Finite Group Databasebeta

← #25177

Subgroups — #25177

Order 256.

547 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#464C211yesmaximal, characteristic, complemented
2C2×#451C211yesmaximal, characteristic, complemented
3C2×#448C211yesmaximal, characteristic
4C2×#440C211yesmaximal, characteristic, complemented
5#2427C211yesmaximal, characteristic
6Q8×C22⋊C4C211yesmaximal, characteristic, complemented
7#2068C211yesmaximal, characteristic, complemented
8#2036C211yesmaximal, characteristic, complemented
9#2034C211yesmaximal, characteristic
10#2015C211yesmaximal, characteristic, complemented
11#1971C211yesmaximal, characteristic, complemented
12#1924C211yesmaximal, characteristic, complemented
13#1878C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic, complemented
15#1865C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27#464—24no—
28#451—24no—
29#448—24no—
30#440—24no—
31C2×#107C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×C4⋊C8C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×#97C2211yescharacteristic
45#397—21no—
46#394—21no—
47#391—21no—
48#391—21no—
49#391C2211yescharacteristic
50#391C2211yescharacteristic
51#391C2211yescharacteristic
52#389C2211yescharacteristic
53#386—21no—
54#380—21no—
55#380—21no—
56#380C2211yescharacteristic
57C4×C22⋊C4—21no—
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—22no—
64C22×Q8—22no—
65C22×Q8—21no—
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×C4—21no—
77C23×C4C2311yescharacteristic
78C2×Q16—22no—
79C2×Q16—22no—
80C2×Q16—22no—
81C2×Q16—22no—
82C2×Q16—22no—
83C2×Q16—41no—
84C2×SD16—22no—
85C2×SD16—22no—
86C2×SD16—22no—
87C2×SD16—22no—
88C2×SD16—41no—
89C2×SD16D414yes—
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94#124—22no—
95#124—41no—
96C22⋊Q8—22no—
97C22⋊Q8—41no—
98#121—22no—
99#121—41no—
100C4×Q8—22no—
101C4×Q8—22no—
102C4×Q8—41no—
103C4×Q8—41no—
104C4×D4—22no—
105C4×D4—41no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
121C2×C22⋊C4D411yescharacteristic
122C2×C42—21no—
123C2×C42—21no—
124C2×C42C2311yescharacteristic
125C2×C42C2311yescharacteristic
126#107—22no—
127#106—41no—
128C4⋊C8—22no—
129#103—22no—
130#103—22no—
131#103—22no—
132#103—22no—
133#103—41no—
134#103D414yescomplemented
135#102—22no—
136#102—22no—
137C22⋊C8—22no—
138C22⋊C8—22no—
139#97—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—24no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
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—42no—
168C2×Q8—42no—
169C2×Q8—44no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8C2×D412yes—
174C2×Q8C2×D412yes—
175C2×Q8#4112yes—
176C2×Q8#4112yes—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—81no—
185C2×D4C2×D412yes—
186C22×C4—21no—
187C22×C4—24no—
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—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—81no—
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2411yescharacteristic
211Q16—44no—
212Q16—44no—
213SD16—28no—
214SD16—44no—
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×C8—21no—
224C2×C8—21no—
225C2×C8C2×D412yes—
226C2×C8C2×D412yes—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—81no—
238C4⋊C4C2×D412yes—
239C22⋊C4—22no—
240C22⋊C4—41no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—42no—
252C42—42no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23#13712yes—
267C23C22×D411yescharacteristic
268Q8—22no—
269Q8—22no—
270Q8—22no—
271Q8—22no—
272Q8—22no—
273Q8—48no—
274Q8—41no—
275Q8—41no—
276Q8—41no—
277Q8—41no—
278Q8—41no—
279Q8—41no—
280Q8#12114yes—
281D4—22no—
282D4—22no—
283D4—41no—
284D4—41no—
285D4—84no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—48no—
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×C4C22×D411yescharacteristic
329C2×C4C22×D411yescharacteristic
330C2×C4C2×#4111yescharacteristic
331C2×C4C2×#4111yescharacteristic
332C2×C4H(5,𝔽2)11yescharacteristic
333C2×C4H(5,𝔽2)11yescharacteristic
334C8—22no—
335C8—22no—
336C8—41no—
337C8—41no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—42no—
346C22—42no—
347C22—42no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22#52511yescharacteristic
353C22D4211yescharacteristic
354C22D4211yescharacteristic
355C22C2×#13711yescharacteristic
356C22#57511yescharacteristic
357C22#57711yescharacteristic
358C22#57811yescharacteristic
359C4—24no—
360C4—41no—
361C4—42no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—42no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4C2×#12111yescharacteristic
372C4#53811yescharacteristic
373C4#54611yescharacteristic
374C4#54611yescharacteristic
375C2—42no—
376C2—82no—
377C2#242411yescharacteristic
378C2#311711yescharacteristic
379C2#312111yescharacteristic
380C2#326611yescharacteristic
381C2#326911yescharacteristic
382C2#327011yescharacteristic
383C2#327511yescharacteristic
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
1. C2×#464C2semidirect
2. C2×#451C2semidirect
4. C2×#440C2semidirect
6. Q8×C22⋊C4C2semidirect
7. #2068C2semidirect
8. #2036C2semidirect
10. #2015C2semidirect
11. #1971C2semidirect
12. #1924C2semidirect
13. #1878C2semidirect
14. #1877C2semidirect
15. #1865C2semidirect
134. #103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4642
C2×#4512
C2×#4482
C2×#4402
#24272
Q8×C22⋊C42
#20682
#20362
#20342
#20152
#19712
#19242
#18782
#18772
#18652