Finite Group Databasebeta

← #24164

Subgroups — #24164

Order 256.

575 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#540C211yesmaximal, characteristic
2C2×#479C211yesmaximal, characteristic, complemented
3C2×#460C211yesmaximal, characteristic, complemented
4#2323C211yesmaximal, characteristic, complemented
5#2034C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, characteristic, complemented
7#2000C211yesmaximal, characteristic, complemented
8#2000C211yesmaximal, characteristic, complemented
9#1989C211yesmaximal, characteristic, complemented
10#1982C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1877C211yesmaximal, characteristic, complemented
13#1864C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15#1798C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18#540—24no—
19#540—24no—
20C2×#127C2211yescharacteristic
21C2×#125C2211yescharacteristic
22C2×#123—21no—
23C2×#123—21no—
24C2×C22⋊Q8C2211yescharacteristic
25C2×#121—21no—
26C2×#121—21no—
27C2×#121C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×C4×Q8C2211yescharacteristic
30C2×C4×D4C2211yescharacteristic
31C2×C4×D4C2211yescharacteristic
32C2×#117C2211yescharacteristic
33#479—24no—
34#460—24no—
35C2×#107C2211yescharacteristic
36C2×#106C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38C2×C4⋊C8C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49#391C2211yescharacteristic
50#389—21no—
51#389C2211yescharacteristic
52#388—21no—
53#388—21no—
54#387—21no—
55#380C2211yescharacteristic
56C4×C4⋊C4C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4C2311yescharacteristic
69C22×D4C2311yescharacteristic
70C23×C4—21no—
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73C2×SD16—22no—
74C2×SD16—22no—
75C2×SD16—22no—
76C2×SD16—22no—
77C2×SD16—41no—
78C2×SD16—41no—
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83#127—22no—
84#127D414yes—
85#125—22no—
86#125D414yes—
87#123—28no—
88#123—28no—
89C22⋊Q8—22no—
90C22⋊Q8—22no—
91#121—28no—
92#121—28no—
93#121—22no—
94#121—22no—
95#121—22no—
96#121—22no—
97C4×Q8—22no—
98C4×Q8—22no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—22no—
103#117—22no—
104#117—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4D411yescharacteristic
111C2×C4⋊C4D411yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
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
128C2×C42C2311yescharacteristic
129#107—22no—
130#106—22no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133#103—22no—
134#103—22no—
135#102—22no—
136#102—22no—
137#102—22no—
138#102—22no—
139#102—41no—
140#102—41no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143#95—21no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148C24—21no—
149C24—21no—
150C24C2×D411yescharacteristic
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8#4112yes—
157C2×D4—24no—
158C2×D4—24no—
159C2×D4—24no—
160C2×D4—24no—
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—81no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4—81no—
176C2×D4—81no—
177C2×D4—81no—
178C2×D4—81no—
179C2×D4C2×D412yes—
180C2×D4C2×D412yes—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—24no—
186C22×C4—24no—
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—81no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2411yescharacteristic
205SD16—44no—
206SD16—44no—
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—21no—
218C2×C8—21no—
219C4⋊C4—24no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—42no—
232C4⋊C4—81no—
233C4⋊C4C2×D412yes—
234C4⋊C4C2×D412yes—
235C4⋊C4#4112yes—
236C22⋊C4—24no—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—81no—
244C42—21no—
245C42—21no—
246C42—42no—
247C42—81no—
248C42C2×D412yes—
249C42C2×D412yes—
250C23—21no—
251C23—22no—
252C23—22no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23C22×D411yescharacteristic
272Q8—22no—
273Q8—22no—
274Q8—41no—
275Q8—41no—
276D4—22no—
277D4—22no—
278D4—22no—
279D4—22no—
280D4—41no—
281D4—41no—
282D4—41no—
283D4—41no—
284D4—84no—
285D4—84no—
286D4—84no—
287D4—84no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
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—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
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—84no—
324C2×C4—84no—
325C2×C4#12012yes—
326C2×C4#12012yes—
327C2×C4C22×D411yescharacteristic
328C2×C4C22×D411yescharacteristic
329C2×C4C2×#4111yescharacteristic
330C2×C4C2×#4111yescharacteristic
331C2×C4H(5,𝔽2)11yescharacteristic
332C2×C42−1+411yescharacteristic
333C8—41no—
334C8—41no—
335C8—41no—
336C8—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—42no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—42no—
345C22—41no—
346C22—42no—
347C22—42no—
348C22—41no—
349C22—42no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22C2×#12011yescharacteristic
359C22#52511yescharacteristic
360C22#53711yescharacteristic
361C22#57611yescharacteristic
362C22#57711yescharacteristic
363C22#57711yescharacteristic
364C22#57711yescharacteristic
365C4—22no—
366C4—22no—
367C4—42no—
368C4—41no—
369C4—42no—
370C4—41no—
371C4—42no—
372C4—41no—
373C4—42no—
374C4—81no—
375C4—81no—
376C4#54611yescharacteristic
377C4#54711yescharacteristic
378C4#54811yescharacteristic
379C4#55011yescharacteristic
380C2—81no—
381C2—82no—
382C2—82no—
383C2#240711yescharacteristic
384C2#300911yescharacteristic
385C2#301211yescharacteristic
386C2#311911yescharacteristic
387C2#312611yescharacteristic
388C2#325711yescharacteristic
389C2#326011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#540C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#479C2semidirect
3. C2×#460C2semidirect
4. #2323C2semidirect
5. #2034C2semidirect
6. #2033C2semidirect
7. #2000C2semidirect
8. #2000C2semidirect
9. #1989C2semidirect
10. #1982C2semidirect
11. #1877C2semidirect
12. #1877C2semidirect
13. #1864C2semidirect
14. #1862C2semidirect
15. #1798C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5402
C2×#4792
C2×#4602
#23232
#20342
#20332
#20002
#19892
#19822
#18772
#18642
#18622
#17982