Finite Group Databasebeta

← #30397

Subgroups — #30397

Order 256.

535 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#465C211yesmaximal, characteristic, complemented
2C2×#449C211yesmaximal, characteristic, complemented
3#2535C211yesmaximal, characteristic, complemented
4#2407C211yesmaximal, characteristic, complemented
5#2092C211yesmaximal, characteristic, complemented
6#2077C211yesmaximal, characteristic, complemented
7#2074C211yesmaximal, characteristic, complemented
8#2036C211yesmaximal, characteristic, complemented
9#2033C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1893C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic, complemented
13#1877C211yesmaximal, characteristic, complemented
14#1869C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#465—24no—
27#449—24no—
28C2×#107C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41#397—21no—
42#394—21no—
43#392—21no—
44#392—21no—
45#391C2211yescharacteristic
46#390C2211yescharacteristic
47#386—21no—
48#386—21no—
49#386—21no—
50#385—21no—
51#382C2211yescharacteristic
52#380C2211yescharacteristic
53#380C2211yescharacteristic
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—41no—
64C2×#41—41no—
65C2×#41—41no—
66C2×#41—41no—
67C22×Q8C2311yescharacteristic
68C22×D4—21no—
69C22×D4—21no—
70C22×D4—21no—
71C22×D4—21no—
72C22×D4D411yescharacteristic
73C22×D4C2311yescharacteristic
74C22×D4C2311yescharacteristic
75C23×C4—21no—
76C23×C4—21no—
77C23×C4C2311yescharacteristic
78C23×C4C2311yescharacteristic
79C2×SD16—41no—
80C2×SD16—41no—
81C2×SD16D414yescomplemented
82C2×D8—22no—
83C2×D8—22no—
84C2×D8—41no—
85C22×C8—22no—
86C22×C8—41no—
87C22×C8—41no—
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22⋊Q8—22no—
93C22⋊Q8—41no—
94#121—22no—
95#121—22no—
96#121—22no—
97#121—41no—
98#121—41no—
99#121—41no—
100C4×D4—41no—
101C4×D4—41no—
102C4×D4—41no—
103C4×D4—41no—
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⋊C4D411yescharacteristic
122C2×C22⋊C4D411yescharacteristic
123C2×C42C2311yescharacteristic
124#107—41no—
125#106—22no—
126#103—22no—
127#103—22no—
128#103—41no—
129#102—22no—
130#102—22no—
131#102—41no—
132#102—41no—
133#102—41no—
134C22⋊C8—22no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144C24—21no—
145C24—21no—
146C24C2×D411yescharacteristic
147C24C2×D411yescharacteristic
148#41—44no—
149#41—44no—
150#41—44no—
151#41—44no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8C2×D412yes—
158C2×D4—22no—
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—21no—
168C2×D4—21no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—81no—
181C2×D4—81no—
182C2×D4—81no—
183C2×D4C2×D412yes—
184C2×D4#4112yes—
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×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4C2×D411yescharacteristic
213C22×C4C2411yescharacteristic
214SD16—44no—
215D8—44no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—44no—
228C2×C8C2×D412yes—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—81no—
250C22⋊C4—81no—
251C22⋊C4—81no—
252C22⋊C4—81no—
253C42—21no—
254C42—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—21no—
264C23—22no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23#13512yes—
283C23C22×D411yescharacteristic
284Q8—22no—
285Q8—22no—
286Q8—41no—
287Q8—41no—
288D4—22no—
289D4—22no—
290D4—22no—
291D4—22no—
292D4—44no—
293D4—48no—
294D4—41no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—84no—
299D4—84no—
300D4—84no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—48no—
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×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—84no—
341C2×C4C22×D411yescharacteristic
342C2×C4C2×#4111yescharacteristic
343C2×C4H(5,𝔽2)11yescharacteristic
344C2×C4H(5,𝔽2)11yescharacteristic
345C2×C4H(5,𝔽2)11yescharacteristic
346C2×C4H(5,𝔽2)11yescharacteristic
347C8—22no—
348C8—41no—
349C8—41no—
350C8—41no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—42no—
359C22—41no—
360C22—42no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—42no—
368C22—41no—
369C22—41no—
370C22—42no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22#53411yescharacteristic
380C22D4211yescharacteristic
381C22#54611yescharacteristic
382C22C2×#13511yescharacteristic
383C22#57511yescharacteristic
384C22#57611yescharacteristic
385C22#57711yescharacteristic
386C4—41no—
387C4—41no—
388C4—42no—
389C4—41no—
390C4—41no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4#53411yescharacteristic
397C4#53811yescharacteristic
398C4#54611yescharacteristic
399C4#56011yescharacteristic
400C2—42no—
401C2—81no—
402C2—82no—
403C2—82no—
404C2#266811yescharacteristic
405C2#319111yescharacteristic
406C2#319811yescharacteristic
407C2#326811yescharacteristic
408C2#328211yescharacteristic
409C2#329711yescharacteristic
410C2#330711yescharacteristic
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×#465C2semidirect
2. C2×#449C2semidirect
3. #2535C2semidirect
4. #2407C2semidirect
5. #2092C2semidirect
6. #2077C2semidirect
7. #2074C2semidirect
8. #2036C2semidirect
9. #2033C2semidirect
10. #1999C2semidirect
11. #1893C2semidirect
12. #1879C2semidirect
13. #1877C2semidirect
14. #1869C2semidirect
15. #1788C2semidirect
81. C2×SD16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4652
C2×#4492
#25352
#24072
#20922
#20772
#20742
#20362
#20332
#19992
#18932
#18792
#18772
#18692
#17882