Finite Group Databasebeta

← #116752

Subgroups — #116752

Order 512.

638 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21993C211yesmaximal, characteristic
2#21303C211yesmaximal, characteristic
3#17449C211yesmaximal, characteristic
4C2×#1479C211yesmaximal, characteristic
5#8084C211yesmaximal, characteristic
6#8065C211yesmaximal, characteristic
7#8060C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#1985—22no—
16#1984—22no—
17#1950—22no—
18#1855—22no—
19#1848—21no—
20#1837—21no—
21#1728—22no—
22#1491—22no—
23#1487—21no—
24#1487—21no—
25#1479—22no—
26#1479—22no—
27#1479C414yes—
28C2×Q8⋊C8—22no—
29#1302—22no—
30#1299—22no—
31#1299—22no—
32C23×Q8Q811yescharacteristic
33C22×M16—21no—
34C22×M16D412yes—
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×Q8—21no—
45C2×C4×Q8C2×C412yes—
46C2×C4×Q8D412yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4Q811yescharacteristic
49C22×C42—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#431—21no—
64#431C2×C412yes—
65#431C2×C412yes—
66#431D412yes—
67C2×#108—24no—
68#423—21no—
69#423—21no—
70#423—21no—
71#423—21no—
72#423—21no—
73#423D411yescharacteristic
74#423D411yescharacteristic
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8D412yes—
79#407—21no—
80#407—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16—21no—
86C4×M16—21no—
87C4×M16—21no—
88C4×M16D412yes—
89C2×#97D412yes—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93#341—24no—
94#336—28no—
95Q8⋊C8—24no—
96Q8⋊C8—24no—
97Q8⋊C8—28no—
98C22×Q8—21no—
99C22×Q8—24no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—42no—
105C22×Q8C4⋊C412yes—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×Q811yescharacteristic
110C23×C4#4111yescharacteristic
111C2×M16—24no—
112C2×M16—22no—
113C2×M16—21no—
114C2×M16—21no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—42no—
118C2×M16—41no—
119C2×M16—41no—
120C2×M16—44no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—41no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4C4⋊C412yes—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—41no—
156C2×C42—41no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42C22×C411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171#108—84no—
172C4⋊C8—22no—
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8—21no—
176C4⋊C8—21no—
177C4⋊C8—22no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185C4⋊C8C22⋊C414yes—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194#97—22no—
195#97—22no—
196#97—22no—
197#97—41no—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—41no—
203C4×C8—41no—
204C4×C8—41no—
205C4×C8—41no—
206C24C22⋊Q811yescharacteristic
207C2×Q8—22no—
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—44no—
215C2×Q8—44no—
216C2×Q8—44no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8#10814yescomplemented
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—44no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—81no—
234C22×C4—82no—
235C22×C4C2×C4⋊C411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4#12011yescharacteristic
239C22×C4#12011yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4C22⋊Q811yescharacteristic
244C22×C4C22⋊Q811yescharacteristic
245C22×C4C22⋊Q811yescharacteristic
246C22×C4C22⋊Q811yescharacteristic
247C22×C4C22⋊Q811yescharacteristic
248C22×C4#12311yescharacteristic
249C22×C4#12411yescharacteristic
250C22×C4#12411yescharacteristic
251C22×C4#12411yescharacteristic
252C22×C4#12611yescharacteristic
253M16—44no—
254M16—44no—
255M16—42no—
256M16—44no—
257M16—42no—
258M16—82no—
259M16—82no—
260C2×C8—24no—
261C2×C8—24no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—44no—
267C2×C8—42no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—82no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C42—42no—
282C42—42no—
283C42—42no—
284C42—42no—
285C42C2×C22⋊C411yescharacteristic
286C42#11711yescharacteristic
287C42C4×D411yescharacteristic
288C42C4×D411yescharacteristic
289C42#12011yescharacteristic
290C42#12111yescharacteristic
291C42#12111yescharacteristic
292C42#12111yescharacteristic
293C42#12111yescharacteristic
294C42#12111yescharacteristic
295C42#12311yescharacteristic
296C42#12311yescharacteristic
297C42#12311yescharacteristic
298C42#12311yescharacteristic
299C42#12611yescharacteristic
300C42#12711yescharacteristic
301C23—22no—
302C23—22no—
303C23—41no—
304C23#38511yescharacteristic
305C23#39311yescharacteristic
306C23#39611yescharacteristic
307Q8—24no—
308Q8—24no—
309Q8—44no—
310Q8—44no—
311Q8—42no—
312Q8—42no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—24no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4#38011yescharacteristic
343C2×C4#38511yescharacteristic
344C2×C4#38611yescharacteristic
345C2×C4#38811yescharacteristic
346C2×C4#38811yescharacteristic
347C2×C4#39011yescharacteristic
348C2×C4#39211yescharacteristic
349C2×C4#39411yescharacteristic
350C2×C4#39411yescharacteristic
351C2×C4#39611yescharacteristic
352C2×C4#39711yescharacteristic
353C2×C4#39911yescharacteristic
354C2×C4C2×#10812yes—
355C2×C4#43012yes—
356C2×C4#45512yes—
357C2×C4#45612yes—
358C8—44no—
359C8—44no—
360C8—82no—
361C8—82no—
362C8—82no—
363C8—81no—
364C8—81no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—44no—
373C22#143911yescharacteristic
374C22#185512yes—
375C22#202812yes—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—44no—
385C4—82no—
386C4#181011yescharacteristic
387C4#181111yescharacteristic
388C4#185512yes—
389C4#188212yes—
390C4#200411yescharacteristic
391C4#200511yescharacteristic
392C2—22no—
393C2—22no—
394C2—41no—
395C2#1106411yescharacteristic
396C2#1109711yescharacteristic
397C2#1111811yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24C22⋊Q8
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
221. C2×Q8#108semidirect

Maximal subgroups

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

subgroupindex
#219932
#213032
#174492
C2×#14792
#80842
#80652
#80602