Finite Group Databasebeta

← #25107

Subgroups — #25107

Order 256.

537 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#468C211yesmaximal, characteristic
2C2×#464C211yesmaximal, characteristic, complemented
3C2×#451C211yesmaximal, characteristic, complemented
4C2×#419C211yesmaximal, characteristic, complemented
5#2436C211yesmaximal, characteristic
6#2362C211yesmaximal, characteristic, complemented
7#2068C211yesmaximal, characteristic, complemented
8#2056C211yesmaximal, characteristic, complemented
9#2042C211yesmaximal, characteristic, complemented
10#2015C211yesmaximal, characteristic, complemented
11#1983C211yesmaximal, characteristic, complemented
12#1928C211yesmaximal, characteristic, complemented
13#1878C211yesmaximal, characteristic, complemented
14#1875C211yesmaximal, characteristic
15#1851C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×M16C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26C2×#117C2211yescharacteristic
27#468—24no—
28#464—24no—
29#451—24no—
30C2×#107C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33#419—24no—
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45#393—21no—
46#393—21no—
47#391C2211yescharacteristic
48#391C2211yescharacteristic
49#391C2211yescharacteristic
50#389C2211yescharacteristic
51#388—21no—
52#388—21no—
53#387—21no—
54#386—21no—
55#386—21no—
56C4×C4⋊C4—21no—
57C4×C4⋊C4C2211yescharacteristic
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—22no—
67C22×Q8—22no—
68C22×Q8—41no—
69C22×Q8—41no—
70C22×Q8C2311yescharacteristic
71C22×Q8C2311yescharacteristic
72C22×Q8C2311yescharacteristic
73C22×D4—21no—
74C22×D4C2311yescharacteristic
75C23×C4—21no—
76C23×C4C2311yescharacteristic
77C2×Q16—22no—
78C2×Q16—22no—
79C2×Q16—22no—
80C2×Q16—22no—
81C2×Q16—41no—
82C2×Q16—41no—
83C2×SD16—22no—
84C2×SD16—22no—
85C2×SD16—41no—
86C2×M16—22no—
87C2×M16—22no—
88C2×M16—41no—
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93#124—22no—
94#124—22no—
95#124—41no—
96#124—41no—
97C22⋊Q8—22no—
98C22⋊Q8D414yes—
99#121—22no—
100#121—22no—
101C4×Q8—22no—
102C4×Q8—41no—
103#117—22no—
104#117—41no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
120C2×C22⋊C4D411yescharacteristic
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42C2311yescharacteristic
125C2×C42C2311yescharacteristic
126#107—22no—
127C4⋊C8—22no—
128C4⋊C8—41no—
129#103—22no—
130#103—22no—
131#103—22no—
132#103—22no—
133#103—22no—
134#103—22no—
135#103D414yescomplemented
136#102—22no—
137#102—22no—
138#102—41no—
139C22⋊C8—22no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95D411yescharacteristic
149#95D411yescharacteristic
150C24—21no—
151C24C2×D411yescharacteristic
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—28no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8—42no—
174C2×Q8C2×D412yes—
175C2×Q8C2×D412yes—
176C2×Q8#4112yes—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
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—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D412yes—
206C22×C4#4112yes—
207C22×C4C2411yescharacteristic
208Q16—44no—
209Q16—44no—
210SD16—44no—
211M16—44no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8C2×D412yes—
224C4⋊C4—24no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4C2×D412yes—
234C4⋊C4C2×D412yes—
235C22⋊C4—24no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—42no—
249C42—42no—
250C42—42no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23#12112yes—
265C23C22×D411yescharacteristic
266Q8—22no—
267Q8—22no—
268Q8—22no—
269Q8—22no—
270Q8—22no—
271Q8—22no—
272Q8—48no—
273Q8—41no—
274Q8—41no—
275Q8—41no—
276Q8—41no—
277Q8—41no—
278Q8—41no—
279D4—41no—
280D4—41no—
281D4—41no—
282D4—41no—
283D4—84no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—48no—
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#12112yes—
325C2×C4C22×D411yescharacteristic
326C2×C4C22×D411yescharacteristic
327C2×C4C2×#4111yescharacteristic
328C2×C4C2×#4111yescharacteristic
329C2×C4H(5,𝔽2)11yescharacteristic
330C2×C4H(5,𝔽2)11yescharacteristic
331C8—22no—
332C8—41no—
333C8—41no—
334C8—41no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22C2×#12111yescharacteristic
350C22D4211yescharacteristic
351C22#54611yescharacteristic
352C22#57511yescharacteristic
353C22#57511yescharacteristic
354C22#57811yescharacteristic
355C22#57811yescharacteristic
356C4—22no—
357C4—42no—
358C4—42no—
359C4—42no—
360C4—41no—
361C4—42no—
362C4—41no—
363C4—42no—
364C4—41no—
365C4—41no—
366C4—81no—
367C4—81no—
368C4#52511yescharacteristic
369C4#53811yescharacteristic
370C4D4211yescharacteristic
371C4#54611yescharacteristic
372C2—42no—
373C2—161no—
374C2#242411yescharacteristic
375C2#304011yescharacteristic
376C2#304411yescharacteristic
377C2#326911yescharacteristic
378C2#327711yescharacteristic
379C2#331311yescharacteristic
380C2#332711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#464C2semidirect
3. C2×#451C2semidirect
4. C2×#419C2semidirect
6. #2362C2semidirect
7. #2068C2semidirect
8. #2056C2semidirect
9. #2042C2semidirect
10. #2015C2semidirect
11. #1983C2semidirect
12. #1928C2semidirect
13. #1878C2semidirect
15. #1851C2semidirect
135. #103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4682
C2×#4642
C2×#4512
C2×#4192
#24362
#23622
#20682
#20562
#20422
#20152
#19832
#19282
#18782
#18752
#18512