Finite Group Databasebeta

← #25533

Subgroups — #25533

Order 256.

722 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#547C211yesmaximal, characteristic, complemented
2C2×#480C211yesmaximal, characteristic, complemented
3C2×#466C211yesmaximal, characteristic, complemented
4#2416C211yesmaximal, characteristic, complemented
5#2057C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, characteristic, complemented
7#2023C211yesmaximal, characteristic, complemented
8#1999C211yesmaximal, characteristic, complemented
9#1879C211yesmaximal, characteristic, complemented
10#1874C211yesmaximal, characteristic, complemented
11#1874C211yesmaximal, characteristic, complemented
12#1869C211yesmaximal, characteristic, complemented
13#1862C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic, complemented
15C2×#340C211yesmaximal, characteristic, complemented
16C22×#41—21no—
17C22×D8C2211yescharacteristic
18C23×C8C2211yescharacteristic
19#547—24no—
20#547—24no—
21C2×C4⋊Q8C2211yescharacteristic
22C2×#123—21no—
23C2×#123—21no—
24C2×C22⋊Q8—21no—
25C2×#121—21no—
26C2×#121C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×#121C2211yescharacteristic
30C2×C4×D4C2211yescharacteristic
31C2×C4×D4C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33C22×C4⋊C4C2211yescharacteristic
34C22×C4⋊C4C2211yescharacteristic
35#480—24no—
36#466—24no—
37C2×#107C2211yescharacteristic
38C2×#107C2211yescharacteristic
39C2×#107C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49#396—21no—
50#394—21no—
51#390C2211yescharacteristic
52#389C2211yescharacteristic
53#386—21no—
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C2×#41—28no—
64C2×#41—44no—
65C2×#41—44no—
66C22×Q8—21no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4—21no—
71C22×D4D411yescharacteristic
72C22×D4C2311yescharacteristic
73C22×D4C2311yescharacteristic
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4C2311yescharacteristic
79C23×C4C2311yescharacteristic
80C2×D8—22no—
81C2×D8—22no—
82C2×D8—41no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—41no—
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C4⋊Q8—22no—
91C4⋊Q8—22no—
92#123—28no—
93#123—28no—
94C22⋊Q8—28no—
95#121—22no—
96#121—22no—
97#121—22no—
98#121—22no—
99#121—22no—
100#121—22no—
101#121—41no—
102#121—44no—
103#121D414yescomplemented
104C4×D4—22no—
105C4×D4—22no—
106C4×D4—22no—
107C4×D4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4D411yescharacteristic
119C2×C4⋊C4D414yescomplemented
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2311yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
123C2×C4⋊C4C2311yescharacteristic
124C2×C4⋊C4C2311yescharacteristic
125C2×C4⋊C4C2311yescharacteristic
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4D411yescharacteristic
135C2×C22⋊C4D411yescharacteristic
136C2×C42C2311yescharacteristic
137#107—22no—
138#107—22no—
139#107—22no—
140#102—22no—
141#102—22no—
142#102—22no—
143#102—22no—
144#102—22no—
145#102—41no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—41no—
149#95—21no—
150#95—21no—
151C24—21no—
152C24—21no—
153C24C2×D411yescharacteristic
154C24C2×D411yescharacteristic
155#41—432no—
156C2×Q8—24no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—24no—
162C2×D4—24no—
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—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—81no—
184C2×D4D814yescomplemented
185C2×D4C2×D412yes—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—24no—
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—21no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—81no—
214C22×C4C2×D411yescharacteristic
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D412yes—
218C22×C4C2×D412yes—
219C22×C4C2×D411yescharacteristic
220C22×C4C2411yescharacteristic
221D8—44no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—44no—
234C2×C8#4112yescomplemented
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—22no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4C2×D412yes—
250C4⋊C4C2×D412yes—
251C4⋊C4C2×D412yes—
252C4⋊C4#4112yescomplemented
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—22no—
256C22⋊C4—24no—
257C22⋊C4—22no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—81no—
263C42—21no—
264C42—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—22no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23#12012yes—
290C23C2×D812yes—
291C23C2×D812yes—
292C23AGL(1,ℤ/8)12yes—
293C23C22×D411yescharacteristic
294Q8—48no—
295D4—28no—
296D4—22no—
297D4—22no—
298D4—48no—
299D4—48no—
300D4—41no—
301D4—41no—
302D4—41no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—84no—
307D4—84no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—48no—
327C2×C4—48no—
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—48no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—84no—
348C2×C4#12012yes—
349C2×C4C2×D812yes—
350C2×C4C22×D411yescharacteristic
351C2×C4C22×D411yescharacteristic
352C2×C4C2×#4111yescharacteristic
353C2×C4C2×#4111yescharacteristic
354C2×C4H(5,𝔽2)11yescharacteristic
355C2×C42−1+411yescharacteristic
356C8—22no—
357C8—41no—
358C8—41no—
359C8—41no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22#44714yes—
388C22C2×#12011yescharacteristic
389C22#54611yescharacteristic
390C22#54711yescharacteristic
391C22C22×D811yescharacteristic
392C22C2×AGL(1,ℤ/8)11yescharacteristic
393C22#57711yescharacteristic
394C22#57811yescharacteristic
395C4—22no—
396C4—22no—
397C4—41no—
398C4—42no—
399C4—42no—
400C4—42no—
401C4—41no—
402C4—42no—
403C4—81no—
404C4—81no—
405C4#53211yescharacteristic
406C4#53711yescharacteristic
407C4#54611yescharacteristic
408C4#54711yescharacteristic
409C2—24no—
410C2—42no—
411C2—82no—
412C2—161no—
413C2#243011yescharacteristic
414C2C2×#44711yescharacteristic
415C2#301311yescharacteristic
416C2#329411yescharacteristic
417C2#330811yescharacteristic
418C2#333411yescharacteristic
419C2#334511yescharacteristic
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×#547C2semidirect
2. C2×#480C2semidirect
3. C2×#466C2semidirect
4. #2416C2semidirect
5. #2057C2semidirect
6. #2033C2semidirect
7. #2023C2semidirect
8. #1999C2semidirect
9. #1879C2semidirect
10. #1874C2semidirect
11. #1874C2semidirect
12. #1869C2semidirect
13. #1862C2semidirect
14. #1852C2semidirect
15. C2×#340C2semidirect
103. #121D4semidirect
119. C2×C4⋊C4D4semidirect
184. C2×D4D8semidirect
234. C2×C8#41semidirect
252. C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#4802
C2×#4662
#24162
#20572
#20332
#20232
#19992
#18792
#18742
#18692
#18622
#18522
C2×#3402