Finite Group Databasebeta

← #29669

Subgroups — #29669

Order 256.

598 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#542C211yesmaximal, characteristic, complemented
2C2×#460C211yesmaximal, characteristic, complemented
3C2×#459C211yesmaximal, characteristic, complemented
4#2619C211yesmaximal, characteristic, complemented
5#2023C212yesmaximal, complemented
6#2012C212yesmaximal, complemented
7#2000C212yesmaximal, complemented
8#1999C212yesmaximal, complemented
9#1805C211yesmaximal, characteristic, complemented
10#1804C211yesmaximal, characteristic, complemented
11#1798C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×D8C2212yes—
14#542—24no—
15#542—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#126—21no—
19C2×#124—21no—
20C2×#124C2211yescharacteristic
21C2×#123—21no—
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#121C2212yes—
26C2×C4×Q8C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×#117C2211yescharacteristic
29#460—24no—
30#459—24no—
31C2×C4⋊C8C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×#103C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×#102C2212yes—
37C2×C22⋊C8C2212yes—
38#394—21no—
39#394—21no—
40#392—21no—
41#392—21no—
42#390C2212yes—
43#382C2211yescharacteristic
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8C2311yescharacteristic
49C22×D4—21no—
50C22×D4—22no—
51C22×D4—21no—
52C22×D4—21no—
53C22×D4C2312yes—
54C22×D4C2311yescharacteristic
55C23×C4—21no—
56C23×C4C2311yescharacteristic
57C2×SD16—24no—
58C2×SD16—42no—
59C2×SD16—42no—
60C2×D8—24no—
61C2×D8—42no—
62C2×D8—42no—
63C22×C8C2312yes—
64C22×C8C2312yes—
65C4⋊Q8—22no—
66C4⋊Q8—22no—
67#127—22no—
68#127—22no—
69#126—28no—
70#124—28no—
71#124—22no—
72#124—22no—
73#123—28no—
74C22⋊Q8—28no—
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77#121—22no—
78#121—22no—
79#121—42no—
80#121—42no—
81C4×Q8—22no—
82C4×Q8D414yescomplemented
83C4×D4—22no—
84C4×D4D414yescomplemented
85#117—22no—
86#117—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4D411yescharacteristic
104C2×C22⋊C4D411yescharacteristic
105C2×C42—21no—
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108C4⋊C8—22no—
109C4⋊C8—22no—
110#103—42no—
111#102—24no—
112#102—24no—
113#102—42no—
114C22⋊C8—42no—
115#95—21no—
116#95—21no—
117C24—21no—
118C24—22no—
119C24C2×D411yescharacteristic
120C2×Q8—24no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—82no—
127C2×Q8C2×D412yes—
128C2×D4—24no—
129C2×D4—21no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—22no—
140C2×D4—44no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—82no—
144C2×D4—82no—
145C2×D4—82no—
146C2×D4—82no—
147C2×D4—82no—
148C2×D4C2×D412yes—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—82no—
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2411yescharacteristic
167SD16—48no—
168D8—48no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C4⋊C4—24no—
176C4⋊C4—24no—
177C4⋊C4—24no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4C2×D412yes—
185C4⋊C4C2×D412yes—
186C22⋊C4—24no—
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—24no—
190C22⋊C4—24no—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C22⋊C4—44no—
194C22⋊C4—82no—
195C22⋊C4—82no—
196C42—24no—
197C42—21no—
198C42—21no—
199C42C2×D412yes—
200C42C2×D412yes—
201C23—21no—
202C23—22no—
203C23—21no—
204C23—21no—
205C23—22no—
206C23—22no—
207C23—41no—
208C23—41no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23C22×D411yescharacteristic
219Q8—22no—
220Q8—22no—
221Q8—41no—
222Q8—41no—
223Q8—84no—
224D4—22no—
225D4—22no—
226D4—41no—
227D4—42no—
228D4—42no—
229D4—41no—
230D4—42no—
231D4—42no—
232D4—84no—
233D4—84no—
234D4—84no—
235D4—88no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—84no—
260C2×C4#12012yes—
261C2×C4#12012yes—
262C2×C4C22×D411yescharacteristic
263C2×C4C22×D411yescharacteristic
264C2×C4H(5,𝔽2)11yescharacteristic
265C2×C4H(5,𝔽2)11yescharacteristic
266C2×C4H(5,𝔽2)11yescharacteristic
267C2×C42−1+411yescharacteristic
268C8—42no—
269C8—42no—
270C22—41no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—42no—
278C22—82no—
279C22—82no—
280C22—82no—
281C22—82no—
282C22—82no—
283C22—82no—
284C22—82no—
285C22—82no—
286C22—82no—
287C22C2×#12011yescharacteristic
288C22#53511yescharacteristic
289C22#53711yescharacteristic
290C22#57612yes—
291C22#57712yes—
292C4—22no—
293C4—22no—
294C4—41no—
295C4—42no—
296C4—41no—
297C4—42no—
298C4—42no—
299C4—41no—
300C4—82no—
301C4#53511yescharacteristic
302C4#53511yescharacteristic
303C4#53711yescharacteristic
304C4#53711yescharacteristic
305C2—81no—
306C2—82no—
307C2—162no—
308C2#261511yescharacteristic
309C2#300811yescharacteristic
310C2#301011yescharacteristic
311C2#322412yes—
312C2#325712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#542C2semidirect
2. C2×#460C2semidirect
3. C2×#459C2semidirect
4. #2619C2semidirect
5. #2023C2semidirect
6. #2012C2semidirect
7. #2000C2semidirect
8. #1999C2semidirect
9. #1805C2semidirect
10. #1804C2semidirect
11. #1798C2semidirect
82. C4×Q8D4semidirect
84. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#4602
C2×#4592
#26192
#20232
#20122
#20002
#19992
#18052
#18042
#17982