Finite Group Databasebeta

← #110520

Subgroups — #110520

Order 512.

628 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21337C211yesmaximal, characteristic
2#20550C211yesmaximal, characteristic
3#17545C211yesmaximal, characteristic
4#17464C211yesmaximal, characteristic
5#8119C211yesmaximal, characteristic
6#8068C211yesmaximal, characteristic
7#7883C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1998—22no—
16#1997C412yescomplemented
17#1997C412yescomplemented
18#1770—21no—
19C8×C4⋊C4—21no—
20#1767—21no—
21C4×C4⋊C8—21no—
22#1753—22no—
23#1742—22no—
24#1508C414yes—
25#1506—22no—
26#1503—21no—
27#1503—21no—
28#1485—22no—
29#1481—21no—
30#1481—21no—
31#1479C414yescomplemented
32#1307—22no—
33#1300—22no—
34#1278—22no—
35C22×M16—21no—
36C22×M16D412yes—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8C2×C412yes—
50C2×C4×Q8—21no—
51C2×C4×Q8C2×C412yes—
52C2×#117—21no—
53C2×#117—21no—
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431C2×C412yes—
65#431C2×C412yes—
66#431D412yes—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71#423C2×C412yes—
72#423D412yes—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8D411yescharacteristic
76C2×C4⋊C8D411yescharacteristic
77#407—21no—
78#407C2×C412yes—
79C2×C22⋊C8—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16C2×C412yes—
86C2×#97—21no—
87C2×#97—22no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8—22no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4D411yescharacteristic
93C4×C4⋊C4D411yescharacteristic
94C4×C22⋊C4—21no—
95C4×C22⋊C4C2×C412yes—
96#376D412yes—
97C43—21no—
98#343—24no—
99#343—24no—
100#336—22no—
101#336—22no—
102#328—24no—
103#328—24no—
104Q8⋊C8—24no—
105Q8⋊C8—24no—
106C22×Q8—21no—
107C22×Q8C4212yes—
108C23×C4C22×C411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4#4111yescharacteristic
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—41no—
117C2×M16—41no—
118C2×M16—41no—
119C2×M16—41no—
120C2×M16—42no—
121C2×M16—42no—
122C2×M16C22⋊C414yes—
123C22×C8—22no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—41no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135#125—41no—
136#125—41no—
137#125—41no—
138#125—41no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C22⋊Q8—41no—
146C22⋊Q8—41no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—41no—
152C4×Q8—41no—
153C4×Q8—41no—
154C4×Q8—41no—
155#117—44no—
156#117—41no—
157#117—41no—
158#117—41no—
159#117—41no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4C4212yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—22no—
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—41no—
195C4⋊C8—41no—
196C4⋊C8—41no—
197C4⋊C8C22⋊C412yes—
198C4⋊C8C22⋊C412yes—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—41no—
205#97—22no—
206#97—41no—
207#97—41no—
208#97—41no—
209#97—82no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—41no—
215C4×C8—41no—
216C4×C8—82no—
217#95—22no—
218#95C22⋊C412yes—
219#95C22⋊C412yes—
220C24C4×D411yescharacteristic
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—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4C2×C4211yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4#11711yescharacteristic
241C22×C4#11711yescharacteristic
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#12311yescharacteristic
252C22×C4#12311yescharacteristic
253C22×C4#12411yescharacteristic
254C22×C4#12611yescharacteristic
255M16—24no—
256M16—42no—
257M16—42no—
258M16—42no—
259M16—44no—
260M16—44no—
261M16—82no—
262M16—82no—
263C2×C8—24no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—41no—
270C2×C8—42no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—42no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C22⋊C4—44no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C42—44no—
295C42—44no—
296C42—42no—
297C42—42no—
298C42—41no—
299C42—41no—
300C42—41no—
301C42—41no—
302C42C2×C22⋊C411yescharacteristic
303C42#11711yescharacteristic
304C42C4×D411yescharacteristic
305C42C4×D411yescharacteristic
306C42C4×D411yescharacteristic
307C42C4×D411yescharacteristic
308C42C4×D411yescharacteristic
309C42C4×D411yescharacteristic
310C42#12111yescharacteristic
311C42#12111yescharacteristic
312C42#12111yescharacteristic
313C42#12311yescharacteristic
314C42#12411yescharacteristic
315C42#12411yescharacteristic
316C42#12611yescharacteristic
317C42#12711yescharacteristic
318C23—21no—
319C23—21no—
320C23—41no—
321C23—41no—
322C23C4×C22⋊C411yescharacteristic
323C23#38611yescharacteristic
324C23#38811yescharacteristic
325Q8—44no—
326Q8—82no—
327Q8—82no—
328Q8—82no—
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—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4C4×C22⋊C411yescharacteristic
362C2×C4C4×C22⋊C411yescharacteristic
363C2×C4#38611yescharacteristic
364C2×C4#38611yescharacteristic
365C2×C4#38811yescharacteristic
366C2×C4#38811yescharacteristic
367C2×C4#38811yescharacteristic
368C2×C4#38811yescharacteristic
369C2×C4#38811yescharacteristic
370C2×C4#39011yescharacteristic
371C2×C4#39011yescharacteristic
372C2×C4#39011yescharacteristic
373C2×C4#45511yescharacteristic
374C2×C4#45511yescharacteristic
375C2×C4#45611yescharacteristic
376C2×C4#45611yescharacteristic
377C8—44no—
378C8—42no—
379C8—42no—
380C8—44no—
381C8—82no—
382C8—82no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—81no—
392C22—81no—
393C22#143711yescharacteristic
394C22#188211yescharacteristic
395C22#188211yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—44no—
405C4—82no—
406C4—82no—
407C4—82no—
408C4#175712yes—
409C4#188211yescharacteristic
410C4#188211yescharacteristic
411C4#188511yescharacteristic
412C4#188611yescharacteristic
413C4#193212yes—
414C2—21no—
415C2—21no—
416C2—41no—
417C2—41no—
418C2#1094611yescharacteristic
419C2#1095711yescharacteristic
420C2#1096111yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24C4×D4
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
16. #1997C4semidirect
17. #1997C4semidirect
31. #1479C4semidirect

Maximal subgroups

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

subgroupindex
#213372
#205502
#175452
#174642
#81192
#80682
#78832