Finite Group Databasebeta

← #27285

Subgroups — #27285

Order 256.

478 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#489C211yesmaximal, characteristic, complemented
2C2×#419C211yesmaximal, characteristic, complemented
3#2439C211yesmaximal, characteristic, complemented
4#2436C211yesmaximal, characteristic
5#2074C211yesmaximal, characteristic, complemented
6#2042C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2024C211yesmaximal, characteristic, complemented
9#1958C211yesmaximal, characteristic, complemented
10#1935C211yesmaximal, characteristic, complemented
11#1933C211yesmaximal, characteristic, complemented
12#1895C211yesmaximal, characteristic
13#1875C211yesmaximal, characteristic
14#1787C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#117C2211yescharacteristic
23#489—24no—
24#419—24no—
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×#97C2211yescharacteristic
39C2×C4×C8C2211yescharacteristic
40#396—21no—
41#393—21no—
42#393—21no—
43#393—21no—
44#391—21no—
45#391C2211yescharacteristic
46#391C2211yescharacteristic
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#387—21no—
50#386—21no—
51#386—21no—
52#382C2211yescharacteristic
53#380C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8—21no—
63C22×Q8—22no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—22no—
67C22×Q8—41no—
68C22×Q8—41no—
69C22×Q8C2311yescharacteristic
70C22×Q8C2311yescharacteristic
71C22×D4—21no—
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4C2311yescharacteristic
75C2×M16—22no—
76C2×M16—22no—
77C2×M16—41no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C4⋊Q8—22no—
83C4⋊Q8—41no—
84#124—22no—
85#124—22no—
86#124—41no—
87#124—41no—
88#121—22no—
89#121—22no—
90#117—22no—
91#117—41no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4D411yescharacteristic
108C2×C22⋊C4D411yescharacteristic
109C2×C42—21no—
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112#103—22no—
113#103—22no—
114#103—22no—
115#103—22no—
116#103—22no—
117#103—41no—
118#103—41no—
119#103D414yescomplemented
120#102—22no—
121#102—22no—
122#102—41no—
123#102—41no—
124C22⋊C8—22no—
125#97—22no—
126C4×C8—41no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95D411yescharacteristic
137#95D411yescharacteristic
138C24—21no—
139C24C2×D411yescharacteristic
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—28no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—44no—
155C2×Q8—44no—
156C2×Q8—44no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—81no—
160C2×Q8C2×D412yes—
161C2×Q8#4112yes—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4#4112yes—
191C22×C4#4112yes—
192C22×C4C2411yescharacteristic
193M16—44no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8C2×D412yes—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—81no—
218C4⋊C4C2×D412yes—
219C22⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—81no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—42no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23#12412yes—
247C23C22×D411yescharacteristic
248Q8—22no—
249Q8—22no—
250Q8—22no—
251Q8—22no—
252Q8—48no—
253Q8—41no—
254Q8—41no—
255Q8—41no—
256Q8—41no—
257Q8—84no—
258D4—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262D4—84no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—48no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4#12412yes—
304C2×C4C22×D411yescharacteristic
305C2×C4C2×#4111yescharacteristic
306C2×C4C2×#4111yescharacteristic
307C2×C4C2×#4111yescharacteristic
308C2×C4H(5,𝔽2)11yescharacteristic
309C2×C4H(5,𝔽2)11yescharacteristic
310C8—22no—
311C8—41no—
312C8—41no—
313C8—41no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22C2×#12411yescharacteristic
329C22D4211yescharacteristic
330C22#54611yescharacteristic
331C22#57511yescharacteristic
332C22#57511yescharacteristic
333C22#57711yescharacteristic
334C22#57811yescharacteristic
335C4—22no—
336C4—42no—
337C4—42no—
338C4—41no—
339C4—42no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4#52511yescharacteristic
348C4#53811yescharacteristic
349C4#54611yescharacteristic
350C4#55111yescharacteristic
351C2—42no—
352C2—161no—
353C2#247211yescharacteristic
354C2#313711yescharacteristic
355C2#314011yescharacteristic
356C2#327511yescharacteristic
357C2#327711yescharacteristic
358C2#331311yescharacteristic
359C2#332111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#489C2semidirect
2. C2×#419C2semidirect
3. #2439C2semidirect
5. #2074C2semidirect
6. #2042C2semidirect
7. #2042C2semidirect
8. #2024C2semidirect
9. #1958C2semidirect
10. #1935C2semidirect
11. #1933C2semidirect
14. #1787C2semidirect
15. #1753C2semidirect
119. #103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4892
C2×#4192
#24392
#24362
#20742
#20422
#20242
#19582
#19352
#19332
#18952
#18752
#17872
#17532