Finite Group Databasebeta

← #203170

Subgroups — #203170

Order 512.

674 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21370C211yesmaximal, characteristic
2#21306C211yesmaximal, characteristic
3#17718C211yesmaximal, characteristic
4#17501C211yesmaximal, characteristic
5M162C211yesmaximal, characteristic
6#9591C214yesmaximal, complemented
7#9589C214yesmaximal, complemented
8#8094C211yesmaximal, characteristic
9#7896C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2285C2211yescharacteristic
17#1986—22no—
18#1986C2212yes—
19#1986C2212yes—
20#1950C2212yes—
21#1950C2212yes—
22#1855—22no—
23C4×#108—22no—
24#1733—22no—
25#1554C414yescomplemented
26#1548C412yescomplemented
27#1546C412yescomplemented
28#1542C2214yes—
29#1493C2214yes—
30#1491—22no—
31#1485—21no—
32#1485—21no—
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—
58C2×#117—21no—
59C22×C42C2311yescharacteristic
60#432—21no—
61#432—21no—
62#432—21no—
63#432C2×C412yes—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431C2×C412yes—
70#431C2×C412yes—
71#431C2312yes—
72C2×#108—22no—
73C2×#108—22no—
74#423C2×C412yes—
75#423C2312yes—
76C2×C4⋊C8C2312yes—
77#407—21no—
78#407—21no—
79C2×C22⋊C8—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16—21no—
86C4×M16D412yes—
87C4×M16D411yescharacteristic
88C4×M16D411yescharacteristic
89C4×M16D412yes—
90C4×M16C2311yescharacteristic
91C4×M16C2311yescharacteristic
92C2×#97—21no—
93C2×#97—21no—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96C2×C4×C8C2312yes—
97#384—21no—
98#382—21no—
99#380—21no—
100#380—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4C2312yes—
103#341—22no—
104#341—22no—
105#339—24no—
106#339—24no—
107#333C2×C418yes—
108#332C2×C418yes—
109Q8⋊C8—24no—
110Q8⋊C8—24no—
111#322—22no—
112#322—22no—
113#322—22no—
114C82—22no—
115C22×Q8—21no—
116C22×Q8—21no—
117C23×C4C2×D411yescharacteristic
118C23×C4#4111yescharacteristic
119C23×C4#4111yescharacteristic
120C2×M16—21no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—21no—
124C2×M16—22no—
125C2×M16—24no—
126C2×M16—22no—
127C2×M16—42no—
128C2×M16—44no—
129C2×M16C2×D412yes—
130C2×M16C2×D412yes—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8C22×C412yes—
141C4⋊Q8C22×C412yes—
142#125—22no—
143#125—22no—
144#125—41no—
145#125—41no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152C22⋊Q8—41no—
153C22⋊Q8—41no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—41no—
159C4×Q8—41no—
160C4×Q8—41no—
161C4×Q8—41no—
162#117—22no—
163#117—22no—
164#117—41no—
165#117—41no—
166#117—44no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C42—22no—
179C2×C42—41no—
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42C2411yescharacteristic
192#108—44no—
193#108—44no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8C22×C414yes—
199C4⋊C8C22×C414yes—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—41no—
205C22⋊C8—41no—
206#97—22no—
207#97—22no—
208#97—22no—
209#97—21no—
210#97—21no—
211#97—21no—
212#97—21no—
213#97—41no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—21no—
218C4×C8—22no—
219C4×C8—21no—
220C4×C8—22no—
221C4×C8—41no—
222C4×C8C2×D412yes—
223C4×C8C2×D412yes—
224#95—22no—
225#95—41no—
226C24#12411yescharacteristic
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—82no—
232C2×Q8—82no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4#11711yescharacteristic
241C22×C4#11711yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12311yescharacteristic
249C22×C4#12311yescharacteristic
250C22×C4#12411yescharacteristic
251C22×C4#12411yescharacteristic
252C22×C4#12411yescharacteristic
253C22×C4#12611yescharacteristic
254C22×C4#12611yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C2×#4111yescharacteristic
257C22×C4C2×#4111yescharacteristic
258M16—24no—
259M16—22no—
260M16—22no—
261M16—22no—
262M16—44no—
263M16—44no—
264M16—42no—
265M16—42no—
266M16—42no—
267M16—82no—
268M16#13714yes—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—44no—
276C2×C8—44no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—44no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—82no—
298C42—42no—
299C42—42no—
300C42—44no—
301C42#11711yescharacteristic
302C42#11711yescharacteristic
303C42C4×D411yescharacteristic
304C42C4×D411yescharacteristic
305C42C4×D411yescharacteristic
306C42C4×D411yescharacteristic
307C42#12111yescharacteristic
308C42#12111yescharacteristic
309C42#12311yescharacteristic
310C42#12311yescharacteristic
311C42#12611yescharacteristic
312C42#12611yescharacteristic
313C42C23×C411yescharacteristic
314C42C22×D411yescharacteristic
315C42C2×#4111yescharacteristic
316C42C2×#4111yescharacteristic
317C23—21no—
318C23—21no—
319C23—41no—
320C23—41no—
321C23#38811yescharacteristic
322C23#38811yescharacteristic
323C23C2×#12411yescharacteristic
324Q8—44no—
325Q8—82no—
326Q8—82no—
327Q8—82no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4#38811yescharacteristic
354C2×C4#38811yescharacteristic
355C2×C4#38811yescharacteristic
356C2×C4#38811yescharacteristic
357C2×C4#38811yescharacteristic
358C2×C4#38811yescharacteristic
359C2×C4C2×#11711yescharacteristic
360C2×C4C2×C4×D411yescharacteristic
361C2×C4C2×C4×D411yescharacteristic
362C2×C4C2×#12111yescharacteristic
363C2×C4C2×#12311yescharacteristic
364C2×C4C2×#12611yescharacteristic
365C2×C4C2×#13712yes—
366C2×C4#57512yes—
367C8—24no—
368C8—24no—
369C8—42no—
370C8—44no—
371C8—44no—
372C8—42no—
373C8—82no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—81no—
383C22—81no—
384C22C2×#38811yescharacteristic
385C22#313911yescharacteristic
386C22#314011yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—82no—
396C4—82no—
397C4—162no—
398C4#193212yes—
399C4C4×#13712yes—
400C4#295712yes—
401C4#305612yes—
402C2—21no—
403C2—21no—
404C2—41no—
405C2—41no—
406C2#2107211yescharacteristic
407C2C2×#193211yescharacteristic
408C2#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. #1554C4semidirect
26. #1548C4semidirect
27. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#213702
#213062
#177182
#175012
M1622
#95912
#95892
#80942
#78962