Finite Group Databasebeta

← #30402

Subgroups — #30402

Order 256.

549 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#466C211yesmaximal, characteristic, complemented
2C2×#449C211yesmaximal, characteristic, complemented
3#2537C211yesmaximal, characteristic, complemented
4#2407C211yesmaximal, characteristic, complemented
5#2095C211yesmaximal, characteristic, complemented
6#2076C211yesmaximal, characteristic, complemented
7#2071C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#2033C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1892C211yesmaximal, 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×#121—21no—
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27#466—24no—
28#449—24no—
29C2×#107C2211yescharacteristic
30C2×#107C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42#394—21no—
43#394—21no—
44#394—21no—
45#392—21no—
46#391C2211yescharacteristic
47#390C2211yescharacteristic
48#386—21no—
49#385—21no—
50#384C2211yescharacteristic
51#380—21no—
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—22no—
80C2×SD16—22no—
81C2×SD16—41no—
82C2×D8—41no—
83C2×D8—41no—
84C2×D8D414yescomplemented
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—
100#121—44no—
101C4×D4—41no—
102C4×D4—41no—
103C4×D4—41no—
104C4×D4—41no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4D411yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
124C2×C22⋊C4D411yescharacteristic
125C2×C22⋊C4D411yescharacteristic
126C2×C42C2311yescharacteristic
127#107—22no—
128#107—41no—
129#103—22no—
130#103—41no—
131#103—41no—
132#102—22no—
133#102—22no—
134#102—22no—
135#102—41no—
136#102—41no—
137C22⋊C8—22no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145C24—21no—
146C24—21no—
147C24C2×D411yescharacteristic
148C24C2×D411yescharacteristic
149#41—44no—
150#41—44no—
151#41—44no—
152#41—44no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8#4112yes—
159C2×D4—22no—
160C2×D4—22no—
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—21no—
170C2×D4—21no—
171C2×D4—41no—
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—42no—
181C2×D4—81no—
182C2×D4—81no—
183C2×D4—81no—
184C2×D4C2×D412yes—
185C2×D4C2×D412yes—
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—21no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4C2×D411yescharacteristic
213C22×C4C2×D411yescharacteristic
214C22×C4C2411yescharacteristic
215SD16—44no—
216D8—44no—
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—21no—
228C2×C8—44no—
229C2×C8C2×D412yes—
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—21no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—81no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—81no—
255C22⋊C4—81no—
256C42—21no—
257C42—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—22no—
266C23—22no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23AGL(1,ℤ/8)12yes—
286C23C22×D411yescharacteristic
287Q8—22no—
288Q8—22no—
289Q8—41no—
290Q8—41no—
291D4—22no—
292D4—22no—
293D4—22no—
294D4—22no—
295D4—44no—
296D4—48no—
297D4—41no—
298D4—41no—
299D4—41no—
300D4—41no—
301D4—84no—
302D4—84no—
303D4—84no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—48no—
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—41no—
341C2×C4—41no—
342C2×C4—84no—
343C2×C4#13512yes—
344C2×C4C22×D411yescharacteristic
345C2×C4C2×#4111yescharacteristic
346C2×C4H(5,𝔽2)11yescharacteristic
347C2×C4H(5,𝔽2)11yescharacteristic
348C2×C4H(5,𝔽2)11yescharacteristic
349C2×C4H(5,𝔽2)11yescharacteristic
350C8—22no—
351C8—41no—
352C8—41no—
353C8—41no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—41no—
363C22—42no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—42no—
368C22—42no—
369C22—41no—
370C22—42no—
371C22—42no—
372C22—41no—
373C22—41no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22#53411yescharacteristic
383C22D4211yescharacteristic
384C22#54611yescharacteristic
385C22C2×#13511yescharacteristic
386C22C2×AGL(1,ℤ/8)11yescharacteristic
387C22#57611yescharacteristic
388C22#57811yescharacteristic
389C4—22no—
390C4—41no—
391C4—41no—
392C4—42no—
393C4—41no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4#53411yescharacteristic
400C4#53811yescharacteristic
401C4#54611yescharacteristic
402C4#56011yescharacteristic
403C2—42no—
404C2—81no—
405C2—82no—
406C2—82no—
407C2#266811yescharacteristic
408C2#318811yescharacteristic
409C2#320311yescharacteristic
410C2#326411yescharacteristic
411C2#328311yescharacteristic
412C2#330511yescharacteristic
413C2#330811yescharacteristic
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×#466C2semidirect
2. C2×#449C2semidirect
3. #2537C2semidirect
4. #2407C2semidirect
5. #2095C2semidirect
6. #2076C2semidirect
7. #2071C2semidirect
8. #2033C2semidirect
9. #2033C2semidirect
10. #2000C2semidirect
11. #1892C2semidirect
12. #1879C2semidirect
13. #1877C2semidirect
14. #1869C2semidirect
15. #1788C2semidirect
84. C2×D8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4662
C2×#4492
#25372
#24072
#20952
#20762
#20712
#20332
#20002
#18922
#18792
#18772
#18692
#17882