Finite Group Databasebeta

← #99320

Subgroups — #99320

Order 512.

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

normal subgroups only

This group has 564 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#21934C211yesmaximal, characteristic
2#11042C211yesmaximal, characteristic
3#8830C212yesmaximal, complemented
4C4×#323C211yesmaximal, characteristic
5#7029C212yesmaximal, complemented
6C22×C4⋊C8C2211yescharacteristic
7C2×#407—21no—
8C22×C22⋊C8—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1851—22no—
11#1851—22no—
12#1797—21no—
13#1797C2212yes—
14C4×C22⋊C8C2212yes—
15C2×#336—21no—
16C2×#336C2211yescharacteristic
17C2×#334—22no—
18C2×#323C412yes—
19#1280C411yescharacteristic
20#1280C411yescharacteristic
21#1277—22no—
22C24×C4D411yescharacteristic
23C22×M16—21no—
24C22×M16—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—
37#407—42no—
38#407—42no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
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—24no—
58C25C4⋊C411yescharacteristic
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—41no—
64C23×C4—41no—
65C23×C4—41no—
66C23×C4—41no—
67C23×C4C4⋊C411yescharacteristic
68C23×C4C4⋊C411yescharacteristic
69C23×C4C4⋊C411yescharacteristic
70C23×C4SD1612yes—
71C23×C4SD1612yes—
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4C2×Q811yescharacteristic
75C2×M16—42no—
76C2×M16—42no—
77C2×M16—81no—
78C2×M16—81no—
79C2×M16—81no—
80C2×M16—81no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—41no—
88C22×C8—41no—
89C22×C8—41no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8—41no—
93C22×C8—41no—
94C22×C8—41no—
95C22×C8—41no—
96C22×C8—41no—
97C22×C8—41no—
98C22×C8—41no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—41no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—41no—
109C2×C42—41no—
110C2×C42—41no—
111C2×C42—41no—
112C2×C42C2×D412yes—
113C2×C42#4112yes—
114C4⋊C8—24no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117C4⋊C8—42no—
118C4⋊C8—81no—
119C4⋊C8—81no—
120C22⋊C8—22no—
121C22⋊C8—22no—
122C22⋊C8—22no—
123C22⋊C8—44no—
124C22⋊C8—48no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127C22⋊C8M1614yescomplemented
128C4×C8—42no—
129C4×C8—42no—
130#95—22no—
131#95—22no—
132C24—21no—
133C24—21no—
134C24—21no—
135C24—21no—
136C24—21no—
137C24—21no—
138C24—41no—
139C24—41no—
140C24—41no—
141C24—41no—
142C24#10612yes—
143C24C2×C4⋊C411yescharacteristic
144C22×C4—21no—
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—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—41no—
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—81no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4#10612yes—
186C22×C4#10812yes—
187C22×C4C4×D412yes—
188C22×C4#12011yescharacteristic
189C22×C4C22⋊Q812yes—
190C22×C4#12311yescharacteristic
191C22×C4C2×M1612yes—
192C22×C4#13111yescharacteristic
193C22×C4#13111yescharacteristic
194C22×C4C2×SD1611yescharacteristic
195C22×C4C2×SD1611yescharacteristic
196C22×C4AGL(1,ℤ/8)11yescharacteristic
197C22×C4#13711yescharacteristic
198M16—84no—
199M16—84no—
200C2×C8—24no—
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—42no—
218C2×C8—81no—
219C2×C8—81no—
220C2×C8—81no—
221C2×C8—81no—
222C2×C8—81no—
223C2×C8—81no—
224C2×C8—81no—
225C2×C8—81no—
226C22⋊C4—44no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C42—22no—
230C42—22no—
231C42—42no—
232C42—41no—
233C42—41no—
234C42—81no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—81no—
257C23—81no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23—81no—
262C23#38511yescharacteristic
263C23#42311yescharacteristic
264C23#42411yescharacteristic
265C23C2×#10611yescharacteristic
266C23#42811yescharacteristic
267C23C2×#10811yescharacteristic
268C23#43011yescharacteristic
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4#43512yes—
311C2×C4#43612yes—
312C2×C4#44811yescharacteristic
313C2×C4#45011yescharacteristic
314C2×C4#45311yescharacteristic
315C2×C4#45511yescharacteristic
316C2×C4B(4,𝔽2)11yescharacteristic
317C2×C4#45811yescharacteristic
318C2×C4#48111yescharacteristic
319C2×C4#48311yescharacteristic
320C8—44no—
321C8—82no—
322C8—82no—
323C8—82no—
324C8—82no—
325C8—82no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—44no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—82no—
343C22—82no—
344C22—82no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22#156611yescharacteristic
350C22#156911yescharacteristic
351C22#157011yescharacteristic
352C22#157411yescharacteristic
353C22#185111yescharacteristic
354C22#185311yescharacteristic
355C22#185511yescharacteristic
356C22#185611yescharacteristic
357C22#185811yescharacteristic
358C22#186711yescharacteristic
359C22#186811yescharacteristic
360C4—22no—
361C4—22no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—42no—
366C4—42no—
367C4—81no—
368C4#159611yescharacteristic
369C4#160111yescharacteristic
370C4#160811yescharacteristic
371C4#161311yescharacteristic
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2—82no—
377C2#883311yescharacteristic
378C2#883711yescharacteristic
379C2#885211yescharacteristic
380C2#885411yescharacteristic
381C2#886011yescharacteristic
382C2#886211yescharacteristic
383C2#886411yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
socleC23#385
prime radicalC25C4⋊C4
LCSG ▷ C23×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
127. C22⋊C8M16semidirect

Maximal subgroups

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

subgroupindex
#219342
#110422
#88302
C4×#3232
#70292