Finite Group Databasebeta

← #116267

Subgroups — #116267

Order 512.

593 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21973C211yesmaximal, characteristic
2#21302C211yesmaximal, characteristic
3#17562C211yesmaximal, characteristic
4#17464C211yesmaximal, characteristic
5#7973C211yesmaximal, characteristic
6#7935C211yesmaximal, characteristic
7#7883C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2283C2211yescharacteristic
15#1985—22no—
16#1984—22no—
17#1939—22no—
18#1850—22no—
19#1843—21no—
20#1832—21no—
21#1729—22no—
22#1515—22no—
23#1511C414yes—
24#1503—22no—
25#1485—22no—
26#1481—21no—
27#1481—21no—
28#1479—22no—
29#1288—22no—
30#1283—22no—
31#1278—22no—
32C22×M16—21no—
33C23×C8D412yes—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42C2×C4⋊Q8Q811yescharacteristic
43C2×C4⋊Q8Q811yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8C2×C412yes—
47C2×C4×Q8—21no—
48C2×C4×Q8D412yes—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#423—21no—
64#423—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—24no—
72C2×C4⋊C8D411yescharacteristic
73C2×C4⋊C8D411yescharacteristic
74#407—21no—
75#407C2×C412yes—
76C2×C22⋊C8—21no—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16D412yes—
83C2×#97—21no—
84C2×C4×C8—21no—
85C2×C4×C8D412yes—
86#383D412yes—
87#382—21no—
88#381C2×C412yes—
89#380—21no—
90C4×C4⋊C4D412yes—
91C4×C22⋊C4—21no—
92#376—21no—
93#336—24no—
94#336—28no—
95#328—28no—
96Q8⋊C8—24no—
97Q8⋊C8—24no—
98C22×Q8—21no—
99C22×Q8C4⋊C412yes—
100C23×C4C22×C411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4#4111yescharacteristic
103C2×M16—24no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—42no—
107C2×M16—41no—
108C2×M16—41no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—22no—
115C22×C8—44no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124#125—41no—
125#125—41no—
126#125—41no—
127#125—41no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144#117—41no—
145#117—41no—
146#117—41no—
147#117—41no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4C4⋊C412yes—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×Q811yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183C4⋊C8—84no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190#97—22no—
191#97—41no—
192#97—41no—
193#97—41no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—22no—
197C4×C8—41no—
198C4×C8—41no—
199C4×C8—41no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95C22⋊C414yes—
204C24C4×D411yescharacteristic
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—44no—
208C2×Q8—44no—
209C2×Q8—41no—
210C2×Q8—41no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4C2×C4⋊C411yescharacteristic
220C22×C4#11711yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4#12011yescharacteristic
225C22×C4#12011yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4C22⋊Q811yescharacteristic
230C22×C4C22⋊Q811yescharacteristic
231C22×C4#12311yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12311yescharacteristic
234C22×C4#12411yescharacteristic
235C22×C4#12611yescharacteristic
236M16—44no—
237M16—44no—
238M16—44no—
239M16—44no—
240C2×C8—24no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—44no—
244C2×C8—42no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—44no—
251C2×C8—42no—
252C2×C8—82no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—44no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C22⋊C4—44no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C42—42no—
270C42—42no—
271C42—42no—
272C42—42no—
273C42#12011yescharacteristic
274C42#12111yescharacteristic
275C42#12111yescharacteristic
276C42#12111yescharacteristic
277C42#12111yescharacteristic
278C42#12111yescharacteristic
279C42C22⋊Q811yescharacteristic
280C42C22⋊Q811yescharacteristic
281C42C22⋊Q811yescharacteristic
282C42C22⋊Q811yescharacteristic
283C42#12311yescharacteristic
284C42#12311yescharacteristic
285C42#12411yescharacteristic
286C42#12411yescharacteristic
287C42#12611yescharacteristic
288C42#12711yescharacteristic
289C23—22no—
290C23—22no—
291C23—41no—
292C23#38511yescharacteristic
293C23#38611yescharacteristic
294C23#38811yescharacteristic
295Q8—44no—
296Q8—82no—
297Q8—82no—
298Q8—82no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4#38011yescharacteristic
326C2×C4#38511yescharacteristic
327C2×C4#38811yescharacteristic
328C2×C4#39011yescharacteristic
329C2×C4#39211yescharacteristic
330C2×C4#39311yescharacteristic
331C2×C4#39411yescharacteristic
332C2×C4#39411yescharacteristic
333C2×C4#39611yescharacteristic
334C2×C4#39611yescharacteristic
335C2×C4#39711yescharacteristic
336C2×C4#39911yescharacteristic
337C2×C4#45512yes—
338C2×C4#45612yes—
339C8—44no—
340C8—82no—
341C8—82no—
342C8—81no—
343C8—82no—
344C8—81no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—44no—
353C22#143911yescharacteristic
354C22#186812yes—
355C22#188212yes—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—84no—
366C4—82no—
367C4—82no—
368C4#200411yescharacteristic
369C4#200511yescharacteristic
370C4#202812yes—
371C4#209812yes—
372C2—22no—
373C2—22no—
374C2—41no—
375C2#1106411yescharacteristic
376C2#1108411yescharacteristic
377C2#1108811yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24C4×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#219732
#213022
#175622
#174642
#79732
#79352
#78832