Finite Group Databasebeta

← #107249

Subgroups — #107249

Order 512.

794 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34378C211yesmaximal, characteristic
2C2×#1998C211yesmaximal, characteristic
3C2×#1470C211yesmaximal, characteristic
4#8042C211yesmaximal, characteristic
5#8035C211yesmaximal, characteristic
6#7899C211yesmaximal, characteristic
7#7896C211yesmaximal, characteristic
8C22×#125C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2382—22no—
14#2370—22no—
15#2345—21no—
16#2344—22no—
17#2344—21no—
18#2336—22no—
19#2330—22no—
20#2316C412yes—
21#2313C412yes—
22#2285C2211yescharacteristic
23#2283C2211yescharacteristic
24#1998—22no—
25#1998C414yescomplemented
26C2×#343—21no—
27C2×#343—21no—
28#1470—22no—
29#1470C414yescomplemented
30C2×#324—22no—
31#1296—22no—
32#1295—22no—
33#1279—22no—
34#1279—22no—
35C22×M16—21no—
36C22×M16—21no—
37C2×#125—21no—
38C2×#125—21no—
39C2×#125—21no—
40C2×#125—21no—
41C2×#125—21no—
42C2×#125—21no—
43C2×#125—21no—
44C2×#125—21no—
45C2×#125C2×C412yes—
46C2×#125C2×C412yes—
47C2×#125D412yes—
48C2×#125Q812yes—
49C2×#117—22no—
50C2×#117—21no—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C42C2311yescharacteristic
56#432—21no—
57#432—21no—
58#432—21no—
59#432C2×C412yes—
60#431—21no—
61#431—21no—
62#431—21no—
63#431C2×C412yes—
64#423—21no—
65#423—21no—
66#423—21no—
67#423C2×C412yes—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C4×M16—21no—
75C4×M16—21no—
76C4×M16—21no—
77C4×M16C2×C412yes—
78C2×#97—21no—
79C2×#97—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82#389—22no—
83#389—24no—
84#389—22no—
85#389—24no—
86#389—24no—
87#387—24no—
88#387—24no—
89#387—24no—
90#387—24no—
91#387—24no—
92#387—24no—
93#384D412yes—
94#383—21no—
95#382—21no—
96#382D412yes—
97#381—22no—
98#381—21no—
99#380—22no—
100#380—21no—
101#380C2×C411yescharacteristic
102#380C2×C412yes—
103#380C2×C411yescharacteristic
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—21no—
112#376—21no—
113#343—24no—
114#343—24no—
115#324—28no—
116C23×C4C22⋊C412yes—
117C23×C4C22⋊C412yes—
118C23×C4C22×C411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C2×M16—24no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—42no—
125C2×M16—41no—
126C2×M16—41no—
127C2×M16—44no—
128C22×C8—41no—
129C22×C8—41no—
130C22×C8—41no—
131C22×C8—41no—
132#125—22no—
133#125—22no—
134#125—22no—
135#125—22no—
136#125—22no—
137#125—22no—
138#125—22no—
139#125—22no—
140#125—22no—
141#125—22no—
142#125—22no—
143#125—22no—
144#125—22no—
145#125—22no—
146#125—22no—
147#125—22no—
148#117—82no—
149#117—84no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C4214yes—
171C2×C4⋊C4C22⋊C414yes—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C414yes—
177C2×C4⋊C4C4⋊C414yes—
178C2×C4⋊C4C4⋊C412yes—
179C2×C4⋊C4C4⋊C412yes—
180C2×C4⋊C4C4⋊C414yes—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—21no—
184C2×C42—22no—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—22no—
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×Q811yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207C4⋊C8—41no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—41no—
211C22⋊C8—41no—
212#97—22no—
213#97—22no—
214#97—41no—
215#97—41no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—41no—
219C4×C8—41no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—21no—
225#95—21no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231C24C2×C22⋊C411yescharacteristic
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—24no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4#9514yes—
248C22×C4#9514yes—
249C22×C4C2×C4211yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C4⋊C411yescharacteristic
255C22×C4C2×C4⋊C411yescharacteristic
256C22×C4C2×C4⋊C411yescharacteristic
257C22×C4#11711yescharacteristic
258C22×C4#11711yescharacteristic
259C22×C4#12011yescharacteristic
260C22×C4#12011yescharacteristic
261C22×C4#12011yescharacteristic
262C22×C4#12011yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12311yescharacteristic
265C22×C4#12311yescharacteristic
266C22×C4#12311yescharacteristic
267M16—24no—
268M16—44no—
269M16—42no—
270M16—44no—
271M16—82no—
272C2×C8—44no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—81no—
278C2×C8—81no—
279C2×C8—81no—
280C2×C8—81no—
281C2×C8—82no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—24no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—82no—
304C4⋊C4—84no—
305C4⋊C4—84no—
306C22⋊C4—42no—
307C22⋊C4—44no—
308C22⋊C4—42no—
309C42—42no—
310C42—42no—
311C42—82no—
312C42—84no—
313C42—84no—
314C42C4×D411yescharacteristic
315C42C4×D411yescharacteristic
316C42C4×D411yescharacteristic
317C42C4×D411yescharacteristic
318C42C4×D411yescharacteristic
319C42C4×D411yescharacteristic
320C42C4×D411yescharacteristic
321C42C4×D411yescharacteristic
322C42#12111yescharacteristic
323C42#12111yescharacteristic
324C42#12111yescharacteristic
325C42#12111yescharacteristic
326C42C22⋊Q811yescharacteristic
327C42C22⋊Q811yescharacteristic
328C42C22⋊Q811yescharacteristic
329C42C22⋊Q811yescharacteristic
330C23—22no—
331C23—21no—
332C23—21no—
333C23—22no—
334C23C2×#9511yescharacteristic
335C23#37911yescharacteristic
336C23#38111yescharacteristic
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—82no—
376C2×C4C4×C22⋊C411yescharacteristic
377C2×C4C4×C22⋊C411yescharacteristic
378C2×C4#38011yescharacteristic
379C2×C4#38011yescharacteristic
380C2×C4#38511yescharacteristic
381C2×C4#38511yescharacteristic
382C2×C4#38511yescharacteristic
383C2×C4#38511yescharacteristic
384C2×C4#38611yescharacteristic
385C2×C4#38611yescharacteristic
386C2×C4#38611yescharacteristic
387C2×C4#38611yescharacteristic
388C2×C4#45412yes—
389C2×C4#45612yes—
390C8—44no—
391C8—82no—
392C8—82no—
393C8—82no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—42no—
401C22—42no—
402C22C22⋊#9511yescharacteristic
403C22#179112yes—
404C22#179312yes—
405C22#179511yescharacteristic
406C22#179611yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—44no—
418C4—44no—
419C4—82no—
420C4—82no—
421C4—84no—
422C4#185712yes—
423C4#188611yescharacteristic
424C4#188811yescharacteristic
425C2—41no—
426C2—41no—
427C2#848512yes—
428C2#848612yes—
429C2#1074711yescharacteristic
430C2#1075011yescharacteristic
431C2#1076511yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1998C4semidirect
29. #1470C4semidirect

Maximal subgroups

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

subgroupindex
#343782
C2×#19982
C2×#14702
#80422
#80352
#78992
#78962