Finite Group Databasebeta

← #117184

Subgroups — #117184

Order 512.

762 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21961C211yesmaximal, characteristic
2#21804C211yesmaximal, characteristic
3#21324C211yesmaximal, characteristic
4C2×#1987C211yesmaximal, characteristic
5#10851C211yesmaximal, characteristic
6#8014C211yesmaximal, characteristic
7C2×#1283C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C22×C4⋊C8C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12#2285C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1993—22no—
16#1990—22no—
17#1987—22no—
18#1987C414yescomplemented
19#1981—22no—
20#1956—22no—
21#1947—22no—
22#1947—22no—
23#1808—22no—
24#1787—22no—
25#1736—22no—
26C2×#340—22no—
27C2×#339—22no—
28#1294—22no—
29#1283—22no—
30C23×Q8—21no—
31C22×M16D412yes—
32C23×C8D412yes—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8D412yes—
44C2×C4⋊Q8Q812yes—
45C2×#117—21no—
46C2×#117—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C42—21no—
51C22×C42C2311yescharacteristic
52#431—21no—
53#431—21no—
54#431—21no—
55#431—21no—
56C2×#107—24no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2×C412yes—
63C2×C4⋊C8D412yes—
64C2×#103—28no—
65C2×C22⋊C8D412yes—
66C4×M16—21no—
67C4×M16—21no—
68C4×M16—21no—
69C4×M16—21no—
70C2×#97—21no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73#384D412yes—
74#384D412yes—
75#382—21no—
76#382—21no—
77#380—21no—
78#380—21no—
79#380—21no—
80#380—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4C2×C412yes—
89C2×#95—21no—
90#340—24no—
91#340—28no—
92#340—28no—
93#339—44no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8—22no—
97C22×Q8—41no—
98C22×Q8—41no—
99C22×Q8—41no—
100C22×Q8—84no—
101C23×C4—41no—
102C23×C4—41no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16Q1614yes—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—44no—
117C22×C8C22⋊C414yes—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—41no—
128C4⋊Q8—41no—
129C4⋊Q8—41no—
130C4⋊Q8—41no—
131C4⋊Q8—41no—
132C4⋊Q8—41no—
133C4⋊Q8C4⋊C414yes—
134#117—42no—
135#117—42no—
136#117—42no—
137#117—42no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C42—22no—
163C2×C42—42no—
164C2×C42—42no—
165C2×C42—41no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42C22×C411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×Q811yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180#107—84no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187#103—88no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190#97—41no—
191#97—41no—
192C4×C8—41no—
193C4×C8—41no—
194C4×C8—41no—
195C4×C8—41no—
196#95—22no—
197#95—41no—
198#95—41no—
199#95—41no—
200#95—41no—
201#95—41no—
202C24#12711yescharacteristic
203C2×Q8—24no—
204C2×Q8—48no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—84no—
209C2×Q8—84no—
210C2×Q8—84no—
211C2×Q8—84no—
212C2×Q8—84no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—82no—
224C22×C4—84no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4#12011yescharacteristic
231C22×C4#12011yescharacteristic
232C22×C4#12011yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4C22⋊Q811yescharacteristic
239C22×C4C22⋊Q811yescharacteristic
240C22×C4C22⋊Q811yescharacteristic
241C22×C4#12311yescharacteristic
242C22×C4#12311yescharacteristic
243C22×C4#12311yescharacteristic
244C22×C4#12411yescharacteristic
245C22×C4C2×Q1612yes—
246C22×C4#13512yes—
247C22×C4AGL(1,ℤ/8)12yes—
248C22×C4#13712yes—
249M16—24no—
250M16—24no—
251M16—42no—
252M16—42no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—44no—
258C2×C8—44no—
259C2×C8—44no—
260C2×C8—42no—
261C2×C8—82no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—42no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—82no—
281C22⋊C4—81no—
282C22⋊C4—81no—
283C22⋊C4—81no—
284C22⋊C4—81no—
285C42—44no—
286C42—82no—
287C42—82no—
288C42—82no—
289C42C2×C4⋊C411yescharacteristic
290C42#11711yescharacteristic
291C42C4×D411yescharacteristic
292C42C4×D411yescharacteristic
293C42#12111yescharacteristic
294C42#12111yescharacteristic
295C42#12111yescharacteristic
296C42C22⋊Q811yescharacteristic
297C42C22⋊Q811yescharacteristic
298C42C22⋊Q811yescharacteristic
299C42#12311yescharacteristic
300C42#12311yescharacteristic
301C42#12411yescharacteristic
302C42#12411yescharacteristic
303C42#12611yescharacteristic
304C42#12611yescharacteristic
305C23—22no—
306C23—22no—
307C23#39011yescharacteristic
308C23#39211yescharacteristic
309C23#39911yescharacteristic
310C23#49411yescharacteristic
311C23#49511yescharacteristic
312C23#49611yescharacteristic
313C23#49711yescharacteristic
314Q8—48no—
315Q8—84no—
316Q8—84no—
317Q8—84no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—82no—
334C2×C4—84no—
335C2×C4—84no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—82no—
339C2×C4—84no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4#38011yescharacteristic
347C2×C4#38511yescharacteristic
348C2×C4#38511yescharacteristic
349C2×C4#38611yescharacteristic
350C2×C4#38811yescharacteristic
351C2×C4#38811yescharacteristic
352C2×C4#39311yescharacteristic
353C2×C4#39411yescharacteristic
354C2×C4#39411yescharacteristic
355C2×C4#39611yescharacteristic
356C2×C4#39611yescharacteristic
357C2×C4#39711yescharacteristic
358C2×C4C4×Q1612yes—
359C2×C4#44012yes—
360C2×C4#44912yes—
361C2×C4#45112yes—
362C2×C4#45512yes—
363C2×C4#45612yes—
364C2×C4#46212yes—
365C2×C4#46312yes—
366C2×C4#46712yes—
367C2×C4#46812yes—
368C2×C4#47712yes—
369C2×C4#47912yes—
370C2×C4#48412yes—
371C2×C4#48512yes—
372C8—44no—
373C8—44no—
374C8—82no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22#143911yescharacteristic
384C22#196911yescharacteristic
385C22#197311yescharacteristic
386C22#200712yes—
387C22#200812yes—
388C22#201211yescharacteristic
389C22#201611yescharacteristic
390C22#209511yescharacteristic
391C22#209711yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—84no—
399C4—84no—
400C4—162no—
401C4—162no—
402C4#186312yes—
403C4#188511yescharacteristic
404C4#188611yescharacteristic
405C4#192812yes—
406C4#202812yes—
407C4#203612yes—
408C4#206812yes—
409C2—21no—
410C2—21no—
411C2—21no—
412C2—21no—
413C2#1019612yes—
414C2#1019912yes—
415C2#1105511yescharacteristic
416C2#1112411yescharacteristic
417C2#1112911yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
18. #1987C4semidirect

Maximal subgroups

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

subgroupindex
#219612
#218042
#213242
C2×#19872
#108512
#80142
C2×#12832