Finite Group Databasebeta

← #25559

Subgroups — #25559

Order 256.

587 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#554C211yesmaximal, characteristic, complemented
2C2×#480C211yesmaximal, characteristic, complemented
3C2×#468C211yesmaximal, characteristic, complemented
4#2414C211yesmaximal, characteristic
5#2057C211yesmaximal, characteristic, complemented
6#2036C211yesmaximal, characteristic, complemented
7#2023C211yesmaximal, characteristic, complemented
8#2000C211yesmaximal, characteristic, complemented
9#1880C211yesmaximal, characteristic, complemented
10#1874C211yesmaximal, characteristic, complemented
11#1873C211yesmaximal, characteristic, complemented
12#1870C211yesmaximal, characteristic, complemented
13#1863C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic, complemented
15#1736C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18#554—24no—
19#554—24no—
20C2×C4⋊Q8C2211yescharacteristic
21C2×#125—21no—
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8—21no—
24C2×C22⋊Q8C2211yescharacteristic
25C2×C22⋊Q8C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×C4×Q8C2211yescharacteristic
29C2×C4×D4—21no—
30C2×C4×D4C2211yescharacteristic
31C2×#117C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33C22×C4⋊C4C2211yescharacteristic
34#480—24no—
35#468—24no—
36C2×#107C2211yescharacteristic
37C2×#107C2211yescharacteristic
38C2×#107C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48#394—21no—
49#392—21no—
50#390—21no—
51#390C2211yescharacteristic
52#389C2211yescharacteristic
53#388—21no—
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8C2311yescharacteristic
65C22×D4—21no—
66C22×D4—21no—
67C22×D4D411yescharacteristic
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C2×SD16—22no—
74C2×SD16—22no—
75C2×SD16—41no—
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—41no—
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C4⋊Q8—22no—
84C4⋊Q8—22no—
85#125—28no—
86C22⋊Q8—28no—
87C22⋊Q8—28no—
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90C22⋊Q8—22no—
91C22⋊Q8D414yescomplemented
92#121—22no—
93#121—22no—
94#121—22no—
95#121—41no—
96C4×Q8—22no—
97C4×Q8—22no—
98C4×D4—28no—
99C4×D4—22no—
100C4×D4—22no—
101#117—22no—
102#117—41no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4D411yescharacteristic
114C2×C4⋊C4D414yescomplemented
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4D411yescharacteristic
130C2×C22⋊C4D411yescharacteristic
131C2×C42—21no—
132C2×C42—21no—
133C2×C42C2311yescharacteristic
134#107—22no—
135#107—22no—
136#107—22no—
137#103—22no—
138#103—22no—
139#103—41no—
140#102—22no—
141#102—22no—
142#102—22no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—41no—
146#95—21no—
147#95—21no—
148C24—21no—
149C24C2×D411yescharacteristic
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—22no—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4—81no—
175C22×C4—21no—
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—24no—
185C22×C4—24no—
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—81no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D412yes—
202C22×C4C2×D412yes—
203C22×C4C2×D411yescharacteristic
204C22×C4C2411yescharacteristic
205SD16—44no—
206M16—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#4112yes—
219C4⋊C4—24no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—22no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—81no—
235C4⋊C4C2×D412yes—
236C4⋊C4C2×D412yes—
237C4⋊C4C2×D412yes—
238C4⋊C4#4112yes—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—81no—
248C22⋊C4—81no—
249C42—24no—
250C42—21no—
251C42—21no—
252C42—42no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23#12012yes—
273C23#13712yes—
274C23C22×D411yescharacteristic
275Q8—22no—
276Q8—22no—
277Q8—41no—
278Q8—41no—
279Q8—84no—
280D4—28no—
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—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
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—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—84no—
327C2×C4—84no—
328C2×C4#12012yes—
329C2×C4#13712yes—
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—41no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
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—
361C22C2×#12011yescharacteristic
362C22#54611yescharacteristic
363C22#54711yescharacteristic
364C22C2×#13711yescharacteristic
365C22C2×#13711yescharacteristic
366C22#57611yescharacteristic
367C22#57611yescharacteristic
368C4—22no—
369C4—22no—
370C4—41no—
371C4—42no—
372C4—41no—
373C4—42no—
374C4—42no—
375C4—42no—
376C4—42no—
377C4—81no—
378C4—81no—
379C4#53211yescharacteristic
380C4#53711yescharacteristic
381C4#54611yescharacteristic
382C4#54711yescharacteristic
383C2—42no—
384C2—42no—
385C2—161no—
386C2#243011yescharacteristic
387C2#300411yescharacteristic
388C2#300811yescharacteristic
389C2#331411yescharacteristic
390C2#333311yescharacteristic
391C2#334611yescharacteristic
392C2#335711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2414C2
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×#554C2semidirect
2. C2×#480C2semidirect
3. C2×#468C2semidirect
5. #2057C2semidirect
6. #2036C2semidirect
7. #2023C2semidirect
8. #2000C2semidirect
9. #1880C2semidirect
10. #1874C2semidirect
11. #1873C2semidirect
12. #1870C2semidirect
13. #1863C2semidirect
14. #1852C2semidirect
15. #1736C2semidirect
91. C22⋊Q8D4semidirect
114. C2×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4802
C2×#4682
#24142
#20572
#20362
#20232
#20002
#18802
#18742
#18732
#18702
#18632
#18522
#17362