Finite Group Databasebeta

← #203156

Subgroups — #203156

Order 512.

703 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21371C211yesmaximal, characteristic
2C2×#1986C211yesmaximal, characteristic
3#17734C211yesmaximal, characteristic
4#17501C211yesmaximal, characteristic
5M162C211yesmaximal, characteristic
6#9591C214yesmaximal, complemented
7#9589C214yesmaximal, complemented
8#8098C211yesmaximal, characteristic
9#7904C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17#1986—22no—
18#1986C2212yes—
19#1986C2212yes—
20#1950C2212yes—
21#1950C2212yes—
22#1855—22no—
23#1778—22no—
24C2×#339—22no—
25#1559C414yescomplemented
26#1558C412yescomplemented
27#1556C412yescomplemented
28#1542C2214yes—
29#1493C2214yes—
30#1491—21no—
31#1491—21no—
32#1485—22no—
33#1455—22no—
34#1451—21no—
35#1451C2212yes—
36C8×M16—21no—
37C8×M16C2212yes—
38#1303C2214yes—
39#1279C2214yes—
40C22×M16—21no—
41C22×M16D412yes—
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—
58C22×C4⋊C4—21no—
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432C2×C412yes—
67#432C2×C412yes—
68#432C2×C412yes—
69#431—21no—
70#431—21no—
71#431—21no—
72#431C2312yes—
73C2×#108—22no—
74C2×#108—22no—
75#423C2×C412yes—
76#423C2312yes—
77C2×C4⋊C8C2312yes—
78#407—21no—
79#407—21no—
80C2×C22⋊C8—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16—21no—
86C4×M16—21no—
87C4×M16D412yes—
88C4×M16D411yescharacteristic
89C4×M16D411yescharacteristic
90C4×M16D412yes—
91C4×M16C2311yescharacteristic
92C4×M16C2311yescharacteristic
93C2×#97—21no—
94C2×#97—21no—
95C2×#97—21no—
96C2×C4×C8—21no—
97C2×C4×C8C2312yes—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4C2312yes—
102C2×#95—21no—
103#341—22no—
104#341—22no—
105#339—24no—
106#339—24no—
107#333C2×C418yescomplemented
108#332C2×C418yescomplemented
109Q8⋊C8—24no—
110Q8⋊C8—24no—
111#322—22no—
112#322—22no—
113#322—22no—
114C82—22no—
115C22×Q8—21no—
116C22×Q8—21no—
117C23×C4—41no—
118C23×C4C2×D411yescharacteristic
119C23×C4#4111yescharacteristic
120C23×C4#4111yescharacteristic
121C2×M16—21no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—21no—
125C2×M16—22no—
126C2×M16—24no—
127C2×M16—22no—
128C2×M16—42no—
129C2×M16—44no—
130C2×M16C2×D412yes—
131C2×M16C2×D412yes—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8C22×C412yes—
142C4⋊Q8C22×C412yes—
143#125—22no—
144#125—22no—
145#125—41no—
146#125—41no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—41no—
152C22⋊Q8—41no—
153C22⋊Q8—41no—
154C22⋊Q8—41no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—41no—
160C4×Q8—41no—
161C4×Q8—41no—
162C4×Q8—41no—
163#117—22no—
164#117—22no—
165#117—41no—
166#117—41no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C42—22no—
178C2×C42—41no—
179C2×C42—44no—
180C2×C42—41no—
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42C2411yescharacteristic
193#108—44no—
194#108—44no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8C22×C414yes—
200C4⋊C8C22×C414yes—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—41no—
206C22⋊C8—41no—
207#97—22no—
208#97—22no—
209#97—22no—
210#97—22no—
211#97—21no—
212#97—21no—
213#97—22no—
214#97—21no—
215#97—21no—
216#97—41no—
217C4×C8—22no—
218C4×C8—21no—
219C4×C8—22no—
220C4×C8—21no—
221C4×C8—22no—
222C4×C8—41no—
223C4×C8C2×D412yes—
224C4×C8C2×D412yes—
225#95—22no—
226#95—41no—
227C24#12411yescharacteristic
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—82no—
233C2×Q8—82no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4#11711yescharacteristic
245C22×C4#11711yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12311yescharacteristic
253C22×C4#12311yescharacteristic
254C22×C4#12411yescharacteristic
255C22×C4#12411yescharacteristic
256C22×C4#12411yescharacteristic
257C22×C4#12611yescharacteristic
258C22×C4#12611yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C2×#4111yescharacteristic
261C22×C4C2×#4111yescharacteristic
262M16—24no—
263M16—22no—
264M16—22no—
265M16—22no—
266M16—44no—
267M16—44no—
268M16—42no—
269M16—42no—
270M16—42no—
271M16—82no—
272M16#13714yes—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—44no—
280C2×C8—44no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C42—42no—
302C42—42no—
303C42—44no—
304C42—82no—
305C42#11711yescharacteristic
306C42#11711yescharacteristic
307C42C4×D411yescharacteristic
308C42C4×D411yescharacteristic
309C42C4×D411yescharacteristic
310C42C4×D411yescharacteristic
311C42#12111yescharacteristic
312C42#12111yescharacteristic
313C42#12311yescharacteristic
314C42#12311yescharacteristic
315C42#12611yescharacteristic
316C42#12611yescharacteristic
317C42C23×C411yescharacteristic
318C42C22×D411yescharacteristic
319C42C2×#4111yescharacteristic
320C42C2×#4111yescharacteristic
321C23—21no—
322C23—21no—
323C23—22no—
324C23—41no—
325C23#38811yescharacteristic
326C23#38811yescharacteristic
327C23C2×#12411yescharacteristic
328Q8—44no—
329Q8—82no—
330Q8—82no—
331Q8—82no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—88no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4#38811yescharacteristic
359C2×C4#38811yescharacteristic
360C2×C4#38811yescharacteristic
361C2×C4#38811yescharacteristic
362C2×C4#38811yescharacteristic
363C2×C4#38811yescharacteristic
364C2×C4C2×#11711yescharacteristic
365C2×C4C2×C4×D411yescharacteristic
366C2×C4C2×C4×D411yescharacteristic
367C2×C4C2×#12111yescharacteristic
368C2×C4C2×#12311yescharacteristic
369C2×C4C2×#12611yescharacteristic
370C2×C4C2×#13712yes—
371C2×C4#57512yes—
372C8—24no—
373C8—24no—
374C8—42no—
375C8—44no—
376C8—44no—
377C8—42no—
378C8—82no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—42no—
387C22—42no—
388C22#193212yes—
389C22C2×#38811yescharacteristic
390C22#313911yescharacteristic
391C22#314011yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—84no—
402C4—82no—
403C4—82no—
404C4C4×#13712yes—
405C4#295612yes—
406C4#305612yes—
407C2—21no—
408C2—21no—
409C2—22no—
410C2—41no—
411C2#2107111yescharacteristic
412C2C2×#193211yescharacteristic
413C2#2109411yescharacteristic
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
25. #1559C4semidirect
26. #1558C4semidirect
27. #1556C4semidirect
107. #333C2×C4semidirect
108. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#213712
C2×#19862
#177342
#175012
M1622
#95912
#95892
#80982
#79042