Finite Group Databasebeta

← #107271

Subgroups — #107271

Order 512.

730 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2#21894C211yesmaximal, characteristic
3#17749C211yesmaximal, characteristic
4#8129C211yesmaximal, characteristic
5#8105C211yesmaximal, characteristic
6#8095C211yesmaximal, characteristic
7#8065C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2986—22no—
10#2985—22no—
11#2984—22no—
12#2983—21no—
13#2982—21no—
14#2962C414yescomplemented
15#2960—22no—
16#2924—22no—
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×#423C2211yescharacteristic
22#2285C2211yescharacteristic
23#1988—21no—
24#1981—21no—
25#1894C414yes—
26#1803—22no—
27#1559C414yescomplemented
28#1550—22no—
29#1548—21no—
30#1546—21no—
31#1309—22no—
32#1305—22no—
33#1303—21no—
34#1303—21no—
35#1299—22no—
36C22×M16—21no—
37C22×M16C2×C412yes—
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×#125—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8C2×C412yes—
51C2×C4×Q8—21no—
52C2×C4×Q8D412yes—
53C2×#117—21no—
54C2×#117—21no—
55C22×C42C2311yescharacteristic
56#446—24no—
57#446—24no—
58#446—28no—
59#446—24no—
60C8×Q8—24no—
61C8×Q8—24no—
62C8×Q8—24no—
63#433—28no—
64#432—24no—
65#432—21no—
66#432—21no—
67#432—21no—
68#432C2×C412yes—
69#431—24no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431D412yes—
77#431D411yescharacteristic
78#431D411yescharacteristic
79#424—24no—
80#423—21no—
81#423—21no—
82#423—21no—
83#423—21no—
84#423—21no—
85#423—21no—
86#423—21no—
87#423—21no—
88#423C2×C412yes—
89#423D412yes—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2×C412yes—
95C2×#103—24no—
96#407—21no—
97#407D412yes—
98C2×C22⋊C8—21no—
99#405—24no—
100C4×M16—21no—
101C4×M16C2×C412yes—
102C2×C4×C8—21no—
103#384—21no—
104#382—21no—
105#380—21no—
106#380C2×C412yes—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109#340—24no—
110#333—24no—
111#332—24no—
112C22×Q8—21no—
113C22×Q8C22⋊C412yes—
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4#4111yescharacteristic
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—41no—
121C2×M16—41no—
122C2×M16—82no—
123C2×M16—81no—
124C2×M16—81no—
125C2×M16C22⋊C414yes—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—41no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8C22⋊C412yes—
134C4⋊Q8C22⋊C412yes—
135C4⋊Q8C4⋊C412yes—
136C4⋊Q8C4⋊C412yes—
137#125—22no—
138#125—22no—
139#125—22no—
140#125—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8C4214yes—
148C22⋊Q8C4⋊C414yes—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8C22⋊C414yes—
156C4×Q8C4⋊C414yes—
157#117—22no—
158#117—22no—
159#117—22no—
160#117—22no—
161#117—81no—
162#117—81no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4C22⋊C412yes—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C42—22no—
175C2×C42—41no—
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×Q811yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C4⋊C8—28no—
189C4⋊C8—28no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—41no—
197C4⋊C8—41no—
198C4⋊C8—41no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203C4⋊C8C22⋊C412yes—
204C4⋊C8C22⋊C412yes—
205#103—84no—
206C22⋊C8—21no—
207C22⋊C8—21no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—41no—
213#97—28no—
214#97—22no—
215#97—22no—
216C4×C8—28no—
217C4×C8—22no—
218C4×C8—21no—
219C4×C8—22no—
220C4×C8—21no—
221C4×C8—41no—
222#95—22no—
223#95—41no—
224C24C4×D411yescharacteristic
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8#9514yes—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4C2×C4211yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C4⋊C411yescharacteristic
249C22×C4#11711yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4C4×D411yescharacteristic
254C22×C4C4×D411yescharacteristic
255C22×C4#12011yescharacteristic
256C22×C4#12011yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4#12311yescharacteristic
262C22×C4#12311yescharacteristic
263M16—44no—
264M16—82no—
265M16—81no—
266M16—81no—
267M16—82no—
268M16—82no—
269M16—82no—
270C2×C8—22no—
271C2×C8—24no—
272C2×C8—22no—
273C2×C8—44no—
274C2×C8—44no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—82no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4#9514yes—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—82no—
298C42—24no—
299C42—24no—
300C42—81no—
301C42—81no—
302C42C2×C22⋊C411yescharacteristic
303C42C2×C22⋊C411yescharacteristic
304C42C2×C22⋊C411yescharacteristic
305C42C2×C4⋊C411yescharacteristic
306C42C2×C4⋊C411yescharacteristic
307C42#11711yescharacteristic
308C42C4×D411yescharacteristic
309C42C4×D411yescharacteristic
310C42#12011yescharacteristic
311C42#12011yescharacteristic
312C42#12111yescharacteristic
313C42#12111yescharacteristic
314C42C22⋊Q811yescharacteristic
315C42C22⋊Q811yescharacteristic
316C42#12311yescharacteristic
317C42#12311yescharacteristic
318C23—21no—
319C23—21no—
320C23—41no—
321C23—41no—
322C23C4×C22⋊C411yescharacteristic
323C23#38511yescharacteristic
324C23#38611yescharacteristic
325Q8—44no—
326Q8—42no—
327Q8—42no—
328Q8—42no—
329Q8—42no—
330Q8—42no—
331Q8—42no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
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—24no—
352C2×C4—24no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4C2×#9511yescharacteristic
366C2×C4C4×C22⋊C411yescharacteristic
367C2×C4#37911yescharacteristic
368C2×C4#38011yescharacteristic
369C2×C4#38011yescharacteristic
370C2×C4#38111yescharacteristic
371C2×C4#38511yescharacteristic
372C2×C4#38511yescharacteristic
373C2×C4#38511yescharacteristic
374C2×C4#38611yescharacteristic
375C2×C4#38611yescharacteristic
376C2×C4#38611yescharacteristic
377C2×C4#47111yescharacteristic
378C2×C4#47111yescharacteristic
379C2×C4#47211yescharacteristic
380C2×C4#47311yescharacteristic
381C8—42no—
382C8—42no—
383C8—44no—
384C8—82no—
385C8—82no—
386C8—82no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—81no—
396C22—81no—
397C22C22⋊#9511yescharacteristic
398C22#190711yescharacteristic
399C22#190811yescharacteristic
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—44no—
409C4—44no—
410C4—162no—
411C4#179612yes—
412C4#181511yescharacteristic
413C4#181611yescharacteristic
414C4#190711yescharacteristic
415C4#190811yescharacteristic
416C4#191411yescharacteristic
417C4#191511yescharacteristic
418C2—21no—
419C2—21no—
420C2—41no—
421C2—41no—
422C2#1075511yescharacteristic
423C2#1077411yescharacteristic
424C2#1077711yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ 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
14. #2962C4semidirect
27. #1559C4semidirect

Maximal subgroups

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

subgroupindex
#602502
#218942
#177492
#81292
#81052
#80952
#80652