Finite Group Databasebeta

← #118164

Subgroups — #118164

Order 512.

659 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21358C211yesmaximal, characteristic
2C2×#1986C211yesmaximal, characteristic
3#17575C211yesmaximal, characteristic
4#17486C211yesmaximal, characteristic
5#10855C211yesmaximal, characteristic
6C2×#1303C211yesmaximal, characteristic
7#7860C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1986—22no—
16#1986—22no—
17#1947—22no—
18#1852—22no—
19C4×#107—22no—
20C2×#344—22no—
21C2×#339—22no—
22#1520—22no—
23#1518—22no—
24#1503—22no—
25#1493—22no—
26#1487—22no—
27#1481C414yescomplemented
28#1303—22no—
29#1275—22no—
30C22×M16—21no—
31C22×M16D412yes—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533C2×C412yes—
39#533D412yes—
40C2×C4⋊Q8Q811yescharacteristic
41C2×C4⋊Q8Q811yescharacteristic
42C2×#125—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×#117—21no—
48C2×#117—21no—
49C22×C4⋊C4—21no—
50C22×C42—21no—
51C22×C42C2311yescharacteristic
52#431—21no—
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431C2×C412yes—
58#431D412yes—
59#431D412yes—
60C2×#107—24no—
61#423—21no—
62#423—21no—
63#423—21no—
64#423—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8C2×C412yes—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C4×M16—21no—
72C4×M16—21no—
73C4×M16—21no—
74C4×M16—21no—
75C2×#97—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8D412yes—
79#384—21no—
80#382—21no—
81#380—21no—
82#380D412yes—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4D412yes—
89C2×#95—21no—
90#344—44no—
91#340—24no—
92#339—28no—
93#328—28no—
94Q8⋊C8—28no—
95C22×Q8—21no—
96C22×Q8—21no—
97C23×C4—41no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4#4111yescharacteristic
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—42no—
109C2×M16—42no—
110C2×M16Q1614yes—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—41no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8C4⋊C412yes—
122C4⋊Q8C4⋊C412yes—
123#125—22no—
124#125—22no—
125#125—41no—
126#125—41no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143#117—22no—
144#117—22no—
145#117—41no—
146#117—41no—
147#117—42no—
148#117—42no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—41no—
163C2×C42—22no—
164C2×C42—41no—
165C2×C42—42no—
166C2×C42—42no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42C22×C411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×Q811yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181#107—48no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—41no—
194#97—41no—
195#97—41no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—41no—
199C4×C8—41no—
200C4×C8—41no—
201C4×C8C22⋊C414yes—
202#95—22no—
203#95—22no—
204#95—41no—
205#95—41no—
206C24#12111yescharacteristic
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—82no—
212C2×Q8—82no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—81no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4#11711yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4#12011yescharacteristic
226C22×C4#12011yescharacteristic
227C22×C4#12011yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4C22⋊Q811yescharacteristic
233C22×C4C22⋊Q811yescharacteristic
234C22×C4C22⋊Q811yescharacteristic
235C22×C4#12311yescharacteristic
236C22×C4#12411yescharacteristic
237C22×C4#12611yescharacteristic
238C22×C4#12611yescharacteristic
239C22×C4#12711yescharacteristic
240C22×C4C2×Q1612yes—
241C22×C4#13512yes—
242C22×C4AGL(1,ℤ/8)12yes—
243C22×C4#13712yes—
244M16—24no—
245M16—24no—
246M16—42no—
247M16—42no—
248M16—82no—
249M16—81no—
250M16—81no—
251C2×C8—24no—
252C2×C8—24no—
253C2×C8—44no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—44no—
259C2×C8—82no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—42no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—44no—
281C42—42no—
282C42—42no—
283C42—44no—
284C42—81no—
285C42—81no—
286C42—81no—
287C42—81no—
288C42C2×C22⋊C411yescharacteristic
289C42C2×C4⋊C411yescharacteristic
290C42C4×D411yescharacteristic
291C42C4×D411yescharacteristic
292C42#12111yescharacteristic
293C42#12111yescharacteristic
294C42#12111yescharacteristic
295C42C22⋊Q811yescharacteristic
296C42C22⋊Q811yescharacteristic
297C42C22⋊Q811yescharacteristic
298C42#12311yescharacteristic
299C42#12311yescharacteristic
300C42#12311yescharacteristic
301C42#12311yescharacteristic
302C42#12411yescharacteristic
303C42#12411yescharacteristic
304C23—41no—
305C23—41no—
306C23#38811yescharacteristic
307C23#39211yescharacteristic
308C23#39611yescharacteristic
309C23#46212yes—
310C23#46312yes—
311Q8—44no—
312Q8—82no—
313Q8—82no—
314Q8—82no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—88no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4#38011yescharacteristic
339C2×C4#38511yescharacteristic
340C2×C4#38511yescharacteristic
341C2×C4#38611yescharacteristic
342C2×C4#38811yescharacteristic
343C2×C4#39011yescharacteristic
344C2×C4#39311yescharacteristic
345C2×C4#39411yescharacteristic
346C2×C4#39411yescharacteristic
347C2×C4#39611yescharacteristic
348C2×C4#39711yescharacteristic
349C2×C4#39911yescharacteristic
350C2×C4C4×Q1612yes—
351C2×C4#44012yes—
352C2×C4#44912yes—
353C2×C4#45112yes—
354C2×C4#46712yes—
355C2×C4#46812yes—
356C2×C4#47712yes—
357C2×C4#47912yes—
358C2×C4#48412yes—
359C2×C4#48512yes—
360C2×C4#49411yescharacteristic
361C2×C4#49511yescharacteristic
362C2×C4#49611yescharacteristic
363C2×C4#49711yescharacteristic
364C8—44no—
365C8—44no—
366C8—82no—
367C8—82no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22#143911yescharacteristic
377C22#192812yes—
378C22#201211yescharacteristic
379C22#201611yescharacteristic
380C22#206812yes—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—84no—
389C4—82no—
390C4—82no—
391C4—162no—
392C4#186312yes—
393C4#196911yescharacteristic
394C4#197311yescharacteristic
395C4#203112yes—
396C4#203612yes—
397C4#209511yescharacteristic
398C4#209711yescharacteristic
399C2—21no—
400C2—22no—
401C2—22no—
402C2—21no—
403C2#1108511yescharacteristic
404C2#1110911yescharacteristic
405C2#1112411yescharacteristic
subgroupG/H
centerC2×C4#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24#121
LCSG ▷ C2×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
27. #1481C4semidirect

Maximal subgroups

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

subgroupindex
#213582
C2×#19862
#175752
#174862
#108552
C2×#13032
#78602