Finite Group Databasebeta

← #107409

Subgroups — #107409

Order 512.

760 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2#21312C211yesmaximal, characteristic
3#17751C211yesmaximal, characteristic
4C2×#1303C211yesmaximal, characteristic
5#8081C211yesmaximal, characteristic
6#8004C211yesmaximal, characteristic
7#7950C211yesmaximal, 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×#423C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1989C414yes—
24#1988—21no—
25#1987—21no—
26#1736—22no—
27#1561—22no—
28#1552C414yescomplemented
29#1549—21no—
30#1546—21no—
31#1303—21no—
32#1303—21no—
33#1301—22no—
34#1292—22no—
35#1285—22no—
36C22×M16—21no—
37C22×M16—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533C2×C412yes—
43#533C2×C412yes—
44#533D412yes—
45#533Q812yes—
46C2×C4⋊Q8D411yescharacteristic
47C2×C4⋊Q8D411yescharacteristic
48C2×#125C2×C412yes—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×#117D412yes—
54C22×C4⋊C4—21no—
55C22×C42—21no—
56C22×C42C2311yescharacteristic
57#446—24no—
58#446—28no—
59#446—24no—
60#446—24no—
61C8×Q8—24no—
62C8×Q8—24no—
63C8×Q8—24no—
64#433—28no—
65#432—24no—
66#432—21no—
67#432—21no—
68#432C2×C412yes—
69#432D412yes—
70#431—24no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431C2×C412yes—
77#431C2×C412yes—
78#431D411yescharacteristic
79#431D411yescharacteristic
80#424—24no—
81#423—21no—
82#423—21no—
83#423—21no—
84#423—21no—
85#423—21no—
86#423—21no—
87#423—21no—
88#423—21no—
89#423—21no—
90#423D412yes—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2×C412yes—
94C2×C4⋊C8C2×C412yes—
95C2×C4⋊C8D412yes—
96#407—21no—
97#407—21no—
98C2×C22⋊C8—21no—
99#405—24no—
100C4×M16—21no—
101C4×M16—21no—
102C2×C4×C8—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4C2×C412yes—
107C2×#95—21no—
108#340—24no—
109#340—24no—
110#332—24no—
111#332—24no—
112C22×Q8—21no—
113C22×Q8—21no—
114C23×C4—41no—
115C23×C4C2×D411yescharacteristic
116C23×C4#4111yescharacteristic
117C23×C4#4111yescharacteristic
118C2×M16—22no—
119C2×M16—44no—
120C2×M16—41no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—81no—
126C2×M16—81no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8C22⋊C412yes—
135C4⋊Q8C22⋊C412yes—
136C4⋊Q8C4⋊C412yes—
137C4⋊Q8C4⋊C412yes—
138#125—22no—
139#125—22no—
140#125C4214yes—
141#125C4⋊C414yes—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158#117—22no—
159#117—22no—
160#117C22⋊C414yes—
161#117C4⋊C414yes—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22⋊C412yes—
172C2×C22⋊C4—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—41no—
176C2×C42—81no—
177C2×C42—81no—
178C2×C42C22×C411yescharacteristic
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×Q811yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C4⋊C8—28no—
191C4⋊C8—28no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8C22⋊C412yes—
205C4⋊C8C22⋊C412yes—
206C4⋊C8C22⋊C414yes—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—41no—
214#97—28no—
215#97—22no—
216#97—22no—
217C4×C8—28no—
218C4×C8—22no—
219C4×C8—22no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—41no—
223#95—22no—
224#95—41no—
225C24#12311yescharacteristic
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—82no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—81no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C4⋊C411yescharacteristic
251C22×C4#11711yescharacteristic
252C22×C4#11711yescharacteristic
253C22×C4C4×D411yescharacteristic
254C22×C4C4×D411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4C4×D411yescharacteristic
257C22×C4#12011yescharacteristic
258C22×C4#12011yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4C22⋊Q811yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12311yescharacteristic
265C22×C4#12311yescharacteristic
266M16—42no—
267M16—42no—
268M16—82no—
269M16—82no—
270M16—82no—
271M16—82no—
272M16—81no—
273M16—81no—
274C2×C8—22no—
275C2×C8—24no—
276C2×C8—22no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—44no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—82no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—24no—
295C4⋊C4—44no—
296C4⋊C4—42no—
297C4⋊C4#9514yes—
298C4⋊C4#9514yes—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C42—24no—
302C42—24no—
303C42—82no—
304C42—81no—
305C42—81no—
306C42C2×C4211yescharacteristic
307C42C2×C22⋊C411yescharacteristic
308C42C2×C22⋊C411yescharacteristic
309C42C2×C22⋊C411yescharacteristic
310C42C2×C4⋊C411yescharacteristic
311C42C2×C4⋊C411yescharacteristic
312C42C4×D411yescharacteristic
313C42C4×D411yescharacteristic
314C42C4×D411yescharacteristic
315C42C4×D411yescharacteristic
316C42#12011yescharacteristic
317C42#12011yescharacteristic
318C42#12111yescharacteristic
319C42#12111yescharacteristic
320C42C22⋊Q811yescharacteristic
321C42C22⋊Q811yescharacteristic
322C23—22no—
323C23—21no—
324C23—21no—
325C23—41no—
326C23#38111yescharacteristic
327C23#38511yescharacteristic
328C23#38611yescharacteristic
329Q8—44no—
330Q8—42no—
331Q8—42no—
332Q8—42no—
333Q8—42no—
334Q8—82no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—88no—
366C2×C4—82no—
367C2×C4C2×#9511yescharacteristic
368C2×C4C4×C22⋊C411yescharacteristic
369C2×C4C4×C22⋊C411yescharacteristic
370C2×C4#37911yescharacteristic
371C2×C4#38011yescharacteristic
372C2×C4#38011yescharacteristic
373C2×C4#38511yescharacteristic
374C2×C4#38511yescharacteristic
375C2×C4#38511yescharacteristic
376C2×C4#38611yescharacteristic
377C2×C4#38611yescharacteristic
378C2×C4#38611yescharacteristic
379C2×C4#45412yes—
380C2×C4#45612yes—
381C2×C4#47111yescharacteristic
382C2×C4#47111yescharacteristic
383C2×C4#47211yescharacteristic
384C2×C4#47311yescharacteristic
385C8—42no—
386C8—42no—
387C8—44no—
388C8—82no—
389C8—82no—
390C8—82no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—42no—
399C22—42no—
400C22C22⋊#9511yescharacteristic
401C22#185712yes—
402C22#190711yescharacteristic
403C22#190811yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—44no—
413C4—44no—
414C4—84no—
415C4#179312yes—
416C4#181511yescharacteristic
417C4#181611yescharacteristic
418C4#188611yescharacteristic
419C4#188811yescharacteristic
420C4#190711yescharacteristic
421C4#190811yescharacteristic
422C2—21no—
423C2—22no—
424C2—21no—
425C2—41no—
426C2#1075011yescharacteristic
427C2#1077411yescharacteristic
428C2#1079311yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #2986C4semidirect
28. #1552C4semidirect

Maximal subgroups

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

subgroupindex
#602502
#213122
#177512
C2×#13032
#80812
#80042
#79502