Finite Group Databasebeta

← #24792

Subgroups — #24792

Order 256.

538 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#469C211yesmaximal, characteristic, complemented
2C2×#469C211yesmaximal, characteristic, complemented
3C2×#436C211yesmaximal, characteristic, complemented
4C2×#428C211yesmaximal, characteristic, complemented
5#2416C211yesmaximal, characteristic, complemented
6#2414C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic, complemented
8#2065C211yesmaximal, characteristic, complemented
9#2034C211yesmaximal, characteristic, complemented
10#2033C211yesmaximal, characteristic, complemented
11#1970C211yesmaximal, characteristic, complemented
12#1941C211yesmaximal, characteristic, complemented
13#1875C211yesmaximal, characteristic, complemented
14#1874C211yesmaximal, characteristic, complemented
15#1809C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C2×#121—21no—
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×#117C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27#469—24no—
28#469—24no—
29#436—24no—
30#428—24no—
31C2×#107C2211yescharacteristic
32C2×#107C2211yescharacteristic
33C2×#106C2211yescharacteristic
34C2×#106C2211yescharacteristic
35C2×#106C2211yescharacteristic
36C2×C4⋊C8C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic
46#396—21no—
47#394—21no—
48#394—21no—
49#392—21no—
50#390—21no—
51#390C2211yescharacteristic
52#390C2211yescharacteristic
53#389C2211yescharacteristic
54#389C2211yescharacteristic
55#388—21no—
56#386—21no—
57#380C2211yescharacteristic
58#380C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4D411yescharacteristic
68C22×D4C2311yescharacteristic
69C22×D4C2311yescharacteristic
70C23×C4—21no—
71C23×C4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C2×D8—22no—
74C2×D8—22no—
75C2×D8—22no—
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—22no—
81C2×M16D414yescomplemented
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86#121—22no—
87#121—22no—
88#121—22no—
89#121—22no—
90#121—22no—
91#121—22no—
92#121—41no—
93#121—41no—
94#121—44no—
95C4×D4—22no—
96C4×D4—22no—
97#117—22no—
98#117—41no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4D411yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
123C2×C22⋊C4D411yescharacteristic
124C2×C42—21no—
125C2×C42C2311yescharacteristic
126#107—22no—
127#107—22no—
128#106—22no—
129#106—22no—
130#106D414yescomplemented
131C4⋊C8—22no—
132#102—22no—
133#102—22no—
134#102—22no—
135#102—22no—
136#102—41no—
137#102—41no—
138C22⋊C8—22no—
139C22⋊C8—22no—
140C4×C8—22no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145C24—21no—
146C24—21no—
147C24C2×D411yescharacteristic
148C24C2×D411yescharacteristic
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
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—41no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4—81no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
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×C4—42no—
193C22×C4—42no—
194C22×C4—81no—
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D412yes—
199C22×C4C2×D411yescharacteristic
200C22×C4#4112yes—
201C22×C4C2411yescharacteristic
202D8—44no—
203M16—28no—
204M16—44no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8C2×D412yes—
214C2×C8C2×D412yes—
215C2×C8C2×D412yes—
216C2×C8#4112yes—
217C4⋊C4—22no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—81no—
231C4⋊C4C2×D412yes—
232C4⋊C4C2×D412yes—
233C22⋊C4—22no—
234C22⋊C4—22no—
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—81no—
244C22⋊C4—81no—
245C22⋊C4—81no—
246C42—21no—
247C42—21no—
248C42—42no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
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#12112yes—
277C23C22×D411yescharacteristic
278D4—44no—
279D4—41no—
280D4—41no—
281D4—41no—
282D4—41no—
283D4—41no—
284D4—41no—
285D4—41no—
286D4—41no—
287D4—84no—
288D4—84no—
289D4—84no—
290D4—84no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
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—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—84no—
328C2×C4#12112yes—
329C2×C4AGL(1,ℤ/8)12yes—
330C2×C4C22×D411yescharacteristic
331C2×C4C22×D411yescharacteristic
332C2×C4C2×#4111yescharacteristic
333C2×C4C2×#4111yescharacteristic
334C2×C4H(5,𝔽2)11yescharacteristic
335C2×C42−1+411yescharacteristic
336C8—22no—
337C8—41no—
338C8—41no—
339C8#12114yescomplemented
340C22—22no—
341C22—22no—
342C22—22no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22C2×#12111yescharacteristic
369C22D4211yescharacteristic
370C22#54711yescharacteristic
371C22C2×AGL(1,ℤ/8)11yescharacteristic
372C22#57511yescharacteristic
373C22#57611yescharacteristic
374C22#57711yescharacteristic
375C4—22no—
376C4—22no—
377C4—41no—
378C4—42no—
379C4—42no—
380C4—41no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4C2×#12111yescharacteristic
386C4#53711yescharacteristic
387C4#54811yescharacteristic
388C4#55311yescharacteristic
389C2—42no—
390C2—81no—
391C2—161no—
392C2—161no—
393C2#241611yescharacteristic
394C2#305511yescharacteristic
395C2#306211yescharacteristic
396C2#327411yescharacteristic
397C2#327511yescharacteristic
398C2#335411yescharacteristic
399C2#335511yescharacteristic
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×#469C2semidirect
2. C2×#469C2semidirect
3. C2×#436C2semidirect
4. C2×#428C2semidirect
5. #2416C2semidirect
6. #2414C2semidirect
7. #2067C2semidirect
8. #2065C2semidirect
9. #2034C2semidirect
10. #2033C2semidirect
11. #1970C2semidirect
12. #1941C2semidirect
13. #1875C2semidirect
14. #1874C2semidirect
15. #1809C2semidirect
81. C2×M16D4semidirect
130. #106D4semidirect
339. C8#121semidirect

Maximal subgroups

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

subgroupindex
C2×#4692
C2×#4362
C2×#4282
#24162
#24142
#20672
#20652
#20342
#20332
#19702
#19412
#18752
#18742
#18092