Finite Group Databasebeta

← #242716

Subgroups — #242716

Order 512.

671 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21993C211yesmaximal, characteristic
2#21829C211yesmaximal, characteristic
3#17809C211yesmaximal, characteristic
4#17803C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic
6C2×#1546C211yesmaximal, characteristic
7#10610C214yesmaximal, complemented
8#10606C214yesmaximal, complemented
9#8004C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×#423C2211yescharacteristic
17#1997—21no—
18#1997—21no—
19#1997C2214yes—
20#1950C2214yes—
21#1855—22no—
22#1848—21no—
23#1837—21no—
24#1821—22no—
25#1574—21no—
26#1574—21no—
27#1572—22no—
28#1571—21no—
29#1571C2212yes—
30#1570—21no—
31#1570C2212yes—
32#1565C2212yes—
33#1565C2212yes—
34#1564—22no—
35#1549—21no—
36#1549—21no—
37#1549C2212yes—
38#1549C2212yes—
39#1546—22no—
40#1546—22no—
41C2×#332—21no—
42C2×#332—21no—
43C2×#332C2214yes—
44#1292C2214yes—
45C23×Q8—21no—
46C22×M16—21no—
47C22×M16—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8Q812yes—
59C2×C4⋊Q8C2312yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4Q812yes—
62C22×C42C2311yescharacteristic
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431D412yes—
76#431D412yes—
77#431D411yescharacteristic
78#431D411yescharacteristic
79#431C2311yescharacteristic
80#431C2311yescharacteristic
81C2×#108—24no—
82#423—21no—
83#423—21no—
84#423—21no—
85#423—21no—
86#423—21no—
87#423—21no—
88#423—21no—
89#423D411yescharacteristic
90#423D411yescharacteristic
91#423C2312yes—
92#423C2312yes—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8D411yescharacteristic
97C2×C4⋊C8D411yescharacteristic
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2312yes—
100#407—21no—
101#407D412yes—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C4×M16—21no—
105C4×M16C2312yes—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108#343—24no—
109#343—24no—
110#341—24no—
111#335—24no—
112#335—22no—
113#335—22no—
114#334—24no—
115#334—22no—
116#334—22no—
117#332—22no—
118#332—22no—
119#332—22no—
120#332—22no—
121#332—28no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—41no—
125C22×Q8—41no—
126C22×Q8—41no—
127C22×Q8—41no—
128C22×Q8—82no—
129C22×Q8—82no—
130C23×C4—22no—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×Q811yescharacteristic
133C23×C4#4111yescharacteristic
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—41no—
137C2×M16—41no—
138C2×M16—41no—
139C2×M16—82no—
140C2×M16—82no—
141C2×M16C2×D412yes—
142C2×M16C2×D412yes—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—41no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—41no—
157C4⋊Q8—41no—
158C4⋊Q8—41no—
159C4⋊Q8—41no—
160C4⋊Q8—41no—
161C4⋊Q8—41no—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4C2×Q814yes—
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×Q811yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42C2411yescharacteristic
188#108—84no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206C4⋊C8C2×D411yescharacteristic
207C4⋊C8C2×D411yescharacteristic
208C4⋊C8C2×D414yes—
209C4⋊C8C2×D411yescharacteristic
210C4⋊C8C2×D411yescharacteristic
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—21no—
218C22⋊C8—21no—
219C22⋊C8—41no—
220C22⋊C8—41no—
221#97—22no—
222#97—41no—
223C4×C8—22no—
224C4×C8—22no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—22no—
229C4×C8—21no—
230C4×C8—41no—
231C24C22⋊Q811yescharacteristic
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—84no—
237C2×Q8—82no—
238C2×Q8—82no—
239C2×Q8—84no—
240C2×Q8—84no—
241C2×Q8—82no—
242C2×Q8—82no—
243C2×Q8—82no—
244C2×Q8—82no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—84no—
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4C22⋊Q811yescharacteristic
263C22×C4C22⋊Q811yescharacteristic
264C22×C4C22⋊Q811yescharacteristic
265C22×C4C22⋊Q811yescharacteristic
266C22×C4C22⋊Q811yescharacteristic
267C22×C4#12611yescharacteristic
268C22×C4#12611yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×Q811yescharacteristic
271C22×C4C2×#4111yescharacteristic
272M16—24no—
273M16—24no—
274M16—44no—
275M16—42no—
276M16—42no—
277M16—82no—
278M16—82no—
279M16—82no—
280C2×C8—24no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—82no—
295C2×C8—82no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C42#12111yescharacteristic
305C42#12111yescharacteristic
306C42#12111yescharacteristic
307C42#12111yescharacteristic
308C42#12111yescharacteristic
309C42#12111yescharacteristic
310C42C22⋊Q811yescharacteristic
311C42C22⋊Q811yescharacteristic
312C42C22⋊Q811yescharacteristic
313C42C22⋊Q811yescharacteristic
314C42#12611yescharacteristic
315C42#12611yescharacteristic
316C42C22×D411yescharacteristic
317C42C22×D411yescharacteristic
318C42C2×#4111yescharacteristic
319C42C2×#4111yescharacteristic
320C23—22no—
321C23—22no—
322C23—41no—
323C23#39611yescharacteristic
324C23#39611yescharacteristic
325C23C2×C22⋊Q811yescharacteristic
326Q8—84no—
327Q8—84no—
328Q8—84no—
329Q8—82no—
330Q8—82no—
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—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—44no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—84no—
355C2×C4#39611yescharacteristic
356C2×C4#39611yescharacteristic
357C2×C4#39611yescharacteristic
358C2×C4#39611yescharacteristic
359C2×C4#39611yescharacteristic
360C2×C4#39611yescharacteristic
361C2×C4#47112yes—
362C2×C4#47212yes—
363C2×C4C2×#12111yescharacteristic
364C2×C4C2×#12111yescharacteristic
365C2×C4C2×#12111yescharacteristic
366C2×C4C2×C22⋊Q811yescharacteristic
367C2×C4C2×C22⋊Q811yescharacteristic
368C2×C4C2×#12611yescharacteristic
369C2×C4#57712yes—
370C2×C4#57812yes—
371C8—44no—
372C8—42no—
373C8—42no—
374C8—41no—
375C8—41no—
376C8—41no—
377C8—41no—
378C8—82no—
379C8—82no—
380C8—82no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—44no—
389C22#206312yes—
390C22C2×#39611yescharacteristic
391C22#308212yes—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—84no—
400C4#206312yes—
401C4#207012yes—
402C4#304612yes—
403C4#305711yescharacteristic
404C4#305911yescharacteristic
405C4C2×#47111yescharacteristic
406C4C2×#47211yescharacteristic
407C2—22no—
408C2—22no—
409C2—41no—
410C2C2×#206311yescharacteristic
411C2#2361311yescharacteristic
412C2C2×#207011yescharacteristic
subgroupG/H
center, socleC22C2×#396
commutator, Frattini, cosocleC2×C42C24
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10610C2semidirect
8. #10606C2semidirect

Maximal subgroups

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

subgroupindex
#219932
#218292
#178092
#178032
C2×#15492
C2×#15462
#106102
#106062
#80042