Finite Group Databasebeta

← #99314

Subgroups — #99314

Order 512.

588 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 132 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#21930C211yesmaximal, characteristic
2#11042C211yesmaximal, characteristic
3#8830C212yesmaximal, complemented
4C4×#323C211yesmaximal, characteristic
5#7029C212yesmaximal, complemented
6C22×C4⋊C8C2211yescharacteristic
7C22×C22⋊C8—21no—
8C22×C22⋊C8—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1850—22no—
11#1850—22no—
12#1797—21no—
13#1797C2212yes—
14C4×C22⋊C8C2212yes—
15C2×#336—21no—
16C2×#336C2211yescharacteristic
17C2×#334—22no—
18C2×#323C411yescharacteristic
19C2×#323C411yescharacteristic
20#1280C412yes—
21#1277—22no—
22C24×C4D411yescharacteristic
23C23×C8—21no—
24C23×C8—21no—
25C23×C8—21no—
26C23×C8—21no—
27C22×C22⋊C4C2×C411yescharacteristic
28C22×C22⋊C4Q811yescharacteristic
29C22×C42D411yescharacteristic
30C22×C42C2311yescharacteristic
31C2×C4⋊C8—22no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8—42no—
35C2×C4⋊C8D412yes—
36C2×C4⋊C8D412yes—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—42no—
40C2×C22⋊C8—42no—
41C2×C22⋊C8—42no—
42C2×C22⋊C8—42no—
43C2×C22⋊C8—42no—
44C2×C22⋊C8C2×C412yes—
45C2×C22⋊C8C2×C412yes—
46C2×C4×C8—22no—
47C4×C22⋊C4—22no—
48C4×C22⋊C4—22no—
49C2×#95C2×C411yescharacteristic
50C2×#95Q811yescharacteristic
51#336—22no—
52#336—22no—
53#336—22no—
54#336—22no—
55#336—42no—
56#334—44no—
57#323—22no—
58#323C814yescomplemented
59C25C4⋊C411yescharacteristic
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—41no—
65C23×C4—41no—
66C23×C4—41no—
67C23×C4—41no—
68C23×C4C4⋊C411yescharacteristic
69C23×C4C4⋊C411yescharacteristic
70C23×C4C4⋊C411yescharacteristic
71C23×C4SD1612yes—
72C23×C4SD1612yes—
73C23×C4C22×C411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C23×C4C2×Q811yescharacteristic
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8—41no—
87C22×C8—42no—
88C22×C8—42no—
89C22×C8—41no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8—41no—
93C22×C8—41no—
94C22×C8—41no—
95C22×C8—41no—
96C22×C8—81no—
97C22×C8—81no—
98C22×C8—81no—
99C22×C8—81no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4C2×C812yes—
106C2×C22⋊C4M1612yes—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—41no—
110C2×C42—41no—
111C2×C42—41no—
112C2×C42—41no—
113C2×C42C2×D412yes—
114C2×C42#4112yes—
115C4⋊C8—24no—
116C4⋊C8—42no—
117C4⋊C8—42no—
118C4⋊C8—42no—
119C4⋊C8—81no—
120C4⋊C8—81no—
121C22⋊C8—22no—
122C22⋊C8—22no—
123C22⋊C8—22no—
124C22⋊C8—44no—
125C22⋊C8—48no—
126C22⋊C8—42no—
127C22⋊C8—42no—
128C22⋊C8C2×C814yescomplemented
129C4×C8—42no—
130C4×C8—42no—
131#95—22no—
132#95—22no—
133C24—21no—
134C24—21no—
135C24—21no—
136C24—21no—
137C24—41no—
138C24—41no—
139C24—41no—
140C24—41no—
141C24C4⋊C812yes—
142C24#10612yes—
143C24#10812yes—
144C24C2×C4⋊C411yescharacteristic
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4—81no—
186C22×C4C4⋊C812yes—
187C22×C4#10612yes—
188C22×C4C4×D412yes—
189C22×C4#12011yescharacteristic
190C22×C4C22⋊Q812yes—
191C22×C4#12311yescharacteristic
192C22×C4C22×C811yescharacteristic
193C22×C4C2×M1611yescharacteristic
194C22×C4#13112yes—
195C22×C4C2×SD1611yescharacteristic
196C22×C4C2×SD1611yescharacteristic
197C22×C4AGL(1,ℤ/8)11yescharacteristic
198C22×C4#13711yescharacteristic
199C2×C8—24no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—81no—
218C2×C8—81no—
219C2×C8—81no—
220C2×C8—81no—
221C2×C8—81no—
222C2×C8—84no—
223C2×C8—84no—
224C2×C8—81no—
225C2×C8—81no—
226C2×C8—81no—
227C22⋊C4—24no—
228C22⋊C4—24no—
229C22⋊C4—44no—
230C42—22no—
231C42—22no—
232C42—42no—
233C42—41no—
234C42—41no—
235C42—81no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—81no—
257C23—81no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23—81no—
262C23#33414yescomplemented
263C23#38511yescharacteristic
264C23C2×C4⋊C811yescharacteristic
265C23#42411yescharacteristic
266C23C2×#10611yescharacteristic
267C23#42811yescharacteristic
268C23C2×#10811yescharacteristic
269C23#43011yescharacteristic
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—22no—
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—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4C8×D412yes—
312C2×C4#43512yes—
313C2×C4#44811yescharacteristic
314C2×C4#45011yescharacteristic
315C2×C4#45311yescharacteristic
316C2×C4#45511yescharacteristic
317C2×C4B(4,𝔽2)11yescharacteristic
318C2×C4#45811yescharacteristic
319C2×C4#48111yescharacteristic
320C2×C4#48311yescharacteristic
321C8—44no—
322C8—82no—
323C8—82no—
324C8—82no—
325C8—82no—
326C8—82no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—24no—
340C22—44no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—82no—
348C22—82no—
349C22—82no—
350C22C2×#33411yescharacteristic
351C22#156511yescharacteristic
352C22#157011yescharacteristic
353C22#157311yescharacteristic
354C22#185011yescharacteristic
355C22#185311yescharacteristic
356C22#185511yescharacteristic
357C22#185611yescharacteristic
358C22#185811yescharacteristic
359C22#186711yescharacteristic
360C22#186811yescharacteristic
361C4—22no—
362C4—22no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—42no—
367C4—42no—
368C4—81no—
369C4#159611yescharacteristic
370C4#160111yescharacteristic
371C4#160811yescharacteristic
372C4#161311yescharacteristic
373C2—21no—
374C2—21no—
375C2—21no—
376C2—21no—
377C2—44no—
378C2#883011yescharacteristic
379C2#883511yescharacteristic
380C2#885111yescharacteristic
381C2#885411yescharacteristic
382C2#885911yescharacteristic
383C2#886211yescharacteristic
384C2#886411yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
socleC23#385
prime radicalC25C4⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8830C2semidirect
5. #7029C2semidirect
58. #323C8semidirect
128. C22⋊C8C2×C8semidirect
262. C23#334semidirect

Maximal subgroups

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

subgroupindex
#219302
#110422
#88302
C4×#3232
#70292