Finite Group Databasebeta

← #110524

Subgroups — #110524

Order 512.

622 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21332C211yesmaximal, characteristic
2#20538C211yesmaximal, characteristic
3#17718C211yesmaximal, characteristic
4#17464C211yesmaximal, characteristic
5#8105C211yesmaximal, characteristic
6#8092C211yesmaximal, characteristic
7#7869C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2285C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1996—22no—
16#1996C414yes—
17#1770—21no—
18#1770—21no—
19C8×C4⋊C4—21no—
20C8×C4⋊C4—21no—
21#1750—22no—
22#1740—22no—
23#1554—22no—
24#1548C412yescomplemented
25#1546C412yescomplemented
26#1542—21no—
27#1542—21no—
28#1485—22no—
29#1481—21no—
30#1481—21no—
31#1479C414yescomplemented
32#1305—22no—
33#1303—22no—
34#1276—22no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8C2×C412yes—
48C2×C4×Q8—21no—
49C2×C4×Q8C2×C412yes—
50C2×#117—21no—
51C2×#117—21no—
52C2×#117D412yes—
53C22×C42C2311yescharacteristic
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431C2×C412yes—
66#431C2×C412yes—
67#431C2×C412yes—
68#431D412yes—
69#431D412yes—
70#423—21no—
71#423C2×C412yes—
72C2×C4⋊C8D411yescharacteristic
73C2×C4⋊C8D411yescharacteristic
74C2×C4⋊C8D411yescharacteristic
75C2×C4⋊C8D411yescharacteristic
76#407—21no—
77#407—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C4×M16—21no—
81C4×M16—21no—
82C2×#97—22no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8—22no—
86#384—21no—
87#382—21no—
88#380C2×C412yes—
89#380D412yes—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C22⋊C4—21no—
95C4×C22⋊C4C2×C412yes—
96#376—21no—
97C43—21no—
98#342—28no—
99#336—22no—
100#336—22no—
101#333—22no—
102#333—22no—
103#332—22no—
104#332—22no—
105Q8⋊C8—24no—
106Q8⋊C8—24no—
107C22×Q8—21no—
108C22×Q8C4212yes—
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C2×M16—22no—
113C2×M16—41no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126#125—41no—
127#125—41no—
128#125—41no—
129#125—41no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146#117—22no—
147#117—22no—
148#117—41no—
149#117—41no—
150#117—41no—
151#117—41no—
152#117—41no—
153#117—41no—
154#117—42no—
155#117—42no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4C4212yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4C22⋊C414yes—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—41no—
173C2×C42—41no—
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—41no—
194C4⋊C8C22⋊C412yes—
195C4⋊C8C22⋊C412yes—
196C4⋊C8C22⋊C412yes—
197C4⋊C8C22⋊C412yes—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205C22⋊C8—41no—
206#97—41no—
207#97—41no—
208#97—82no—
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—41no—
215C4×C8—82no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—41no—
220C24C2×C22⋊C411yescharacteristic
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—44no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—82no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4C2×C4211yescharacteristic
238C22×C4#11711yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4#12011yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12311yescharacteristic
253C22×C4#12411yescharacteristic
254C22×C4#12711yescharacteristic
255M16—82no—
256M16—82no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—42no—
270C2×C8—42no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C4⋊C4—82no—
288C22⋊C4—24no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—44no—
293C42—42no—
294C42—42no—
295C42—44no—
296C42—42no—
297C42—42no—
298C42—42no—
299C42—42no—
300C42—82no—
301C42C2×C22⋊C411yescharacteristic
302C42C2×C22⋊C411yescharacteristic
303C42#11711yescharacteristic
304C42#11711yescharacteristic
305C42C4×D411yescharacteristic
306C42C4×D411yescharacteristic
307C42C4×D411yescharacteristic
308C42C4×D411yescharacteristic
309C42#12111yescharacteristic
310C42#12111yescharacteristic
311C42#12311yescharacteristic
312C42#12311yescharacteristic
313C42#12411yescharacteristic
314C42#12411yescharacteristic
315C42#12611yescharacteristic
316C42#12611yescharacteristic
317C23—21no—
318C23—21no—
319C23—41no—
320C23—41no—
321C23C4×C22⋊C411yescharacteristic
322C23#38611yescharacteristic
323C23#39011yescharacteristic
324Q8—44no—
325Q8—82no—
326Q8—82no—
327Q8—82no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4C4×C22⋊C411yescharacteristic
361C2×C4C4×C22⋊C411yescharacteristic
362C2×C4#38611yescharacteristic
363C2×C4#38611yescharacteristic
364C2×C4#38811yescharacteristic
365C2×C4#38811yescharacteristic
366C2×C4#38811yescharacteristic
367C2×C4#38811yescharacteristic
368C2×C4#38811yescharacteristic
369C2×C4#38811yescharacteristic
370C2×C4#39011yescharacteristic
371C2×C4#39011yescharacteristic
372C2×C4#45511yescharacteristic
373C2×C4#45511yescharacteristic
374C2×C4#45611yescharacteristic
375C2×C4#45611yescharacteristic
376C8—42no—
377C8—42no—
378C8—42no—
379C8—42no—
380C8—82no—
381C8—82no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—81no—
391C22—81no—
392C22#143711yescharacteristic
393C22#188511yescharacteristic
394C22#188611yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—84no—
404C4—84no—
405C4—82no—
406C4—82no—
407C4#175712yes—
408C4#188211yescharacteristic
409C4#188211yescharacteristic
410C4#188211yescharacteristic
411C4#188211yescharacteristic
412C4#197612yes—
413C2—21no—
414C2—21no—
415C2—41no—
416C2—41no—
417C2#1094411yescharacteristic
418C2#1095711yescharacteristic
419C2#1096111yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ 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
24. #1548C4semidirect
25. #1546C4semidirect
31. #1479C4semidirect

Maximal subgroups

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

subgroupindex
#213322
#205382
#177182
#174642
#81052
#80922
#78692