7#7944C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9#2920C2212yes—
10#2356C412yescomplemented
11#2353C2212yes—
12#2353C2212yes—
13#2349C412yescomplemented
14#2346—21no—
15#2343—22no—
16#2339—21no—
17#2295—21no—
18#2290—22no—
19#2283C412yescomplemented
20#2283C2212yes—
21#2283C2211yescharacteristic
22C42×D4—21no—
23C4×#117C2211yescharacteristic
24C2×#376C412yescomplemented
25#1820—21no—
26#1799—22no—
27#1782—21no—
28C4×#99C2214yes—
29#1467—21no—
30#1466—21no—
31#1465C2214yes—
32#1458—22no—
33#1297C2218yes—
34#1284C2214yes—
35#1284C2214yes—
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×#117—21no—
49C2×#117D411yescharacteristic
50C2×#117D412yes—
51C22×C42—22no—
52C22×C42C2×C412yes—
53C22×C42C2311yescharacteristic
54#432—22no—
55#432—22no—
56#431—22no—
57#431—22no—
58#423—22no—
59C2×#99—24no—
60#407C2314yes—
61C4×M16—22no—
62#391—22no—
63#389—22no—
64#388—22no—
65#388—22no—
66#387—22no—
67#387—22no—
68#387—22no—
69#387—22no—
70#386—22no—
71#386—22no—
72#385—22no—
73#385—22no—
74#383—21no—
75#383—22no—
76#383C2×C412yes—
77#383C2×C412yes—
78#382—22no—
79#382—22no—
80#382C2×C412yes—
81#381—22no—
82#381—22no—
83#381C2311yescharacteristic
84#381C2311yescharacteristic
85#380—21no—
86#380—22no—
87#380C2×C412yes—
88#380C2×C412yes—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4C2×C412yes—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4C2314yes—
101C4×C22⋊C4C2312yes—
102#376—22no—
103#376—22no—
104#376—22no—
105#376C2×C412yes—
106#376C2×C412yes—
107#376C2×C411yescharacteristic
108#376C2×C412yes—
109#376C2×C411yescharacteristic
110C2×#95—22no—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113C43—21no—
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—
149#117—44no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4C22⋊C411yescharacteristic
156C2×C4⋊C4C22⋊C411yescharacteristic
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4C22⋊C412yes—
166C2×C22⋊C4C2×D412yes—
167C2×C22⋊C4C2×D412yes—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—24no—
180C2×C42—48no—
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—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—22no—
210#95—22no—
211#95—21no—
212#95—21no—
213#95—24no—
214#95—42no—
215#95—42no—
216#95—42no—
217#95C22×C412yes—
218#95C22×C412yes—
219#95C22×C412yes—
220#95C22×C412yes—
221C24C2×C22⋊C412yes—
222C24C22×D411yescharacteristic
223#41—48no—
224#41—48no—
225C2×Q8—22no—
226C2×Q8—42no—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—42no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—81no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4#12312yes—
264C22×C4#12312yes—
265C22×C4#12312yes—
266C22×C4#12312yes—
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C2×#4112yes—
270C22×C4C2×#4112yes—
271M16—84no—
272C2×C8—44no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—84no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—48no—
289C22⋊C4—81no—
290C22⋊C4—81no—
291C42—24no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—44no—
299C42—44no—
300C42—44no—
301C42—42no—
302C42—44no—
303C42—82no—
304C42—84no—
305C42#11711yescharacteristic
306C42#11711yescharacteristic
307C42#11711yescharacteristic
308C42#11711yescharacteristic
309C42#11712yes—
310C42#11712yes—
311C42#12312yes—
312C42#12312yes—
313C42#12312yes—
314C42#12312yes—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—41no—
319C23—41no—
320C23—42no—
321C23—42no—
322C23#42114yes—
323C23C22×C22⋊C411yescharacteristic
324C23C2×#12312yes—
325Q8—44no—
326D4—28no—
327D4—48no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—88no—
366C2×C4—88no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4#38111yescharacteristic
371C2×C4#38111yescharacteristic
372C2×C4#38111yescharacteristic
373C2×C4#38112yes—
374C2×C4#38112yes—
375C2×C4#38111yescharacteristic
376C2×C4C2×#11711yescharacteristic
377C2×C4C2×#11711yescharacteristic
378C2×C4C2×#12312yes—
379C8—84no—
380C22—21no—
381C22—21no—
382C22—22no—
383C22—24no—
384C22—22no—
385C22—22no—
386C22—44no—
387C22—44no—
388C22—42no—
389C22—42no—
390C22—82no—
391C22C2×#38111yescharacteristic
392C22C2×#42112yes—
393C4—21no—
394C4—21no—
395C4—22no—
396C4—21no—
397C4—22no—
398C4—22no—
399C4—21no—
400C4—21no—
401C4—44no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4—82no—
406C4—82no—
407C4—82no—
408C4—164no—
409C4#182712yes—
410C2—22no—
411C2—42no—
412C2—44no—
413C2#2070511yescharacteristic
414C2C2×#182711yescharacteristic
415C2#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. #8052C2semidirect
7. #7944C2semidirect
10. #2356C4semidirect
13. #2349C4semidirect
19. #2283C4semidirect
24. C2×#376C4semidirect

Maximal subgroups

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

subgroupindex
#341232
#207232
#173612
#87772
#87762
#80522
#79442