Finite Group Databasebeta

← #142169

Subgroups — #142169

Order 512.

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

normal subgroups only

This group has 680 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#15195C211yesmaximal, characteristic
2C4×#342C211yesmaximal, characteristic
3#8557C211yesmaximal, characteristic
4#7250C214yesmaximal
5C2×#384C411yescharacteristic, complemented
6C2×#380—21no—
7C2×C4×C22⋊C4—21no—
8C2×C4×C22⋊C4C2211yescharacteristic
9#1797—21no—
10C2×#342—21no—
11C2×#342C2212yes—
12C2×#323C2212yes—
13#1442—21no—
14#1442C412yescomplemented
15#1439—21no—
16#1439C2212yes—
17C4⋊#95—21no—
18C4⋊#95C411yescharacteristic, complemented
19#1280—21no—
20C24×C4D411yescharacteristic
21C22×C4⋊C4—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4C2×C412yes—
25C22×C22⋊C4—22no—
26C22×C22⋊C4—22no—
27C22×C22⋊C4D411yescharacteristic
28C22×C22⋊C4C2311yescharacteristic
29C22×C42—41no—
30C22×C42C2×C411yescharacteristic
31C22×C42D411yescharacteristic
32C2×C4⋊C8—41no—
33C2×C22⋊C8—22no—
34#384—41no—
35#384—41no—
36#380—42no—
37#380—42no—
38C4×C22⋊C4—24no—
39C4×C22⋊C4—41no—
40C4×C22⋊C4—41no—
41C4×C22⋊C4—42no—
42C2×#95—21no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—41no—
47C2×#95C2×C412yes—
48C2×#95C2×C411yescharacteristic
49C2×#95D411yescharacteristic
50#342—24no—
51#342—22no—
52#342Q814yes—
53#336—41no—
54#323—24no—
55C25C2×D411yescharacteristic
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—42no—
62C23×C4—42no—
63C23×C4—42no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4D812yes—
69C23×C4SD1612yes—
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C22×C8—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—82no—
83C2×C4⋊C4—82no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—82no—
98C2×C22⋊C4—82no—
99C2×C22⋊C4C2×Q812yes—
100C2×C22⋊C4#4112yes—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—42no—
106C2×C42—42no—
107C2×C42—44no—
108C2×C42—82no—
109C2×C42—82no—
110C4⋊C8—82no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113#95—41no—
114#95—41no—
115#95—42no—
116#95—44no—
117#95—44no—
118#95—42no—
119#95—82no—
120#95—82no—
121C24—21no—
122C24—21no—
123C24—21no—
124C24—21no—
125C24—41no—
126C24—41no—
127C24—41no—
128C24—41no—
129C24C2×C22⋊C411yescharacteristic
130C24C22⋊Q812yes—
131C24C2×D811yescharacteristic
132C24C2×SD1611yescharacteristic
133C24#13712yes—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—84no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4—82no—
186C22×C4—82no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4#9911yescharacteristic
190C22×C4#9911yescharacteristic
191C22×C4#10212yes—
192C22×C4#10212yes—
193C22×C4#10412yes—
194C22×C4#10412yes—
195C22×C4C4×Q812yes—
196C22×C4#12411yescharacteristic
197C22×C4C4⋊Q811yescharacteristic
198C2×C8—82no—
199C2×C8—82no—
200C2×C8—82no—
201C4⋊C4—84no—
202C4⋊C4—84no—
203C4⋊C4—84no—
204C4⋊C4—82no—
205C4⋊C4—82no—
206C22⋊C4—28no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—44no—
210C22⋊C4—44no—
211C22⋊C4—44no—
212C22⋊C4—44no—
213C22⋊C4—84no—
214C22⋊C4—84no—
215C42—42no—
216C42—42no—
217C42—42no—
218C42—81no—
219C42—88no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—21no—
230C23—21no—
231C23—22no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—81no—
245C23—81no—
246C23—81no—
247C23—81no—
248C23—81no—
249C23—81no—
250C23#39111yescharacteristic
251C23C2×#9911yescharacteristic
252C23#41011yescharacteristic
253C23C2×#10211yescharacteristic
254C23#41911yescharacteristic
255C23C2×#10411yescharacteristic
256C23C2×#10411yescharacteristic
257C23#47412yes—
258C23#47512yes—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—48no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—84no—
295C2×C4—84no—
296C2×C4—84no—
297C2×C4—84no—
298C2×C4—84no—
299C2×C4—84no—
300C2×C4—81no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4#32712yes—
316C2×C4#32712yes—
317C2×C4#35212yes—
318C2×C4#35412yes—
319C8—162no—
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—22no—
333C22—22no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—42no—
343C22—42no—
344C22—81no—
345C22—81no—
346C22—82no—
347C22—82no—
348C22C2×#32711yescharacteristic
349C22#150211yescharacteristic
350C22#150611yescharacteristic
351C22#150811yescharacteristic
352C22#198811yescharacteristic
353C22#199411yescharacteristic
354C22#199611yescharacteristic
355C22C2×#35211yescharacteristic
356C22#212211yescharacteristic
357C22C2×#35411yescharacteristic
358C22#212811yescharacteristic
359C4—22no—
360C4—42no—
361C4—42no—
362C4—42no—
363C4—41no—
364C4—41no—
365C4—88no—
366C4—81no—
367C4—162no—
368C4—162no—
369C4#134414yes—
370C2—21no—
371C2—21no—
372C2—21no—
373C2—21no—
374C2—42no—
375C2—42no—
376C2#990911yescharacteristic
377C2C2×#134411yescharacteristic
378C2#1169311yescharacteristic
379C2#1178611yescharacteristic
380C2#1178711yescharacteristic
381C2#1329511yescharacteristic
382C2#1329911yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#384C4semidirect
14. #1442C4semidirect
18. C4⋊#95C4semidirect

Maximal subgroups

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

subgroupindex
#151952
C4×#3422
#85572
#72502