Finite Group Databasebeta

← #110702

Subgroups — #110702

Order 512.

634 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21329C211yesmaximal, characteristic
2#20550C211yesmaximal, characteristic
3#17718C211yesmaximal, characteristic
4#17545C211yesmaximal, characteristic
5#8092C211yesmaximal, characteristic
6#8061C211yesmaximal, characteristic
7#8048C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C4×#117C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1995C414yescomplemented
16#1994—22no—
17#1770—21no—
18C8×C4⋊C4—21no—
19#1767—21no—
20C4×C4⋊C8—21no—
21#1753—22no—
22#1733—22no—
23#1554—22no—
24#1548C412yescomplemented
25#1546C412yescomplemented
26#1542—21no—
27#1542—21no—
28#1508—22no—
29#1506C414yescomplemented
30#1503—21no—
31#1503—21no—
32#1303—22no—
33#1299—22no—
34#1297—22no—
35C22×M16D412yes—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533C2×C412yes—
43#533C2×C412yes—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117—21no—
52C2×#117—21no—
53C2×#117—21no—
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432C2×C412yes—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431C2×C412yes—
66#431D412yes—
67#423—21no—
68#423—21no—
69#423—21no—
70#423C2×C412yes—
71#423C2×C412yes—
72#423D412yes—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8D411yescharacteristic
76C2×C4⋊C8D411yescharacteristic
77#407—21no—
78#407—21no—
79C2×C22⋊C8—21no—
80C4×M16—21no—
81C4×M16—21no—
82C2×#97—22no—
83C2×C4×C8—21no—
84C2×C4×C8—22no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4D411yescharacteristic
89C4×C4⋊C4D411yescharacteristic
90C4×C22⋊C4—21no—
91C4×C22⋊C4—21no—
92C4×C22⋊C4—21no—
93C4×C22⋊C4D412yes—
94#376C2×C412yes—
95#376C2×C412yes—
96C43—21no—
97C43—21no—
98#339—28no—
99#336—22no—
100#336—22no—
101#333—22no—
102#333—22no—
103#332—22no—
104#332—22no—
105#328—24no—
106#328—24no—
107C22×Q8—21no—
108C22×Q8—21no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—41no—
115C2×M16—42no—
116C2×M16—42no—
117C2×M16C22⋊C414yes—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8C4212yes—
128C4⋊Q8C4212yes—
129#125—22no—
130#125—22no—
131#125—41no—
132#125—41no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—41no—
149#117—22no—
150#117—22no—
151#117—22no—
152#117—22no—
153#117—44no—
154#117—41no—
155#117—41no—
156#117—41no—
157#117—41no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—22no—
175C2×C42—41no—
176C2×C42—41no—
177C2×C42—41no—
178C2×C42C22×C411yescharacteristic
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—41no—
200C4⋊C8C22⋊C412yes—
201C4⋊C8C22⋊C412yes—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—41no—
206C22⋊C8—41no—
207C22⋊C8—41no—
208#97—22no—
209#97—41no—
210#97—82no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—41no—
214C4×C8—41no—
215C4×C8—82no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95C22⋊C412yes—
220#95C22⋊C412yes—
221C24#12111yescharacteristic
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—82no—
227C2×Q8—82no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4#11711yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12311yescharacteristic
252C22×C4#12311yescharacteristic
253C22×C4#12411yescharacteristic
254C22×C4#12411yescharacteristic
255C22×C4#12611yescharacteristic
256C22×C4#12711yescharacteristic
257M16—44no—
258M16—82no—
259M16—82no—
260M16—82no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—24no—
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—41no—
267C2×C8—42no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—42no—
271C2×C8—42no—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C22⋊C4—44no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C42—24no—
294C42—44no—
295C42—42no—
296C42—44no—
297C42—42no—
298C42—42no—
299C42—41no—
300C42—41no—
301C42—41no—
302C42—41no—
303C42—82no—
304C42—82no—
305C42C2×C4211yescharacteristic
306C42C2×C22⋊C411yescharacteristic
307C42#11711yescharacteristic
308C42#11711yescharacteristic
309C42C4×D411yescharacteristic
310C42C4×D411yescharacteristic
311C42C4×D411yescharacteristic
312C42C4×D411yescharacteristic
313C42C4×D411yescharacteristic
314C42C4×D411yescharacteristic
315C42#12111yescharacteristic
316C42#12111yescharacteristic
317C42#12111yescharacteristic
318C42#12311yescharacteristic
319C42#12411yescharacteristic
320C42#12611yescharacteristic
321C23—21no—
322C23—21no—
323C23—41no—
324C23—41no—
325C23#38611yescharacteristic
326C23#38811yescharacteristic
327C23#39011yescharacteristic
328Q8—82no—
329Q8—82no—
330Q8—82no—
331Q8—82no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
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—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4C4×C22⋊C411yescharacteristic
368C2×C4C4×C22⋊C411yescharacteristic
369C2×C4C4×C22⋊C411yescharacteristic
370C2×C4#38611yescharacteristic
371C2×C4#38611yescharacteristic
372C2×C4#38811yescharacteristic
373C2×C4#38811yescharacteristic
374C2×C4#38811yescharacteristic
375C2×C4#38811yescharacteristic
376C2×C4#38811yescharacteristic
377C2×C4#39011yescharacteristic
378C2×C4#39011yescharacteristic
379C2×C4#45511yescharacteristic
380C2×C4#45511yescharacteristic
381C2×C4#45611yescharacteristic
382C2×C4#45611yescharacteristic
383C8—42no—
384C8—42no—
385C8—44no—
386C8—82no—
387C8—82no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—81no—
397C22—81no—
398C22#143711yescharacteristic
399C22#188211yescharacteristic
400C22#188211yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—44no—
410C4—82no—
411C4—84no—
412C4—82no—
413C4—82no—
414C4#188211yescharacteristic
415C4#188211yescharacteristic
416C4#188511yescharacteristic
417C4#188611yescharacteristic
418C4#193212yes—
419C4#197612yes—
420C2—21no—
421C2—21no—
422C2—41no—
423C2—41no—
424C2#1094411yescharacteristic
425C2#1094611yescharacteristic
426C2#1095711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24#121
LCSG ▷ C42 ▷ 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
15. #1995C4semidirect
24. #1548C4semidirect
25. #1546C4semidirect
29. #1506C4semidirect

Maximal subgroups

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

subgroupindex
#213292
#205502
#177182
#175452
#80922
#80612
#80482