Finite Group Databasebeta

← #11121

Subgroups — #11121

Order 256.

627 conjugacy classes of subgroups.

The subgroup lattice has 355 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#106C211yesmaximal, characteristic, complemented
2C22×#102C211yesmaximal, characteristic
3C22×#102C211yesmaximal, characteristic
4C2×#390C211yesmaximal, characteristic
5C2×#340C211yesmaximal, characteristic, complemented
6C2×#336C211yesmaximal, characteristic, complemented
7C4⋊#95C211yesmaximal, characteristic, complemented
8C23×D4C2211yescharacteristic
9C23×C8C2211yescharacteristic
10C23×C8C2211yescharacteristic
11C22×C4⋊C4C2211yescharacteristic
12C22×C4⋊C4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C22⋊C4—22no—
15C22×C42C2211yescharacteristic
16C2×#106—22no—
17C2×#106—22no—
18C2×#106—22no—
19C2×#102—22no—
20C2×#102—22no—
21C2×#102—22no—
22C2×#102—22no—
23C2×#102—22no—
24C2×#102C414yescomplemented
25#390—22no—
26#390—22no—
27C2×#95—22no—
28#340—22no—
29#336—22no—
30C25—21no—
31C22×D4—21no—
32C22×D4—21no—
33C22×D4—21no—
34C22×D4—21no—
35C22×D4—21no—
36C22×D4—21no—
37C22×D4—21no—
38C22×D4—21no—
39C22×D4—21no—
40C22×D4—21no—
41C22×D4—21no—
42C22×D4C2×C412yes—
43C22×D4D412yes—
44C22×D4Q812yes—
45C23×C4—21no—
46C23×C4—21no—
47C23×C4—21no—
48C23×C4D412yes—
49C23×C4C2311yescharacteristic
50C22×C8—21no—
51C22×C8—21no—
52C22×C8—21no—
53C22×C8—21no—
54C22×C8—21no—
55C22×C8—21no—
56C22×C8—21no—
57C22×C8—21no—
58C22×C8—21no—
59C22×C8—21no—
60C22×C8C2×C412yes—
61C22×C8D412yes—
62C22×C8D412yes—
63C22×C8D412yes—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4C2×C412yes—
80C2×C4⋊C4D412yes—
81C2×C4⋊C4D412yes—
82C2×C22⋊C4—28no—
83C2×C22⋊C4—44no—
84C2×C22⋊C4—44no—
85C2×C42—21no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—21no—
91#106—28no—
92#102—28no—
93#102—28no—
94#95—44no—
95C24—22no—
96C24—22no—
97C24—22no—
98C24—41no—
99C24—41no—
100C24—41no—
101C24—41no—
102C24—42no—
103C24—42no—
104C24—42no—
105C24—42no—
106C24C2×D411yescharacteristic
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4—41no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—41no—
134C2×D4C4⋊C414yes—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4SD1614yes—
157C22×C4C22×C411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×Q811yescharacteristic
165C22×C4#4111yescharacteristic
166C22×C4#4111yescharacteristic
167C22×C4#4111yescharacteristic
168C22×C4#4111yescharacteristic
169C22×C4#4111yescharacteristic
170C22×C4#4111yescharacteristic
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8C22⋊C414yescomplemented
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C22⋊C4—416no—
198C42—41no—
199C42—41no—
200C42—41no—
201C42—41no—
202C23—24no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—44no—
210C23—44no—
211C23—44no—
212C23—44no—
213C23—44no—
214C23—44no—
215C23—82no—
216C23—82no—
217C23—82no—
218C23—82no—
219C23—82no—
220C23—82no—
221C23—81no—
222C23—81no—
223C23—81no—
224C23—81no—
225C23C4×D411yescharacteristic
226C23#12011yescharacteristic
227C23#12111yescharacteristic
228C23#12111yescharacteristic
229C23C22⋊Q811yescharacteristic
230C23#12311yescharacteristic
231C23#12711yescharacteristic
232C23C2×SD1612yes—
233C23#13512yes—
234C23AGL(1,ℤ/8)12yes—
235C23AGL(1,ℤ/8)12yes—
236D4—24no—
237D4—24no—
238D4—42no—
239D4—42no—
240D4—42no—
241D4—42no—
242D4—42no—
243D4—42no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4C2×C22⋊C411yescharacteristic
267C2×C4C2×C4⋊C411yescharacteristic
268C2×C4#11711yescharacteristic
269C2×C4C4×D411yescharacteristic
270C2×C4C4×D411yescharacteristic
271C2×C4C4×D411yescharacteristic
272C2×C4#12011yescharacteristic
273C2×C4#12011yescharacteristic
274C2×C4#12111yescharacteristic
275C2×C4#12111yescharacteristic
276C2×C4#12111yescharacteristic
277C2×C4#12111yescharacteristic
278C2×C4#12111yescharacteristic
279C2×C4#12111yescharacteristic
280C2×C4C22⋊Q811yescharacteristic
281C2×C4C22⋊Q811yescharacteristic
282C2×C4C22⋊Q811yescharacteristic
283C2×C4C22⋊Q811yescharacteristic
284C2×C4C22⋊Q811yescharacteristic
285C2×C4#12311yescharacteristic
286C2×C4#12311yescharacteristic
287C2×C4#12311yescharacteristic
288C2×C4#12311yescharacteristic
289C2×C4#12411yescharacteristic
290C2×C4#12411yescharacteristic
291C2×C4#12411yescharacteristic
292C2×C4#12611yescharacteristic
293C2×C4#12611yescharacteristic
294C8—24no—
295C8—42no—
296C22—48no—
297C22—44no—
298C22—84no—
299C22—82no—
300C22—82no—
301C22—84no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—84no—
307C22#38511yescharacteristic
308C22#38811yescharacteristic
309C22#39011yescharacteristic
310C22#39211yescharacteristic
311C22#39411yescharacteristic
312C22#39611yescharacteristic
313C22#39911yescharacteristic
314C22C4×SD1612yes—
315C22#44212yes—
316C22#44912yes—
317C22#45012yes—
318C22#46012yes—
319C22#46312yes—
320C22#46512yes—
321C22#46912yes—
322C22#47612yes—
323C22#47812yes—
324C22#48112yes—
325C22#48212yes—
326C22#49211yescharacteristic
327C22#49511yescharacteristic
328C22#49611yescharacteristic
329C22#49611yescharacteristic
330C4—44no—
331C4—82no—
332C4—82no—
333C4—82no—
334C4#38011yescharacteristic
335C4#38511yescharacteristic
336C4#38611yescharacteristic
337C4#38811yescharacteristic
338C4#39311yescharacteristic
339C4#39411yescharacteristic
340C4#39611yescharacteristic
341C4#39711yescharacteristic
342C2—84no—
343C2#143911yescharacteristic
344C2#186412yes—
345C2#192612yes—
346C2#196811yescharacteristic
347C2#197211yescharacteristic
348C2#201211yescharacteristic
349C2#201411yescharacteristic
350C2#203412yes—
351C2#206512yes—
352C2#209211yescharacteristic
353C2#209411yescharacteristic
subgroupG/H
center, socleC24C2×D4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#106C2semidirect
5. C2×#340C2semidirect
6. C2×#336C2semidirect
7. C4⋊#95C2semidirect
24. C2×#102C4semidirect
184. C2×C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#1062
C22×#1022
C2×#3902
C2×#3402
C2×#3362
C4⋊#952