Finite Group Databasebeta

← #102792

Subgroups — #102792

Order 512.

695 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 695 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30291C211yesmaximal, characteristic
2#25813C211yesmaximal, characteristic
3#25722C211yesmaximal, characteristic
4#11384C211yesmaximal, characteristic
5#9884C211yesmaximal, characteristic
6#9679C211yesmaximal, characteristic
7#7016C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#500—21no—
10C2×#494—21no—
11C2×#451—22no—
12#2857C2211yescharacteristic
13#2857C2211yescharacteristic
14#2083—21no—
15#2083—21no—
16#2057—21no—
17#2057—21no—
18#2057—22no—
19#2016—22no—
20#1990—21no—
21#1956—21no—
22#1852—22no—
23#1808—22no—
24C4×C22⋊C8C2211yescharacteristic
25C4×C22⋊C8C2211yescharacteristic
26C2×#340—22no—
27C2×#336C2211yescharacteristic
28C2×#328—22no—
29C4⋊#95C2211yescharacteristic
30#1294—22no—
31C23×Q8—21no—
32C22×Q16—24no—
33C23×C8D412yes—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—44no—
38C2×C4⋊Q8D411yescharacteristic
39C2×C4⋊Q8D411yescharacteristic
40C2×C4⋊Q8D412yes—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C42C2311yescharacteristic
47#500—44no—
48#494—44no—
49#451—44no—
50#451—44no—
51C2×#107—24no—
52C2×#107—22no—
53C2×#107—22no—
54C2×#103—24no—
55C2×C22⋊C8D412yes—
56C2×C22⋊C8D412yes—
57C2×C4×C8—21no—
58C2×C4×C8—21no—
59#391—21no—
60#391—21no—
61#389—21no—
62#389—21no—
63#389—21no—
64#389—21no—
65#384D411yescharacteristic
66#384D411yescharacteristic
67#380—21no—
68#380—21no—
69#380—21no—
70#380—21no—
71C2×#95—21no—
72C2×#95—21no—
73#340—22no—
74#340—22no—
75#340—24no—
76#340—44no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#328—44no—
82C22×Q8—22no—
83C22×Q8—21no—
84C22×Q8—21no—
85C22×Q8—44no—
86C22×Q8—41no—
87C22×Q8—42no—
88C22×Q8—42no—
89C22×Q8—82no—
90C23×C4—41no—
91C23×C4—41no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C2×Q16—48no—
96C2×Q16—48no—
97C2×Q16—84no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—42no—
103C22×C8—42no—
104C22×C8Q1614yescomplemented
105C4⋊Q8—24no—
106C4⋊Q8—24no—
107C4⋊Q8—42no—
108C4⋊Q8—42no—
109C4⋊Q8—42no—
110C4⋊Q8—42no—
111C4⋊Q8—44no—
112C4⋊Q8—44no—
113C22⋊Q8—42no—
114C22⋊Q8—42no—
115C22⋊Q8—42no—
116C22⋊Q8—42no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C42—42no—
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42#4111yescharacteristic
147#107—48no—
148#107—82no—
149#107—82no—
150#103—48no—
151C22⋊C8—22no—
152C22⋊C8—22no—
153C22⋊C8D814yes—
154C22⋊C8Q1614yescomplemented
155C4×C8—42no—
156C4×C8—42no—
157#95—41no—
158#95—41no—
159#95—41no—
160#95—41no—
161#95—42no—
162#95—42no—
163C24#12711yescharacteristic
164C2×Q8—24no—
165C2×Q8—44no—
166C2×Q8—42no—
167C2×Q8—44no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—44no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—82no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×Q8—88no—
177C2×Q8—88no—
178C2×Q8—82no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—44no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—82no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4—81no—
201C22×C4—81no—
202C22×C4#12011yescharacteristic
203C22×C4#12011yescharacteristic
204C22×C4#12011yescharacteristic
205C22×C4#12111yescharacteristic
206C22×C4#12111yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4C2×D812yes—
209C22×C4C2×Q1612yes—
210C22×C4C2×Q1612yes—
211C22×C4AGL(1,ℤ/8)12yes—
212C22×C4#13712yes—
213C22×C4#13712yes—
214Q16—816no—
215C2×C8—24no—
216C2×C8—24no—
217C2×C8—24no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—44no—
225C2×C8—82no—
226C4⋊C4—24no—
227C4⋊C4—24no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—44no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—42no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—81no—
244C4⋊C4—81no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C22⋊C4—81no—
249C22⋊C4—81no—
250C22⋊C4—81no—
251C22⋊C4—81no—
252C42—22no—
253C42—22no—
254C42—41no—
255C42—41no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23#39211yescharacteristic
261C23#49311yescharacteristic
262C23#49411yescharacteristic
263C23#49411yescharacteristic
264C23#49611yescharacteristic
265C23#49711yescharacteristic
266C23#49711yescharacteristic
267Q8—48no—
268Q8—84no—
269Q8—88no—
270Q8—84no—
271Q8—162no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—84no—
292C2×C4—84no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—161no—
311C2×C4—161no—
312C2×C4—161no—
313C2×C4—161no—
314C2×C4#44712yes—
315C2×C4#45112yes—
316C2×C4#45112yes—
317C2×C4#45511yescharacteristic
318C2×C4#45611yescharacteristic
319C2×C4B(4,𝔽2)12yes—
320C2×C4#45912yes—
321C2×C4#46212yes—
322C2×C4#46212yes—
323C8—44no—
324C8—44no—
325C8—44no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22#171311yescharacteristic
339C22#172211yescharacteristic
340C22#172311yescharacteristic
341C22#172311yescharacteristic
342C22#200711yescharacteristic
343C22#200811yescharacteristic
344C22#200911yescharacteristic
345C22#201111yescharacteristic
346C22#201611yescharacteristic
347C22#201611yescharacteristic
348C22#202111yescharacteristic
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—84no—
355C4—162no—
356C4—161no—
357C4—161no—
358C4—161no—
359C4—161no—
360C4#160212yes—
361C4#160512yes—
362C2—21no—
363C2—21no—
364C2—21no—
365C2—21no—
366C2#994611yescharacteristic
367C2#995711yescharacteristic
368C2#996711yescharacteristic
369C2#1019911yescharacteristic
370C2#1020411yescharacteristic
371C2#1022411yescharacteristic
372C2#1030911yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#127
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
104. C22×C8Q16semidirect
154. C22⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#302912
#258132
#257222
#113842
#98842
#96792
#70162