Finite Group Databasebeta

← #26889

Subgroups — #26889

Order 256.

506 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#474C211yesmaximal, characteristic, complemented
2C2×C4×D8C211yesmaximal, characteristic, complemented
3#2452C211yesmaximal, characteristic, complemented
4#2366C211yesmaximal, characteristic, complemented
5#2054C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic, complemented
7#2023C211yesmaximal, characteristic, complemented
8#2011C211yesmaximal, characteristic
9#1934C211yesmaximal, characteristic, complemented
10#1925C211yesmaximal, characteristic, complemented
11#1918C211yesmaximal, characteristic, complemented
12#1916C211yesmaximal, characteristic, complemented
13#1879C211yesmaximal, characteristic, complemented
14#1879C211yesmaximal, characteristic, complemented
15#1862C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#127C2211yescharacteristic
20C2×C22⋊Q8—21no—
21C2×C4×D4—21no—
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25#474—24no—
26C4×D8—24no—
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C4×C8C2211yescharacteristic
39#396—21no—
40#393—21no—
41#390—21no—
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#389C2211yescharacteristic
47#389C2211yescharacteristic
48#386—21no—
49#386—21no—
50#385—21no—
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C22×D4C2311yescharacteristic
66C22×D4C2311yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C2×D8—22no—
71C2×D8—22no—
72C2×D8—22no—
73C2×D8—22no—
74C2×D8—22no—
75C2×D8—41no—
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C4⋊Q8—22no—
81C4⋊Q8—41no—
82#127—22no—
83#127—22no—
84C22⋊Q8—44no—
85C4×D4—22no—
86C4×D4—22no—
87C4×D4—22no—
88C4×D4—41no—
89C4×D4—41no—
90C4×D4—41no—
91C4×D4—44no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4D411yescharacteristic
96C2×C4⋊C4D411yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C42—21no—
114C2×C42C2311yescharacteristic
115C2×C42C2311yescharacteristic
116C2×C42C2311yescharacteristic
117#107—22no—
118#107D414yescomplemented
119C4⋊C8—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—41no—
128C4×C8—22no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95D411yescharacteristic
133#95D411yescharacteristic
134C24—21no—
135C24—21no—
136C24—21no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—81no—
140C2×D4—24no—
141C2×D4—24no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—81no—
160C2×D4—81no—
161C2×D4#4112yes—
162C2×D4#4112yes—
163C2×D4#4112yes—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2411yescharacteristic
185D8—28no—
186D8—44no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8C2×D412yes—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4D814yescomplemented
212C4⋊C4C2×D412yes—
213C4⋊C4C2×D412yes—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—81no—
223C22⋊C4—81no—
224C22⋊C4—81no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—42no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23C22×D411yescharacteristic
254Q8—84no—
255D4—22no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—22no—
260D4—22no—
261D4—48no—
262D4—41no—
263D4—41no—
264D4—41no—
265D4—41no—
266D4—41no—
267D4—41no—
268D4—84no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—84no—
300C2×C4—84no—
301C2×C4—84no—
302C2×C4C2×D812yes—
303C2×C4C2×D812yes—
304C2×C4C2×D812yes—
305C2×C4C22×D411yescharacteristic
306C2×C4C2×#4111yescharacteristic
307C2×C4C2×#4111yescharacteristic
308C2×C4C2×#4111yescharacteristic
309C2×C4H(5,𝔽2)11yescharacteristic
310C2×C42−1+411yescharacteristic
311C8—22no—
312C8—41no—
313C8—41no—
314C8—41no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22#52511yescharacteristic
337C22D4211yescharacteristic
338C22#54811yescharacteristic
339C22C22×D811yescharacteristic
340C22#57511yescharacteristic
341C22#57611yescharacteristic
342C22#57711yescharacteristic
343C4—22no—
344C4—22no—
345C4—22no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—42no—
350C4—42no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4#52511yescharacteristic
355C4#53911yescharacteristic
356C4#54811yescharacteristic
357C4#55111yescharacteristic
358C2—82no—
359C2—82no—
360C2—82no—
361C2#246311yescharacteristic
362C2#310011yescharacteristic
363C2#311811yescharacteristic
364C2#327111yescharacteristic
365C2D4×D811yescharacteristic
366C2#335811yescharacteristic
367C2#337011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2011C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#474C2semidirect
2. C2×C4×D8C2semidirect
3. #2452C2semidirect
4. #2366C2semidirect
5. #2054C2semidirect
6. #2041C2semidirect
7. #2023C2semidirect
9. #1934C2semidirect
10. #1925C2semidirect
11. #1918C2semidirect
12. #1916C2semidirect
13. #1879C2semidirect
14. #1879C2semidirect
15. #1862C2semidirect
118. #107D4semidirect
211. C4⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#4742
C2×C4×D82
#24522
#23662
#20542
#20412
#20232
#20112
#19342
#19252
#19182
#19162
#18792
#18622