Finite Group Databasebeta

← #142155

Subgroups — #142155

Order 512.

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

normal subgroups only

This group has 726 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#15135C211yesmaximal, characteristic
2C4×#342C211yesmaximal, characteristic
3#8557C211yesmaximal, characteristic
4#7250C214yesmaximal
5C2×#399—21no—
6C2×#389—21no—
7C2×#384C411yescharacteristic, complemented
8C2×#380—21no—
9C2×C4×C22⋊C4—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11#1797—21no—
12C2×#342—21no—
13C2×#342C2212yes—
14C2×#323C2212yes—
15#1445—21no—
16#1444C412yescomplemented
17#1439C2212yes—
18C4⋊#95C411yescharacteristic, complemented
19#1280—21no—
20C24×C4D411yescharacteristic
21C22×C4⋊C4—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4C2×C412yes—
26C22×C22⋊C4—22no—
27C22×C22⋊C4—22no—
28C22×C22⋊C4D411yescharacteristic
29C22×C22⋊C4C2311yescharacteristic
30C22×C42—41no—
31C22×C42C2×C411yescharacteristic
32C22×C42D411yescharacteristic
33C2×C4⋊C8—41no—
34C2×C22⋊C8—22no—
35#399—44no—
36#389—44no—
37#384—41no—
38#384—41no—
39#380—44no—
40C4×C22⋊C4—24no—
41C4×C22⋊C4—41no—
42C4×C22⋊C4—41no—
43C4×C22⋊C4—42no—
44C2×#95—21no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—41no—
48C2×#95C2×C412yes—
49C2×#95C2×C411yescharacteristic
50C2×#95D411yescharacteristic
51#342—24no—
52#342—22no—
53#342Q814yes—
54#336—41no—
55#323—24no—
56C25C2×D411yescharacteristic
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—42no—
63C23×C4—42no—
64C23×C4—42no—
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4SD1612yes—
70C23×C4Q1612yes—
71C23×C4C22×C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C22×C8—42no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—44no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—44no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—82no—
88C2×C4⋊C4—82no—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—24no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—44no—
101C2×C22⋊C4—44no—
102C2×C22⋊C4—82no—
103C2×C22⋊C4C2×Q812yes—
104C2×C22⋊C4#4112yes—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—42no—
110C2×C42—42no—
111C2×C42—44no—
112C2×C42—82no—
113C2×C42—82no—
114C4⋊C8—82no—
115C22⋊C8—42no—
116C22⋊C8—42no—
117#95—41no—
118#95—41no—
119#95—42no—
120#95—44no—
121#95—44no—
122#95—42no—
123#95—44no—
124C24—21no—
125C24—21no—
126C24—21no—
127C24—21no—
128C24—41no—
129C24—41no—
130C24—41no—
131C24—41no—
132C24C2×C22⋊C411yescharacteristic
133C24C22⋊Q812yes—
134C24C2×SD1611yescharacteristic
135C24C2×Q1611yescharacteristic
136C24AGL(1,ℤ/8)12yes—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—24no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—44no—
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—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—84no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4—82no—
191C22×C4—82no—
192C22×C4#9911yescharacteristic
193C22×C4#9911yescharacteristic
194C22×C4#10312yes—
195C22×C4#10312yes—
196C22×C4#10412yes—
197C22×C4#10412yes—
198C22×C4C4×Q812yes—
199C22×C4#12411yescharacteristic
200C22×C4C4⋊Q811yescharacteristic
201C2×C8—82no—
202C2×C8—82no—
203C2×C8—82no—
204C4⋊C4—48no—
205C4⋊C4—82no—
206C4⋊C4—84no—
207C4⋊C4—88no—
208C4⋊C4—84no—
209C4⋊C4—82no—
210C22⋊C4—28no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—44no—
214C22⋊C4—44no—
215C22⋊C4—44no—
216C22⋊C4—44no—
217C22⋊C4—88no—
218C42—42no—
219C42—42no—
220C42—42no—
221C42—81no—
222C42—88no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—21no—
229C23—21no—
230C23—22no—
231C23—22no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—42no—
236C23—41no—
237C23—41no—
238C23—42no—
239C23—42no—
240C23—41no—
241C23—41no—
242C23—42no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—81no—
248C23—81no—
249C23—81no—
250C23—81no—
251C23—81no—
252C23—81no—
253C23#39111yescharacteristic
254C23C2×#9911yescharacteristic
255C23#41011yescharacteristic
256C23C2×#10311yescharacteristic
257C23#41811yescharacteristic
258C23C2×#10411yescharacteristic
259C23C2×#10411yescharacteristic
260C23#47612yes—
261C23#47712yes—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—48no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—84no—
304C2×C4—84no—
305C2×C4—84no—
306C2×C4—84no—
307C2×C4—84no—
308C2×C4—84no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4#32812yes—
319C2×C4#32812yes—
320C2×C4#35212yes—
321C2×C4#35312yes—
322C8—162no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—22no—
336C22—22no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—82no—
348C22—82no—
349C22—81no—
350C22—81no—
351C22C2×#32811yescharacteristic
352C22#150311yescharacteristic
353C22#150511yescharacteristic
354C22#150711yescharacteristic
355C22#198711yescharacteristic
356C22#199411yescharacteristic
357C22#199611yescharacteristic
358C22C2×#35211yescharacteristic
359C22#212211yescharacteristic
360C22C2×#35311yescharacteristic
361C22#212911yescharacteristic
362C4—22no—
363C4—42no—
364C4—42no—
365C4—41no—
366C4—41no—
367C4—42no—
368C4—84no—
369C4—88no—
370C4—81no—
371C4—162no—
372C4#135214yes—
373C2—21no—
374C2—21no—
375C2—21no—
376C2—21no—
377C2—42no—
378C2—42no—
379C2#990811yescharacteristic
380C2C2×#135211yescharacteristic
381C2#1171711yescharacteristic
382C2#1177311yescharacteristic
383C2#1177611yescharacteristic
384C2#1329511yescharacteristic
385C2#1329811yescharacteristic
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
7. C2×#384C4semidirect
16. #1444C4semidirect
18. C4⋊#95C4semidirect

Maximal subgroups

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

subgroupindex
#151352
C4×#3422
#85572
#72502