Finite Group Databasebeta

← #30854

Subgroups — #30854

Order 256.

485 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#467C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic
3C2×#464C211yesmaximal, characteristic, complemented
4C2×#462C211yesmaximal, characteristic, complemented
5#2429C211yesmaximal, characteristic, complemented
6#2426C211yesmaximal, characteristic
7#2067C211yesmaximal, characteristic, complemented
8#2066C211yesmaximal, characteristic, complemented
9#2015C211yesmaximal, characteristic, complemented
10#2013C211yesmaximal, characteristic, complemented
11#2001C211yesmaximal, characteristic, complemented
12#2000C211yesmaximal, characteristic
13#1865C211yesmaximal, characteristic, complemented
14#1863C211yesmaximal, characteristic, complemented
15#1850C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C23×C8C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×C22⋊Q8C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28C2×C4×Q8C2211yescharacteristic
29#467—24no—
30#465—24no—
31#464—24no—
32#462—24no—
33C2×#107C2211yescharacteristic
34C2×#106C2211yescharacteristic
35C2×C4⋊C8C2211yescharacteristic
36C2×C4⋊C8C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46#398—21no—
47#396—21no—
48#396—21no—
49#394—21no—
50#391C2211yescharacteristic
51#390—21no—
52#390C2211yescharacteristic
53#389—21no—
54#389C2211yescharacteristic
55#389C2211yescharacteristic
56#388—21no—
57#388—21no—
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×D4—21no—
67C22×D4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C2×Q16—22no—
70C2×Q16—22no—
71C2×Q16—22no—
72C2×Q16—22no—
73C2×Q16—41no—
74C2×Q16—41no—
75C2×SD16—22no—
76C2×SD16—22no—
77C2×SD16—41no—
78C2×SD16—41no—
79C2×SD16—41no—
80C2×SD16—41no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—41no—
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C4⋊Q8—22no—
89C4⋊Q8—41no—
90#124—22no—
91#124—41no—
92C22⋊Q8—22no—
93C22⋊Q8—22no—
94C22⋊Q8—22no—
95C22⋊Q8—41no—
96C22⋊Q8—41no—
97C22⋊Q8—41no—
98#121—22no—
99#121—41no—
100C4×Q8—22no—
101C4×Q8—22no—
102C4×Q8—41no—
103C4×Q8—41no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4D411yescharacteristic
126C2×C22⋊C4D411yescharacteristic
127C2×C42—21no—
128C2×C42—21no—
129C2×C42C2311yescharacteristic
130C2×C42C2311yescharacteristic
131#107—22no—
132#106—22no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135#103—22no—
136#103—22no—
137#103—41no—
138#103—41no—
139#103D414yes—
140#103D414yescomplemented
141#102—22no—
142#102—22no—
143C22⋊C8—41no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95D411yescharacteristic
149#95D411yescharacteristic
150C24—21no—
151C24C2×D411yescharacteristic
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8C2×D412yes—
169C2×Q8C2×D412yes—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—81no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2411yescharacteristic
198Q16—44no—
199Q16—44no—
200SD16—44no—
201SD16—44no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—44no—
213C2×C8C2×D412yes—
214C2×C8C2×D412yes—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—81no—
230C4⋊C4C2×D412yes—
231C4⋊C4C2×D412yes—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—81no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—42no—
247C42—42no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23#13712yes—
262C23C22×D411yescharacteristic
263Q8—22no—
264Q8—22no—
265Q8—22no—
266Q8—22no—
267Q8—41no—
268Q8—41no—
269Q8—41no—
270Q8—41no—
271Q8—41no—
272Q8—41no—
273Q8—41no—
274Q8—41no—
275D4—41no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—84no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4#13512yes—
320C2×C4C22×D411yescharacteristic
321C2×C4C22×D411yescharacteristic
322C2×C4H(5,𝔽2)11yescharacteristic
323C2×C4H(5,𝔽2)11yescharacteristic
324C2×C4H(5,𝔽2)11yescharacteristic
325C2×C42−1+411yescharacteristic
326C8—22no—
327C8—22no—
328C8—41no—
329C8—41no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22#53711yescharacteristic
345C22D4211yescharacteristic
346C22D4211yescharacteristic
347C22C2×#13511yescharacteristic
348C22C2×#13711yescharacteristic
349C22#57711yescharacteristic
350C22#57811yescharacteristic
351C4—22no—
352C4—41no—
353C4—42no—
354C4—42no—
355C4—41no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—42no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4#53711yescharacteristic
364C4#53711yescharacteristic
365C4D4211yescharacteristic
366C4#56011yescharacteristic
367C2—42no—
368C2—161no—
369C2#267511yescharacteristic
370C2#324811yescharacteristic
371C2#325811yescharacteristic
372C2#326611yescharacteristic
373C2#327011yescharacteristic
374C2#328411yescharacteristic
375C2#328511yescharacteristic
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×#467C2semidirect
3. C2×#464C2semidirect
4. C2×#462C2semidirect
5. #2429C2semidirect
7. #2067C2semidirect
8. #2066C2semidirect
9. #2015C2semidirect
10. #2013C2semidirect
11. #2001C2semidirect
13. #1865C2semidirect
14. #1863C2semidirect
15. #1850C2semidirect
140. #103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4672
C2×#4652
C2×#4642
C2×#4622
#24292
#24262
#20672
#20662
#20152
#20132
#20012
#20002
#18652
#18632
#18502