Finite Group Databasebeta

← #203147

Subgroups — #203147

Order 512.

668 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21970C211yesmaximal, characteristic
2#21844C211yesmaximal, characteristic
3#17708C211yesmaximal, characteristic
4#17501C211yesmaximal, characteristic
5#17327C211yesmaximal, characteristic
6#9591C214yesmaximal, complemented
7#9589C214yesmaximal, complemented
8#8092C211yesmaximal, characteristic
9#7903C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16C4×#117C2211yescharacteristic
17#1994—22no—
18#1986C2214yes—
19#1950—22no—
20#1950C2214yes—
21#1826—22no—
22#1811—22no—
23#1550C414yescomplemented
24#1549C412yescomplemented
25#1548C412yescomplemented
26#1542C2212yes—
27#1542C2212yes—
28#1493C2214yes—
29#1491—22no—
30#1485—21no—
31#1485—21no—
32#1453—22no—
33#1451—21no—
34#1451C2212yes—
35C8×M16—21no—
36C8×M16C2212yes—
37#1303C2212yes—
38#1303C2212yes—
39#1279C2214yes—
40C22×M16—21no—
41C22×M16—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533C2×C412yes—
49#533C2312yes—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×#117—21no—
58C2×#117—21no—
59C22×C42C2311yescharacteristic
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431C2×C412yes—
71#431C2311yescharacteristic
72#431C2311yescharacteristic
73C2×#108—24no—
74#423—21no—
75#423—21no—
76#423C2×C412yes—
77#423C2×C412yes—
78#423D412yes—
79#423C2312yes—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2×C412yes—
82C2×C4⋊C8C2312yes—
83#407—21no—
84#407D412yes—
85C2×C22⋊C8—21no—
86C4×M16—21no—
87C4×M16—21no—
88C4×M16—21no—
89C4×M16D411yescharacteristic
90C4×M16D411yescharacteristic
91C4×M16D412yes—
92C4×M16C2312yes—
93C2×#97—21no—
94C2×C4×C8—21no—
95C2×C4×C8C2312yes—
96C4×C4⋊C4—21no—
97C4×C4⋊C4C2312yes—
98C4×C22⋊C4—21no—
99C4×C22⋊C4—21no—
100#376—21no—
101C43—21no—
102#341—24no—
103#339—28no—
104#333C2×C414yes—
105#333C2×C414yes—
106#332C2×C414yes—
107#332C2×C414yes—
108Q8⋊C8—24no—
109Q8⋊C8—24no—
110#322—22no—
111#322—22no—
112#322—22no—
113C82—22no—
114C22×Q8—21no—
115C22×Q8—21no—
116C23×C4C2×D411yescharacteristic
117C23×C4#4111yescharacteristic
118C23×C4#4111yescharacteristic
119C2×M16—21no—
120C2×M16—22no—
121C2×M16—21no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—44no—
126C2×M16—82no—
127C2×M16C2×D414yes—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8C22×C412yes—
138C4⋊Q8C22×C412yes—
139#125—22no—
140#125—22no—
141#125—41no—
142#125—41no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—41no—
156C4×Q8—41no—
157C4×Q8—41no—
158C4×Q8—41no—
159#117—22no—
160#117—22no—
161#117—44no—
162#117—41no—
163#117—41no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C42—22no—
174C2×C42—41no—
175C2×C42—41no—
176C2×C42—41no—
177C2×C42C22×C411yescharacteristic
178C2×C42C22×C411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42C2411yescharacteristic
189#108—84no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8C22×C412yes—
199C4⋊C8C22×C412yes—
200C4⋊C8C22×C414yes—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—41no—
207C22⋊C8—41no—
208#97—22no—
209#97—21no—
210#97—21no—
211#97—22no—
212#97—41no—
213C4×C8—22no—
214C4×C8—21no—
215C4×C8—22no—
216C4×C8—21no—
217C4×C8—22no—
218C4×C8—41no—
219C4×C8C2×D412yes—
220C4×C8C2×D412yes—
221#95—22no—
222#95—41no—
223C24#12411yescharacteristic
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—82no—
229C2×Q8—82no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4#11711yescharacteristic
238C22×C4#11711yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12311yescharacteristic
246C22×C4#12311yescharacteristic
247C22×C4#12411yescharacteristic
248C22×C4#12411yescharacteristic
249C22×C4#12411yescharacteristic
250C22×C4#12611yescharacteristic
251C22×C4#12611yescharacteristic
252C22×C4C22×D411yescharacteristic
253C22×C4C2×#4111yescharacteristic
254C22×C4C2×#4111yescharacteristic
255M16—24no—
256M16—24no—
257M16—24no—
258M16—44no—
259M16—42no—
260M16—42no—
261M16—42no—
262M16—82no—
263M16—82no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—22no—
270C2×C8—44no—
271C2×C8—44no—
272C2×C8—42no—
273C2×C8—82no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C22⋊C4—44no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C42—44no—
292C42—42no—
293C42—42no—
294C42—42no—
295C42—42no—
296C42—82no—
297C42#11711yescharacteristic
298C42#11711yescharacteristic
299C42C4×D411yescharacteristic
300C42C4×D411yescharacteristic
301C42C4×D411yescharacteristic
302C42C4×D411yescharacteristic
303C42#12111yescharacteristic
304C42#12111yescharacteristic
305C42#12311yescharacteristic
306C42#12311yescharacteristic
307C42#12611yescharacteristic
308C42#12611yescharacteristic
309C42C23×C411yescharacteristic
310C42C22×D411yescharacteristic
311C42C2×#4111yescharacteristic
312C42C2×#4111yescharacteristic
313C23—22no—
314C23—41no—
315C23—41no—
316C23#38811yescharacteristic
317C23#38811yescharacteristic
318C23C2×#12411yescharacteristic
319Q8—44no—
320Q8—82no—
321Q8—82no—
322Q8—82no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4#38811yescharacteristic
350C2×C4#38811yescharacteristic
351C2×C4#38811yescharacteristic
352C2×C4#38811yescharacteristic
353C2×C4#38811yescharacteristic
354C2×C4#38811yescharacteristic
355C2×C4C2×#11711yescharacteristic
356C2×C4C2×C4×D411yescharacteristic
357C2×C4C2×C4×D411yescharacteristic
358C2×C4C2×#12111yescharacteristic
359C2×C4C2×#12311yescharacteristic
360C2×C4C2×#12611yescharacteristic
361C2×C4#57712yes—
362C2×C4#57812yes—
363C8—24no—
364C8—42no—
365C8—44no—
366C8—42no—
367C8—42no—
368C8—44no—
369C8—82no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—82no—
379C22C2×#38811yescharacteristic
380C22#313712yes—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—84no—
390C4—82no—
391C4—82no—
392C4#193212yes—
393C4#294911yescharacteristic
394C4#295011yescharacteristic
395C4#295612yes—
396C4#306312yes—
397C2—22no—
398C2—41no—
399C2—41no—
400C2#2107111yescharacteristic
401C2C2×#193211yescharacteristic
402C2#2108811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#388
prime radicalC24#124
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9591C2semidirect
7. #9589C2semidirect
23. #1550C4semidirect
24. #1549C4semidirect
25. #1548C4semidirect

Maximal subgroups

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

subgroupindex
#219702
#218442
#177082
#175012
#173272
#95912
#95892
#80922
#79032