Finite Group Databasebeta

← #121868

Subgroups — #121868

Order 512.

537 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 116 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#21949C211yesmaximal, characteristic
2#8055C211yesmaximal, characteristic
3#8009C211yesmaximal, characteristic
4#8009C211yesmaximal, characteristic
5#7929C211yesmaximal, characteristic
6#7929C211yesmaximal, characteristic
7#7911C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C2×#407C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1855—22no—
16#1855—22no—
17#1811C414yescomplemented
18#1298—22no—
19#1293—22no—
20#1293—22no—
21#1282—22no—
22#1282—22no—
23#1280—22no—
24C24×C4C2311yescharacteristic
25C22×M16—21no—
26C22×M16—21no—
27C22×M16—21no—
28C22×M16—21no—
29C22×M16—21no—
30C22×M16C2×C412yes—
31C22×C22⋊C4Q812yes—
32C22×C42—21no—
33C2×#108—24no—
34#407—21no—
35#407—21no—
36#407—21no—
37#407—21no—
38#407—21no—
39#407—21no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407—21no—
45#407—21no—
46#407—21no—
47#407—21no—
48#407—21no—
49#407—21no—
50#407—21no—
51#407—21no—
52#407—21no—
53#407—21no—
54#407—21no—
55#407C2×C412yes—
56#407C2×C412yes—
57#407Q812yes—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8D412yes—
67C2×C22⋊C8D412yes—
68C2×C22⋊C8D412yes—
69C2×C22⋊C8D412yes—
70C4×C22⋊C4—21no—
71C4×C22⋊C4—21no—
72C4×C22⋊C4—21no—
73C4×C22⋊C4—21no—
74C2×#95—21no—
75#341—24no—
76C25C2×Q811yescharacteristic
77C23×C4—22no—
78C23×C4—82no—
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×Q811yescharacteristic
85C23×C4#4111yescharacteristic
86C23×C4#4111yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4#4111yescharacteristic
89C23×C4#4111yescharacteristic
90C23×C4#4111yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4#4111yescharacteristic
93C2×M16—22no—
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—22no—
97C2×M16—22no—
98C2×M16—41no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—41no—
103C2×M16—41no—
104C2×M16—41no—
105C2×M16—82no—
106C2×M16—82no—
107C2×M16—82no—
108C2×M16—82no—
109C2×M16—82no—
110C2×M16—82no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—41no—
114C22×C8—41no—
115C22×C8—41no—
116C22×C8C22⋊C414yes—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—44no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—82no—
123#108—84no—
124C22⋊C8—22no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148#95—41no—
149#95—41no—
150C24—42no—
151C24—81no—
152C24—81no—
153C24C4×Q811yescharacteristic
154C24C22⋊Q811yescharacteristic
155C24C22⋊Q811yescharacteristic
156C24C22⋊Q811yescharacteristic
157C24#12511yescharacteristic
158C24#12511yescharacteristic
159C24C4⋊Q811yescharacteristic
160C22×C4—24no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—82no—
178C22×C4—82no—
179C22×C4C2×C22⋊C411yescharacteristic
180C22×C4#11711yescharacteristic
181C22×C4#11711yescharacteristic
182C22×C4C4×D411yescharacteristic
183C22×C4C4×D411yescharacteristic
184C22×C4C4×Q811yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4C22⋊Q811yescharacteristic
188C22×C4C22⋊Q811yescharacteristic
189C22×C4C22⋊Q811yescharacteristic
190C22×C4#12311yescharacteristic
191C22×C4#12311yescharacteristic
192C22×C4#12311yescharacteristic
193C22×C4#12311yescharacteristic
194C22×C4#12311yescharacteristic
195C22×C4#12311yescharacteristic
196C22×C4#12311yescharacteristic
197C22×C4#12311yescharacteristic
198C22×C4#12411yescharacteristic
199C22×C4#12411yescharacteristic
200C22×C4#12511yescharacteristic
201C22×C4#12511yescharacteristic
202C22×C4#12611yescharacteristic
203C22×C4#12611yescharacteristic
204C22×C4#12611yescharacteristic
205C22×C4#12611yescharacteristic
206C22×C4#12711yescharacteristic
207M16—44no—
208M16—44no—
209M16—82no—
210M16—82no—
211M16—82no—
212M16—82no—
213M16—82no—
214M16—82no—
215M16—82no—
216M16—82no—
217M16—82no—
218M16—82no—
219C2×C8—24no—
220C2×C8—44no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—82no—
227C2×C8—82no—
228C2×C8—82no—
229C2×C8—82no—
230C2×C8—82no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C42—82no—
236C42—82no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—84no—
259C23—84no—
260C23#38711yescharacteristic
261C23#38711yescharacteristic
262C23#39111yescharacteristic
263C23#39811yescharacteristic
264C23#39811yescharacteristic
265C23#39911yescharacteristic
266C23#40011yescharacteristic
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—44no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—84no—
299C2×C4—84no—
300C2×C4#38111yescharacteristic
301C2×C4#38711yescharacteristic
302C2×C4#38711yescharacteristic
303C2×C4#39011yescharacteristic
304C2×C4#39711yescharacteristic
305C2×C4#39711yescharacteristic
306C2×C4#39911yescharacteristic
307C2×C4#40011yescharacteristic
308C8—44no—
309C8—82no—
310C8—82no—
311C8—82no—
312C8—82no—
313C8—82no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—42no—
335C22—42no—
336C22—82no—
337C22—82no—
338C22—82no—
339C22—82no—
340C22—82no—
341C22—82no—
342C22#144511yescharacteristic
343C22#209112yes—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—84no—
352C4#182812yes—
353C2—22no—
354C2—41no—
355C2—41no—
356C2—41no—
357C2—41no—
358C2—41no—
359C2—41no—
360C2#1135111yescharacteristic
361C2#1136311yescharacteristic
362C2#1137511yescharacteristic
subgroupG/H
center, socleC22#1445
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×Q8
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1811C4semidirect

Maximal subgroups

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

subgroupindex
#219492
#80552
#80092
#79292
#79112