Finite Group Databasebeta

← #107439

Subgroups — #107439

Order 512.

770 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2C2×#1986C211yesmaximal, characteristic
3#17715C211yesmaximal, characteristic
4#8098C211yesmaximal, characteristic
5#8084C211yesmaximal, characteristic
6#7892C211yesmaximal, characteristic
7#7883C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2986C414yescomplemented
10#2985—22no—
11#2984—22no—
12#2983—21no—
13#2982—21no—
14#2962—22no—
15#2960—22no—
16#2924—22no—
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1986—22no—
24#1986C414yes—
25C2×#339—22no—
26#1552—22no—
27#1549C412yescomplemented
28#1546C412yescomplemented
29#1544—22no—
30#1303—22no—
31#1302—22no—
32#1279—22no—
33#1278—21no—
34#1278—21no—
35C22×M16—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533C2×C412yes—
41#533C2×C412yes—
42#533D412yes—
43#533D412yes—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125Q812yes—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117D412yes—
52C22×C4⋊C4—21no—
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55#446—24no—
56#446—28no—
57#446—24no—
58#446—24no—
59C8×Q8—24no—
60C8×Q8—24no—
61C8×Q8—24no—
62#433—28no—
63#432—24no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432D412yes—
68#431—24no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431—21no—
77#431C2×C412yes—
78#431C2×C412yes—
79#431C2×C412yes—
80#431D411yescharacteristic
81#431D411yescharacteristic
82#424—24no—
83#423C2×C412yes—
84#423D412yes—
85C2×C4⋊C8—21no—
86C2×C4⋊C8D412yes—
87#407—21no—
88#407—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91#405—24no—
92C4×M16—21no—
93C4×M16—21no—
94C4×M16—21no—
95C4×M16—21no—
96C4×M16—21no—
97C4×M16C2×C412yes—
98C2×#97—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4C2×C412yes—
106C2×#95—21no—
107#339—28no—
108#332—24no—
109#332—24no—
110C22×Q8—21no—
111C22×Q8—21no—
112C23×C4—41no—
113C23×C4C2×D411yescharacteristic
114C23×C4#4111yescharacteristic
115C23×C4#4111yescharacteristic
116C2×M16—22no—
117C2×M16—24no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—42no—
121C2×M16—41no—
122C22×C8—22no—
123C22×C8—41no—
124C22×C8—41no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8C4212yes—
130C4⋊Q8C4212yes—
131C4⋊Q8C22⋊C412yes—
132C4⋊Q8C22⋊C412yes—
133#125—22no—
134#125—22no—
135#125C4⋊C414yes—
136#125C4⋊C414yes—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153#117—22no—
154#117—22no—
155#117C22⋊C414yes—
156#117C22⋊C414yes—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4C4⋊C412yes—
166C2×C4⋊C4C4⋊C412yes—
167C2×C22⋊C4—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—44no—
171C2×C42—41no—
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×Q811yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C4⋊C8—28no—
185C4⋊C8—28no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—41no—
192C4⋊C8C22⋊C412yes—
193C4⋊C8C22⋊C412yes—
194C4⋊C8C22⋊C414yes—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—21no—
198C22⋊C8—21no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—41no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204#97—28no—
205#97—22no—
206#97—22no—
207#97—41no—
208#97—41no—
209C4×C8—28no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—41no—
217C4×C8—41no—
218C4×C8—41no—
219#95—22no—
220#95—41no—
221C24#12311yescharacteristic
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—82no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—81no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C4⋊C411yescharacteristic
243C22×C4#11711yescharacteristic
244C22×C4#11711yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4C22⋊Q811yescharacteristic
254C22×C4C22⋊Q811yescharacteristic
255C22×C4#12311yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4#12311yescharacteristic
258M16—24no—
259M16—42no—
260M16—44no—
261M16—42no—
262M16—42no—
263M16—42no—
264M16—42no—
265C2×C8—22no—
266C2×C8—24no—
267C2×C8—22no—
268C2×C8—44no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—82no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4#9514yes—
289C4⋊C4#9514yes—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C42—24no—
293C42—24no—
294C42—44no—
295C42—82no—
296C42C2×C4211yescharacteristic
297C42C2×C22⋊C411yescharacteristic
298C42C2×C22⋊C411yescharacteristic
299C42C2×C22⋊C411yescharacteristic
300C42C2×C4⋊C411yescharacteristic
301C42C2×C4⋊C411yescharacteristic
302C42C4×D411yescharacteristic
303C42C4×D411yescharacteristic
304C42C4×D411yescharacteristic
305C42C4×D411yescharacteristic
306C42#12011yescharacteristic
307C42#12011yescharacteristic
308C42#12111yescharacteristic
309C42#12111yescharacteristic
310C42C22⋊Q811yescharacteristic
311C42C22⋊Q811yescharacteristic
312C23—22no—
313C23—21no—
314C23—21no—
315C23—41no—
316C23#38111yescharacteristic
317C23#38511yescharacteristic
318C23#38611yescharacteristic
319Q8—44no—
320Q8—44no—
321Q8—42no—
322Q8—42no—
323Q8—82no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—88no—
350C2×C4—82no—
351C2×C4C2×#9511yescharacteristic
352C2×C4C4×C22⋊C411yescharacteristic
353C2×C4C4×C22⋊C411yescharacteristic
354C2×C4#37911yescharacteristic
355C2×C4#38011yescharacteristic
356C2×C4#38011yescharacteristic
357C2×C4#38511yescharacteristic
358C2×C4#38511yescharacteristic
359C2×C4#38511yescharacteristic
360C2×C4#38611yescharacteristic
361C2×C4#38611yescharacteristic
362C2×C4#38611yescharacteristic
363C2×C4#44312yes—
364C2×C4#44412yes—
365C2×C4#45412yes—
366C2×C4#45612yes—
367C8—44no—
368C8—42no—
369C8—42no—
370C8—44no—
371C8—82no—
372C8—82no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—42no—
381C22—42no—
382C22C22⋊#9511yescharacteristic
383C22#186612yes—
384C22#188611yescharacteristic
385C22#188811yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—44no—
395C4—44no—
396C4—84no—
397C4#176411yescharacteristic
398C4#176511yescharacteristic
399C4#179312yes—
400C4#185712yes—
401C4#187612yes—
402C2—21no—
403C2—22no—
404C2—21no—
405C2—41no—
406C2#1075011yescharacteristic
407C2#1077811yescharacteristic
408C2#1079111yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC24#123
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #2986C4semidirect
27. #1549C4semidirect
28. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#602502
C2×#19862
#177152
#80982
#80842
#78922
#78832