Finite Group Databasebeta

← #24167

Subgroups — #24167

Order 256.

571 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#542C211yesmaximal, characteristic, complemented
2C2×#474C211yesmaximal, characteristic, complemented
3C2×#463C211yesmaximal, characteristic, complemented
4#2324C211yesmaximal, characteristic, complemented
5#2036C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, characteristic, complemented
7#2000C211yesmaximal, characteristic, complemented
8#1999C211yesmaximal, characteristic
9#1987C211yesmaximal, characteristic, complemented
10#1983C211yesmaximal, characteristic, complemented
11#1880C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic, complemented
13#1863C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15#1798C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18#542—24no—
19#542—24no—
20C2×C4⋊Q8C2211yescharacteristic
21C2×#126—21no—
22C2×#124—21no—
23C2×#124C2211yescharacteristic
24C2×#123—21no—
25C2×C22⋊Q8—21no—
26C2×C22⋊Q8C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×C4×Q8C2211yescharacteristic
30C2×C4×D4C2211yescharacteristic
31C2×C4×D4C2211yescharacteristic
32C2×#117C2211yescharacteristic
33#474—24no—
34#463—24no—
35C2×#107C2211yescharacteristic
36C2×#107C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38C2×C4⋊C8C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49#390—21no—
50#390C2211yescharacteristic
51#389C2211yescharacteristic
52#388—21no—
53#386—21no—
54#385—21no—
55#380C2211yescharacteristic
56C4×C4⋊C4C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8C2311yescharacteristic
65C22×D4—21no—
66C22×D4—21no—
67C22×D4C2311yescharacteristic
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4C2311yescharacteristic
72C2×SD16—22no—
73C2×SD16—22no—
74C2×SD16—41no—
75C2×D8—22no—
76C2×D8—22no—
77C2×D8—41no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C4⋊Q8—22no—
83C4⋊Q8D414yescomplemented
84#126—28no—
85#124—28no—
86#124—22no—
87#124D414yescomplemented
88#123—28no—
89C22⋊Q8—28no—
90C22⋊Q8—22no—
91C22⋊Q8—22no—
92#121—22no—
93#121—22no—
94#121—22no—
95#121—22no—
96C4×Q8—22no—
97C4×Q8—22no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—22no—
102#117—22no—
103#117—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4D411yescharacteristic
109C2×C4⋊C4D411yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
125C2×C22⋊C4D411yescharacteristic
126C2×C22⋊C4D411yescharacteristic
127C2×C42—21no—
128C2×C42—21no—
129C2×C42C2311yescharacteristic
130C2×C42C2311yescharacteristic
131#107—22no—
132#107—22no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135#103—22no—
136#103—22no—
137#103—41no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102—41no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145#95—21no—
146#95—21no—
147#95—21no—
148C24—21no—
149C24—21no—
150C24C2×D411yescharacteristic
151C2×Q8—24no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—81no—
158C2×Q8—81no—
159C2×Q8C2×D412yes—
160C2×D4—24no—
161C2×D4—24no—
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—21no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4—81no—
176C2×D4C2×D412yes—
177C2×D4#4112yes—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—24no—
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—42no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2411yescharacteristic
202SD16—44no—
203D8—44no—
204C2×C8—21no—
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×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—81no—
229C4⋊C4C2×D412yes—
230C4⋊C4C2×D412yes—
231C4⋊C4#4112yes—
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—24no—
235C22⋊C4—24no—
236C22⋊C4—24no—
237C22⋊C4—24no—
238C22⋊C4—22no—
239C22⋊C4—22no—
240C22⋊C4—42no—
241C22⋊C4—81no—
242C22⋊C4—81no—
243C22⋊C4—81no—
244C42—24no—
245C42—21no—
246C42—21no—
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—
276Q8—84no—
277D4—22no—
278D4—22no—
279D4—22no—
280D4—22no—
281D4—41no—
282D4—41no—
283D4—41no—
284D4—41no—
285D4—84no—
286D4—84no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
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—84no—
323C2×C4—84no—
324C2×C4#12012yes—
325C2×C4#12012yes—
326C2×C4C22×D411yescharacteristic
327C2×C4C22×D411yescharacteristic
328C2×C4C2×#4111yescharacteristic
329C2×C4C2×#4111yescharacteristic
330C2×C4H(5,𝔽2)11yescharacteristic
331C2×C42−1+411yescharacteristic
332C8—41no—
333C8—41no—
334C8—41no—
335C8—41no—
336C22—41no—
337C22—42no—
338C22—41no—
339C22—42no—
340C22—41no—
341C22—42no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—42no—
346C22—41no—
347C22—41no—
348C22—42no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22C2×#12011yescharacteristic
358C22#52511yescharacteristic
359C22#53711yescharacteristic
360C22#57611yescharacteristic
361C22#57611yescharacteristic
362C22#57711yescharacteristic
363C22#57811yescharacteristic
364C4—22no—
365C4—22no—
366C4—42no—
367C4—41no—
368C4—42no—
369C4—41no—
370C4—42no—
371C4—42no—
372C4—41no—
373C4—81no—
374C4—81no—
375C4#54611yescharacteristic
376C4#54711yescharacteristic
377C4#54811yescharacteristic
378C4#55011yescharacteristic
379C2—81no—
380C2—82no—
381C2—82no—
382C2#240711yescharacteristic
383C2#300811yescharacteristic
384C2#301311yescharacteristic
385C2#311811yescharacteristic
386C2#312411yescharacteristic
387C2#325711yescharacteristic
388C2#326211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#1999C2
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
1. C2×#542C2semidirect
2. C2×#474C2semidirect
3. C2×#463C2semidirect
4. #2324C2semidirect
5. #2036C2semidirect
6. #2033C2semidirect
7. #2000C2semidirect
9. #1987C2semidirect
10. #1983C2semidirect
11. #1880C2semidirect
12. #1879C2semidirect
13. #1863C2semidirect
14. #1862C2semidirect
15. #1798C2semidirect
83. C4⋊Q8D4semidirect
87. #124D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#4742
C2×#4632
#23242
#20362
#20332
#20002
#19992
#19872
#19832
#18802
#18792
#18632
#18622
#17982