Finite Group Databasebeta

← #30708

Subgroups — #30708

Order 256.

491 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic, complemented
3#2540C211yesmaximal, characteristic, complemented
4#2414C211yesmaximal, characteristic, complemented
5#2042C211yesmaximal, characteristic, complemented
6#2036C211yesmaximal, characteristic, complemented
7#2012C211yesmaximal, characteristic, complemented
8#2000C211yesmaximal, characteristic, complemented
9#1999C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1925C211yesmaximal, characteristic, complemented
12#1919C211yesmaximal, characteristic, complemented
13#1893C211yesmaximal, characteristic, complemented
14#1873C211yesmaximal, characteristic, complemented
15#1869C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27C2×#117C2211yescharacteristic
28#482—24no—
29#465—24no—
30C2×#107C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43#399—21no—
44#394—21no—
45#392—21no—
46#392—21no—
47#390—21no—
48#390—21no—
49#390—21no—
50#390C2211yescharacteristic
51#389C2211yescharacteristic
52#388—21no—
53#380C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4D411yescharacteristic
67C22×D4C2311yescharacteristic
68C22×D4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C2×SD16—22no—
72C2×SD16—41no—
73C2×SD16—41no—
74C2×D8—41no—
75C2×D8—41no—
76C2×D8—41no—
77C22×C8—22no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84#124—41no—
85#124—41no—
86C22⋊Q8—22no—
87C22⋊Q8—22no—
88C22⋊Q8—41no—
89C22⋊Q8—41no—
90#121—22no—
91#121—22no—
92#121—22no—
93#121—41no—
94#121—41no—
95#121—41no—
96#121—41no—
97#121—41no—
98C4×D4—41no—
99C4×D4—41no—
100#117—22no—
101#117—41no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4D411yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
124C2×C22⋊C4D411yescharacteristic
125C2×C42—21no—
126C2×C42—21no—
127C2×C42C2311yescharacteristic
128#107—22no—
129#106D414yescomplemented
130#103—22no—
131#103—41no—
132#103—41no—
133#102—22no—
134#102—22no—
135#102—22no—
136#102—22no—
137#102—41no—
138C22⋊C8—22no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141#95—21no—
142#95—21no—
143#95—21no—
144C24—21no—
145C24—21no—
146C24C2×D411yescharacteristic
147C24C2×D411yescharacteristic
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×D4—22no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—81no—
177C2×D4—81no—
178C2×D4—81no—
179C2×D4—81no—
180C2×D4—81no—
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×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2411yescharacteristic
204SD16—44no—
205D8—44no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—44no—
218C2×C8C2×D412yes—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—81no—
230C4⋊C4C2×D412yes—
231C4⋊C4C2×D412yes—
232C4⋊C4#4112yes—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—81no—
246C22⋊C4—81no—
247C42—21no—
248C42—21no—
249C42—42no—
250C42—81no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23#13512yes—
279C23C22×D411yescharacteristic
280Q8—41no—
281Q8—41no—
282Q8—41no—
283Q8—41no—
284D4—44no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4—41no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4—84no—
294D4—84no—
295D4—84no—
296D4—84no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4C22×D411yescharacteristic
336C2×C4C2×#4111yescharacteristic
337C2×C4H(5,𝔽2)11yescharacteristic
338C2×C4H(5,𝔽2)11yescharacteristic
339C2×C4H(5,𝔽2)11yescharacteristic
340C2×C4H(5,𝔽2)11yescharacteristic
341C8—22no—
342C8—41no—
343C8—41no—
344C8—41no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22#53411yescharacteristic
374C22D4211yescharacteristic
375C22#54611yescharacteristic
376C22C2×#13511yescharacteristic
377C22#57511yescharacteristic
378C22#57611yescharacteristic
379C22#57711yescharacteristic
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—42no—
385C4—42no—
386C4—42no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4#53511yescharacteristic
391C4#54011yescharacteristic
392C4#55011yescharacteristic
393C4#56011yescharacteristic
394C2—42no—
395C2—81no—
396C2—161no—
397C2—161no—
398C2#267311yescharacteristic
399C2#319111yescharacteristic
400C2#319811yescharacteristic
401C2#326811yescharacteristic
402C2#328211yescharacteristic
403C2#329711yescharacteristic
404C2#332111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#482C2semidirect
2. C2×#465C2semidirect
3. #2540C2semidirect
4. #2414C2semidirect
5. #2042C2semidirect
6. #2036C2semidirect
7. #2012C2semidirect
8. #2000C2semidirect
9. #1999C2semidirect
10. #1999C2semidirect
11. #1925C2semidirect
12. #1919C2semidirect
13. #1893C2semidirect
14. #1873C2semidirect
15. #1869C2semidirect
129. #106D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4652
#25402
#24142
#20422
#20362
#20122
#20002
#19992
#19252
#19192
#18932
#18732
#18692