Finite Group Databasebeta

← #115877

Subgroups — #115877

Order 512.

724 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21313C211yesmaximal, characteristic
2#20793C211yesmaximal, characteristic
3C2×#1548C211yesmaximal, characteristic
4C22×#332C211yesmaximal, characteristic
5C2×#1301C211yesmaximal, characteristic
6C2×#1292C211yesmaximal, characteristic
7#7891C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#424—22no—
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C4×#117C2211yescharacteristic
16#1988—22no—
17#1987C414yes—
18#1843—22no—
19#1842—22no—
20#1736—22no—
21#1548—22no—
22#1548—22no—
23C2×#333—22no—
24C2×#332—22no—
25C2×#332—22no—
26C2×#332—22no—
27#1301—22no—
28#1292—22no—
29#1279—22no—
30C23×Q8—21no—
31C22×M16—21no—
32C22×M16—21no—
33C23×C8—21no—
34C23×C8—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⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8D412yes—
46C2×C4⋊Q8Q812yes—
47C2×#117D412yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C42C2311yescharacteristic
51#431—21no—
52#431—21no—
53#431—21no—
54#431D412yes—
55#424—44no—
56#424—44no—
57#423—21no—
58#423—21no—
59#423—21no—
60#423—21no—
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⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—28no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8C2×C412yes—
80C2×C4⋊C8D412yes—
81C2×C4⋊C8D412yes—
82C2×C4⋊C8D412yes—
83C2×C22⋊C8—21no—
84C2×C4×C8—21no—
85C4×C4⋊C4C2×C412yes—
86C4×C4⋊C4D412yes—
87C4×C22⋊C4—21no—
88C4×C22⋊C4—21no—
89#376—21no—
90C43—21no—
91#340—28no—
92#333—28no—
93#332—28no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8—22no—
97C22×Q8—41no—
98C22×Q8—41no—
99C22×Q8—41no—
100C22×Q8—82no—
101C22×Q8—82no—
102C23×C4—41no—
103C23×C4C2×D411yescharacteristic
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—41no—
109C2×M16—41no—
110C2×M16—42no—
111C2×M16—42no—
112C2×M16—42no—
113C2×M16—42no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8—41no—
121C22×C8—41no—
122C22×C8—41no—
123C22×C8—42no—
124C22×C8—42no—
125C22×C8—82no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—41no—
136C4⋊Q8—41no—
137C4⋊Q8—41no—
138C4⋊Q8—41no—
139C4⋊Q8—41no—
140C4⋊Q8—41no—
141C4⋊Q8C4⋊C414yes—
142#117—22no—
143#117—22no—
144#117SD1614yes—
145#117Q1614yes—
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—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4C22⋊C414yes—
159C2×C22⋊C4—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42C22×C411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×Q811yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—416no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C4⋊C8—41no—
191C4⋊C8—41no—
192C4⋊C8—41no—
193C4⋊C8—41no—
194C22⋊C8—22no—
195C22⋊C8—41no—
196C4×C8—22no—
197C4×C8—41no—
198#95—22no—
199#95—22no—
200C24#12311yescharacteristic
201C2×Q8—24no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—84no—
206C2×Q8—82no—
207C2×Q8—82no—
208C2×Q8—84no—
209C2×Q8—82no—
210C2×Q8—82no—
211C2×Q8—84no—
212C2×Q8—82no—
213C2×Q8—82no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—84no—
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4#12011yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4C22⋊Q811yescharacteristic
230C22×C4C22⋊Q811yescharacteristic
231C22×C4C22⋊Q811yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12311yescharacteristic
234C22×C4#12311yescharacteristic
235C22×C4#12311yescharacteristic
236C22×C4#12411yescharacteristic
237C22×C4#12611yescharacteristic
238C22×C4#12611yescharacteristic
239C22×C4#12711yescharacteristic
240C22×C4C2×SD1612yes—
241C22×C4C2×Q1612yes—
242C22×C4AGL(1,ℤ/8)12yes—
243C22×C4#13712yes—
244M16—82no—
245M16—82no—
246M16—81no—
247M16—81no—
248M16—81no—
249M16—81no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—44no—
256C2×C8—44no—
257C2×C8—44no—
258C2×C8—82no—
259C2×C8—82no—
260C2×C8—82no—
261C2×C8—82no—
262C2×C8—81no—
263C2×C8—81no—
264C2×C8—82no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4#10318yes—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C42—24no—
278C42—24no—
279C42—42no—
280C42—42no—
281C42—82no—
282C42C2×C4⋊C411yescharacteristic
283C42#11711yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42#12011yescharacteristic
287C42#12011yescharacteristic
288C42#12111yescharacteristic
289C42#12111yescharacteristic
290C42#12111yescharacteristic
291C42#12111yescharacteristic
292C42#12111yescharacteristic
293C42C22⋊Q811yescharacteristic
294C42C22⋊Q811yescharacteristic
295C42C22⋊Q811yescharacteristic
296C42#12411yescharacteristic
297C42#12411yescharacteristic
298C23—22no—
299C23—22no—
300C23#38611yescharacteristic
301C23#39711yescharacteristic
302C23#39911yescharacteristic
303C23#48312yes—
304C23#48412yes—
305Q8—84no—
306Q8—84no—
307Q8—84no—
308Q8—82no—
309Q8—82no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—28no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—84no—
328C2×C4—84no—
329C2×C4—84no—
330C2×C4#38011yescharacteristic
331C2×C4#38511yescharacteristic
332C2×C4#38511yescharacteristic
333C2×C4#38811yescharacteristic
334C2×C4#38811yescharacteristic
335C2×C4#39011yescharacteristic
336C2×C4#39211yescharacteristic
337C2×C4#39311yescharacteristic
338C2×C4#39411yescharacteristic
339C2×C4#39411yescharacteristic
340C2×C4#39611yescharacteristic
341C2×C4#39611yescharacteristic
342C2×C4C2×#10312yes—
343C2×C4#41712yes—
344C2×C4#45012yes—
345C2×C4#45112yes—
346C2×C4#46012yes—
347C2×C4#46212yes—
348C2×C4#47112yes—
349C2×C4#47212yes—
350C2×C4#47512yes—
351C2×C4#47712yes—
352C2×C4#48712yes—
353C2×C4#48912yes—
354C2×C4#49211yescharacteristic
355C2×C4#49411yescharacteristic
356C2×C4#49611yescharacteristic
357C2×C4#49711yescharacteristic
358C8—82no—
359C8—82no—
360C8—82no—
361C8—82no—
362C8—82no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—24no—
368C22—24no—
369C22#143911yescharacteristic
370C22#187212yes—
371C22#190212yes—
372C22#190712yes—
373C22#207212yes—
374C22#209511yescharacteristic
375C22#209611yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—48no—
383C4—84no—
384C4#180312yes—
385C4#195611yescharacteristic
386C4#195911yescharacteristic
387C4#201411yescharacteristic
388C4#201611yescharacteristic
389C4#202412yes—
390C4#204511yescharacteristic
391C4#204611yescharacteristic
392C4#206312yes—
393C2—22no—
394C2—21no—
395C2—21no—
396C2#916714yes—
397C2#1105011yescharacteristic
398C2#1105911yescharacteristic
399C2#1106111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ 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
#213132
#207932
C2×#15482
C22×#3322
C2×#13012
C2×#12922
#78912