Finite Group Databasebeta

← #25429

Subgroups — #25429

Order 256.

517 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×#495C211yesmaximal, characteristic, complemented
2C2×#468C211yesmaximal, characteristic, complemented
3C2×#468C211yesmaximal, characteristic, complemented
4C2×#440C211yesmaximal, characteristic, complemented
5#2427C211yesmaximal, characteristic, complemented
6#2426C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2042C211yesmaximal, characteristic, complemented
9#2001C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic
11#1973C211yesmaximal, characteristic, complemented
12#1942C211yesmaximal, characteristic, complemented
13#1879C211yesmaximal, characteristic, complemented
14#1878C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C22×M16C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29#495—24no—
30#468—24no—
31#468—24no—
32#440—24no—
33C2×#107C2211yescharacteristic
34C2×#107C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×#97C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic
46#397—21no—
47#396—21no—
48#394—21no—
49#394—21no—
50#391C2211yescharacteristic
51#390—21no—
52#390C2211yescharacteristic
53#389C2211yescharacteristic
54#389C2211yescharacteristic
55#388—21no—
56#386—21no—
57#380—21no—
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×D4—21no—
66C22×D4—21no—
67C22×D4C2311yescharacteristic
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C23×C4C2311yescharacteristic
71C2×Q16—22no—
72C2×Q16—22no—
73C2×Q16—41no—
74C2×SD16—22no—
75C2×SD16—22no—
76C2×SD16—22no—
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16D414yescomplemented
80C2×D8—22no—
81C2×D8—22no—
82C2×D8—41no—
83C2×M16—22no—
84C2×M16—22no—
85C2×M16—22no—
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90#124—22no—
91#124—22no—
92#124—41no—
93#124—41no—
94C22⋊Q8—22no—
95C22⋊Q8—22no—
96C22⋊Q8—41no—
97C22⋊Q8—41no—
98#121—22no—
99#121—22no—
100#121—41no—
101#121—41no—
102C4×Q8—22no—
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⋊C4—21no—
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⋊C4—21no—
126C2×C22⋊C4D411yescharacteristic
127C2×C22⋊C4D411yescharacteristic
128C2×C42—21no—
129C2×C42C2311yescharacteristic
130C2×C42C2311yescharacteristic
131#107—22no—
132#107D414yescomplemented
133#103—22no—
134#103—22no—
135#103—22no—
136#103—41no—
137#102—22no—
138#102—22no—
139#102—22no—
140#102—41no—
141C22⋊C8—22no—
142#97—22no—
143C4×C8—22no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95D411yescharacteristic
149#95D411yescharacteristic
150C24—21no—
151C24—21no—
152C24C2×D411yescharacteristic
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×Q8C2×D412yes—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—81no—
181C2×D4—81no—
182C2×D4C2×D412yes—
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—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—81no—
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2411yescharacteristic
204Q16—44no—
205SD16—28no—
206SD16—44no—
207D8—44no—
208M16—44no—
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×C8C2×D412yes—
218C2×C8C2×D412yes—
219C2×C8#4112yes—
220C2×C8#4112yes—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—81no—
231C4⋊C4—81no—
232C4⋊C4C2×D412yes—
233C4⋊C4C2×D412yes—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—81no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—42no—
252C23—21no—
253C23—22no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23C22×D411yescharacteristic
274Q8—22no—
275Q8—22no—
276Q8—41no—
277Q8—41no—
278Q8—41no—
279Q8—41no—
280Q8—41no—
281Q8—41no—
282D4—22no—
283D4—22no—
284D4—41no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4—41no—
290D4—84no—
291D4—84no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
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—41no—
328C2×C4—41no—
329C2×C4—84no—
330C2×C4C22×D411yescharacteristic
331C2×C4C22×D411yescharacteristic
332C2×C4C2×#4111yescharacteristic
333C2×C4C2×#4111yescharacteristic
334C2×C4H(5,𝔽2)11yescharacteristic
335C2×C4H(5,𝔽2)11yescharacteristic
336C8—22no—
337C8—41no—
338C8—41no—
339C8#12114yes—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—42no—
344C22—41no—
345C22—42no—
346C22—41no—
347C22—42no—
348C22—41no—
349C22—41no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22#53211yescharacteristic
362C22D4211yescharacteristic
363C22D4211yescharacteristic
364C22#57511yescharacteristic
365C22#57511yescharacteristic
366C22#57611yescharacteristic
367C22#57811yescharacteristic
368C4—41no—
369C4—42no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—42no—
374C4—42no—
375C4—41no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4C2×#12111yescharacteristic
380C4#54011yescharacteristic
381C4#54611yescharacteristic
382C4#55011yescharacteristic
383C2—81no—
384C2—82no—
385C2—161no—
386C2#242811yescharacteristic
387C2#317511yescharacteristic
388C2#318311yescharacteristic
389C2#326811yescharacteristic
390C2#326911yescharacteristic
391C2#327311yescharacteristic
392C2#327711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#1999C2
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×#495C2semidirect
2. C2×#468C2semidirect
3. C2×#468C2semidirect
4. C2×#440C2semidirect
5. #2427C2semidirect
6. #2426C2semidirect
7. #2042C2semidirect
8. #2042C2semidirect
9. #2001C2semidirect
11. #1973C2semidirect
12. #1942C2semidirect
13. #1879C2semidirect
14. #1878C2semidirect
15. #1753C2semidirect
79. C2×SD16D4semidirect
132. #107D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4952
C2×#4682
C2×#4402
#24272
#24262
#20422
#20012
#19992
#19732
#19422
#18792
#18782
#17532