Finite Group Databasebeta

← #118530

Subgroups — #118530

Order 512.

667 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21358C211yesmaximal, characteristic
2#21306C211yesmaximal, characteristic
3#17575C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#10851C211yesmaximal, characteristic
6#7981C211yesmaximal, characteristic
7#7918C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12#2285C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1986—22no—
16#1986—22no—
17#1947—22no—
18#1852—22no—
19C4×#107—22no—
20#1733—22no—
21C2×#339—22no—
22#1520—22no—
23#1518C414yescomplemented
24#1503—22no—
25#1503—22no—
26#1503—22no—
27C2×#328—22no—
28#1289—22no—
29#1281—22no—
30C22×M16D412yes—
31#533—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533C2×C412yes—
38#533D412yes—
39C2×C4⋊Q8Q811yescharacteristic
40C2×C4⋊Q8Q811yescharacteristic
41C2×#125—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×#117—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#431—21no—
58#431C2×C412yes—
59#431D412yes—
60C2×#107—24no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C22⋊C8D412yes—
64C2×C22⋊C8D412yes—
65C4×M16—21no—
66C4×M16—21no—
67C4×M16—21no—
68C4×M16—21no—
69C2×#97—21no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73#384—21no—
74#384—21no—
75#382—21no—
76#382C2×C412yes—
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⋊C4D412yes—
88C4×C4⋊C4D412yes—
89C2×#95—21no—
90#340—24no—
91#339—28no—
92#339—44no—
93#328—28no—
94#328—28no—
95C22×Q8—21no—
96C22×Q8—21no—
97C23×C4—41no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16Q1614yes—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8C4⋊C412yes—
117C4⋊Q8C4⋊C412yes—
118#125—22no—
119#125—22no—
120#125—41no—
121#125—41no—
122C22⋊Q8—22no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—41no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138#117—22no—
139#117—22no—
140#117—41no—
141#117—41no—
142#117—42no—
143#117—42no—
144#117—42no—
145#117—42no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—44no—
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×C4⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—41no—
167C2×C42—42no—
168C2×C42—42no—
169C2×C42—41no—
170C2×C42—41no—
171C2×C42C22×C411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×Q811yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183#107—48no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8C22⋊C414yes—
192#97—41no—
193#97—41no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—41no—
197C4×C8—41no—
198C4×C8—41no—
199C4×C8—41no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—41no—
205#95—41no—
206C24#12711yescharacteristic
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—42no—
220C22×C4—81no—
221C22×C4—82no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4#12011yescharacteristic
228C22×C4#12011yescharacteristic
229C22×C4#12011yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4C22⋊Q811yescharacteristic
236C22×C4C22⋊Q811yescharacteristic
237C22×C4C22⋊Q811yescharacteristic
238C22×C4#12311yescharacteristic
239C22×C4#12311yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12411yescharacteristic
242C22×C4C2×Q1612yes—
243C22×C4#13512yes—
244C22×C4AGL(1,ℤ/8)12yes—
245C22×C4#13712yes—
246M16—24no—
247M16—24no—
248M16—42no—
249M16—42no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—44no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—82no—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—42no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—82no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—81no—
281C22⋊C4—81no—
282C22⋊C4—81no—
283C22⋊C4—81no—
284C42—42no—
285C42—42no—
286C42—44no—
287C42—44no—
288C42—82no—
289C42—82no—
290C42C2×C4⋊C411yescharacteristic
291C42#11711yescharacteristic
292C42C4×D411yescharacteristic
293C42C4×D411yescharacteristic
294C42#12111yescharacteristic
295C42#12111yescharacteristic
296C42#12111yescharacteristic
297C42C22⋊Q811yescharacteristic
298C42C22⋊Q811yescharacteristic
299C42C22⋊Q811yescharacteristic
300C42#12311yescharacteristic
301C42#12311yescharacteristic
302C42#12411yescharacteristic
303C42#12411yescharacteristic
304C42#12611yescharacteristic
305C42#12611yescharacteristic
306C23—41no—
307C23—41no—
308C23#39011yescharacteristic
309C23#39211yescharacteristic
310C23#39911yescharacteristic
311C23#49411yescharacteristic
312C23#49511yescharacteristic
313C23#49611yescharacteristic
314C23#49711yescharacteristic
315Q8—82no—
316Q8—82no—
317Q8—82no—
318Q8—82no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—82no—
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#46212yes—
363C2×C4#46312yes—
364C2×C4#46712yes—
365C2×C4#46812yes—
366C2×C4#47712yes—
367C2×C4#47912yes—
368C2×C4#48412yes—
369C2×C4#48512yes—
370C8—44no—
371C8—44no—
372C8—82no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22#143911yescharacteristic
384C22#196911yescharacteristic
385C22#197311yescharacteristic
386C22#201211yescharacteristic
387C22#201611yescharacteristic
388C22#209511yescharacteristic
389C22#209711yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—84no—
398C4—82no—
399C4—82no—
400C4—162no—
401C4—162no—
402C4#186312yes—
403C4#192812yes—
404C4#203112yes—
405C4#203612yes—
406C4#206812yes—
407C2—21no—
408C2—21no—
409C2—21no—
410C2—21no—
411C2—21no—
412C2—21no—
413C2#1109111yescharacteristic
414C2#1111911yescharacteristic
415C2#1112411yescharacteristic
subgroupG/H
centerC2×C4#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24#127
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
23. #1518C4semidirect

Maximal subgroups

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

subgroupindex
#213582
#213062
#175752
C2×#15032
#108512
#79812
#79182