Finite Group Databasebeta

← #242706

Subgroups — #242706

Order 512.

615 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21970C211yesmaximal, characteristic
2#21879C211yesmaximal, characteristic
3#17814C211yesmaximal, characteristic
4#17810C211yesmaximal, characteristic
5C2×#1559C211yesmaximal, characteristic
6#17718C211yesmaximal, characteristic
7#10605C214yesmaximal, complemented
8#10604C214yesmaximal, complemented
9#8003C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×#423C2211yescharacteristic
17#1997C2212yes—
18#1966C2212yes—
19#1950—22no—
20#1950C2214yes—
21#1901—22no—
22#1811—22no—
23#1801—22no—
24#1573—22no—
25#1572—22no—
26#1571—22no—
27#1571C2214yes—
28#1567—22no—
29#1565C2214yes—
30#1559—22no—
31#1559—22no—
32#1554—22no—
33#1548C2212yes—
34#1546C2212yes—
35#1542—22no—
36C2×#333C2212yes—
37C2×#332C2212yes—
38#1292C2214yes—
39C22×M16—21no—
40C22×M16—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49C2×C4⋊Q8C2311yescharacteristic
50C2×C4⋊Q8C2311yescharacteristic
51C2×#125—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8Q812yes—
54C2×C4×Q8—21no—
55C2×C4×Q8Q812yes—
56C2×#117—21no—
57C22×C42C2311yescharacteristic
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431D412yes—
73#431C2312yes—
74C2×#108—24no—
75#423—21no—
76#423—21no—
77#423—21no—
78#423—21no—
79#423—21no—
80#423D412yes—
81#423D412yes—
82#423D412yes—
83#423C2312yes—
84#423C2312yes—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8D412yes—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×#101—24no—
92#407—21no—
93#407D412yes—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C4×M16—21no—
97C4×M16C2312yes—
98C2×#97—21no—
99C2×C4×C8—21no—
100#343—24no—
101#341—24no—
102#335—24no—
103#335—28no—
104#334—24no—
105#333—24no—
106#333—24no—
107#332—24no—
108#332—24no—
109C22×Q8—21no—
110C22×Q8C2×Q812yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×Q811yescharacteristic
113C23×C4#4111yescharacteristic
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—41no—
119C2×M16—82no—
120C2×M16—82no—
121C2×M16C2×D414yes—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—41no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134#125—41no—
135#125—41no—
136#125—41no—
137#125—41no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C22⋊Q8—41no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—41no—
151C4×Q8—41no—
152C4×Q8—41no—
153C4×Q8—41no—
154#117—41no—
155#117—41no—
156#117—41no—
157#117—41no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4C2×Q812yes—
164C2×C22⋊C4—22no—
165C2×C42—22no—
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×Q811yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42C2411yescharacteristic
178#108—84no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—41no—
192C4⋊C8—41no—
193C4⋊C8C2×D414yes—
194C4⋊C8C2×D414yes—
195#101—84no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—41no—
201C22⋊C8—41no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204#97—22no—
205#97—41no—
206#97—41no—
207#97—41no—
208C4×C8—22no—
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—41no—
212C24C22⋊Q811yescharacteristic
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—82no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—81no—
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4C22⋊Q811yescharacteristic
233C22×C4C22⋊Q811yescharacteristic
234C22×C4C22⋊Q811yescharacteristic
235C22×C4C22⋊Q811yescharacteristic
236C22×C4C22⋊Q811yescharacteristic
237C22×C4C22⋊Q811yescharacteristic
238C22×C4C22⋊Q811yescharacteristic
239C22×C4#12611yescharacteristic
240C22×C4#12611yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C22×Q811yescharacteristic
243C22×C4C2×#4111yescharacteristic
244M16—24no—
245M16—24no—
246M16—44no—
247M16—42no—
248M16—42no—
249M16—82no—
250M16—82no—
251M16—82no—
252C2×C8—24no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—44no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—82no—
260C2×C8—82no—
261C2×C8—82no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C42—42no—
277C42—42no—
278C42#12111yescharacteristic
279C42#12111yescharacteristic
280C42#12111yescharacteristic
281C42#12111yescharacteristic
282C42#12111yescharacteristic
283C42#12111yescharacteristic
284C42C22⋊Q811yescharacteristic
285C42C22⋊Q811yescharacteristic
286C42C22⋊Q811yescharacteristic
287C42C22⋊Q811yescharacteristic
288C42#12611yescharacteristic
289C42#12611yescharacteristic
290C42C22×D411yescharacteristic
291C42C22×D411yescharacteristic
292C42C2×#4111yescharacteristic
293C42C2×#4111yescharacteristic
294C23—22no—
295C23—41no—
296C23—41no—
297C23#39611yescharacteristic
298C23#39611yescharacteristic
299C23C2×C22⋊Q811yescharacteristic
300Q8—82no—
301Q8—82no—
302Q8—81no—
303Q8—81no—
304Q8—82no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—82no—
328C2×C4#39611yescharacteristic
329C2×C4#39611yescharacteristic
330C2×C4#39611yescharacteristic
331C2×C4#39611yescharacteristic
332C2×C4#39611yescharacteristic
333C2×C4#39611yescharacteristic
334C2×C4C2×#12111yescharacteristic
335C2×C4C2×#12111yescharacteristic
336C2×C4C2×#12111yescharacteristic
337C2×C4C2×C22⋊Q811yescharacteristic
338C2×C4C2×C22⋊Q811yescharacteristic
339C2×C4C2×#12611yescharacteristic
340C2×C4#57612yes—
341C2×C4#57812yes—
342C8—44no—
343C8—44no—
344C8—44no—
345C8—82no—
346C8—82no—
347C8—82no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—82no—
357C22C2×#39611yescharacteristic
358C22#308312yes—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—82no—
369C4—82no—
370C4#304111yescharacteristic
371C4#304411yescharacteristic
372C4#306112yes—
373C4#306912yes—
374C2—22no—
375C2—41no—
376C2—41no—
377C2#2360311yescharacteristic
378C2#2361111yescharacteristic
379C2#2362211yescharacteristic
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. #10605C2semidirect
8. #10604C2semidirect

Maximal subgroups

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

subgroupindex
#219702
#218792
#178142
#178102
C2×#15592
#177182
#106052
#106042
#80032