Finite Group Databasebeta

← #124309

Subgroups — #124309

Order 512.

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

normal subgroups only

This group has 680 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#13924C211yesmaximal, characteristic
2#8557C211yesmaximal, characteristic
3C4×#323C211yesmaximal, characteristic
4#7132C214yesmaximal, complemented
5C23×C4⋊C4C411yescharacteristic
6C2×#380C411yescharacteristic
7C2×C4×C22⋊C4C2211yescharacteristic
8#1797—21no—
9C4×C22⋊C8—21no—
10C2×#323—21no—
11C2×#323—21no—
12#1445C412yes—
13#1445C2212yes—
14C4⋊#95—21no—
15C4⋊#95—21no—
16C22⋊#95—21no—
17C22⋊#95—21no—
18#1280C2212yes—
19#1280C2212yes—
20C24×C4—21no—
21C24×C4C2×C411yescharacteristic
22C22×C4⋊C4—21no—
23C22×C4⋊C4—22no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—41no—
26C22×C4⋊C4—41no—
27C22×C4⋊C4—42no—
28C22×C4⋊C4C814yescomplemented
29C22×C4⋊C4C2×C412yes—
30C22×C22⋊C4—21no—
31C22×C22⋊C4D411yescharacteristic
32C22×C22⋊C4D411yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42D411yescharacteristic
35C2×C4⋊C8—41no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C4×C8—41no—
39#380—42no—
40C4×C22⋊C4—41no—
41C4×C22⋊C4—41no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95C2×C412yes—
47C2×#95C2×C411yescharacteristic
48C2×#95C2311yescharacteristic
49#336—41no—
50#336—41no—
51#323—24no—
52#323—42no—
53C25C22⋊C411yescharacteristic
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—24no—
59C23×C4—22no—
60C23×C4—42no—
61C23×C4—42no—
62C23×C4—42no—
63C23×C4—42no—
64C23×C4—82no—
65C23×C4—82no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C2×C812yes—
70C23×C4M1612yes—
71C23×C4C22×C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C22×C8—42no—
75C22×C8—42no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—44no—
81C2×C4⋊C4—44no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—44no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4C2×C814yes—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4SD1612yes—
100C2×C22⋊C4Q1612yes—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—42no—
106C2×C42—42no—
107C4⋊C8—82no—
108C22⋊C8—24no—
109C22⋊C8—42no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112C4×C8—82no—
113#95—41no—
114#95—41no—
115#95—42no—
116#95—82no—
117#95—82no—
118#95—82no—
119#95—82no—
120C24—21no—
121C24—21no—
122C24—21no—
123C24—21no—
124C24—41no—
125C24—41no—
126C24—41no—
127C24—41no—
128C24C22⋊C812yes—
129C24#10312yes—
130C24#10412yes—
131C24C2×C22⋊C411yescharacteristic
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—24no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—48no—
153C22×C4—44no—
154C22×C4—44no—
155C22×C4—42no—
156C22×C4—42no—
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—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—48no—
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—88no—
182C22×C4—88no—
183C22×C4—84no—
184C22×C4—84no—
185C22×C4C22⋊C812yes—
186C22×C4#9911yescharacteristic
187C22×C4#9911yescharacteristic
188C22×C4#10012yes—
189C22×C4#10312yes—
190C22×C4C22×C811yescharacteristic
191C22×C4C2×M1611yescharacteristic
192C22×C4C2×SD1611yescharacteristic
193C22×C4C2×Q1611yescharacteristic
194C22×C4#13712yes—
195C2×C8—44no—
196C2×C8—82no—
197C2×C8—82no—
198C2×C8—82no—
199C2×C8—82no—
200C2×C8—82no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—84no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—44no—
209C22⋊C4—84no—
210C42—42no—
211C42—42no—
212C42—81no—
213C42—81no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23Q8⋊C814yes—
241C23C2×C22⋊C811yescharacteristic
242C23C2×#9911yescharacteristic
243C23C2×#10011yescharacteristic
244C23C2×#10311yescharacteristic
245C23#41911yescharacteristic
246C23C2×#10411yescharacteristic
247C23#42111yescharacteristic
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—44no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—816no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—81no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4#32312yes—
301C2×C4#32312yes—
302C2×C4#35212yes—
303C2×C4#35212yes—
304C8—84no—
305C8—162no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—24no—
319C22—44no—
320C22—44no—
321C22—44no—
322C22—44no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22C2×#32311yescharacteristic
330C22C2×Q8⋊C811yescharacteristic
331C22#147911yescharacteristic
332C22#150811yescharacteristic
333C22#152011yescharacteristic
334C22#152711yescharacteristic
335C22#154811yescharacteristic
336C22C2×#35211yescharacteristic
337C22C2×#35211yescharacteristic
338C22#212811yescharacteristic
339C22#213411yescharacteristic
340C4—22no—
341C4—42no—
342C4—41no—
343C4—41no—
344C4—42no—
345C4—41no—
346C4—41no—
347C4—81no—
348C4—84no—
349C4—162no—
350C4#131714yes—
351C2—21no—
352C2—21no—
353C2—21no—
354C2—21no—
355C2—44no—
356C2#757511yescharacteristic
357C2C2×#131711yescharacteristic
358C2#1143311yescharacteristic
359C2#1178611yescharacteristic
360C2#1179011yescharacteristic
361C2#1189911yescharacteristic
362C2#1190311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7132C2semidirect
28. C22×C4⋊C4C8semidirect

Maximal subgroups

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

subgroupindex
#139242
#85572
C4×#3232
#71322