Finite Group Databasebeta

← #25379

Subgroups — #25379

Order 256.

479 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#497C211yesmaximal, characteristic
2C2×#464C211yesmaximal, characteristic
3C2×#462C211yesmaximal, characteristic, complemented
4C2×#462C211yesmaximal, characteristic, complemented
5#2545C211yesmaximal, characteristic
6#2365C211yesmaximal, characteristic, complemented
7#2056C211yesmaximal, characteristic, complemented
8#2055C211yesmaximal, characteristic
9#2016C211yesmaximal, characteristic, complemented
10#2015C211yesmaximal, characteristic
11#1983C211yesmaximal, characteristic
12#1940C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic
14#1878C211yesmaximal, characteristic, complemented
15#1763C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×Q16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×SD16C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23C2×C4⋊Q8C2211yescharacteristic
24C2×#124C2211yescharacteristic
25C2×#124C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28C2×C4×Q8C2211yescharacteristic
29#497—24no—
30#464—24no—
31#462—24no—
32#462—24no—
33C2×C4⋊C8C2211yescharacteristic
34C2×C4⋊C8C2211yescharacteristic
35C2×C4⋊C8C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#97C2211yescharacteristic
45#396—21no—
46#396—21no—
47#391C2211yescharacteristic
48#391C2211yescharacteristic
49#391C2211yescharacteristic
50#390C2211yescharacteristic
51#389—21no—
52#388—21no—
53#388—21no—
54#387—21no—
55#387—21no—
56C4×C4⋊C4—21no—
57C4×C4⋊C4C2211yescharacteristic
58C4×C4⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×Q8C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C2×Q16—22no—
69C2×Q16—22no—
70C2×Q16—22no—
71C2×Q16—22no—
72C2×Q16—22no—
73C2×Q16—22no—
74C2×Q16—22no—
75C2×Q16—22no—
76C2×Q16—41no—
77C2×Q16—41no—
78C2×Q16—41no—
79C2×Q16—41no—
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—22no—
83C2×SD16—22no—
84C2×SD16—41no—
85C2×SD16—41no—
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C4⋊Q8—22no—
91C4⋊Q8—22no—
92C4⋊Q8—41no—
93C4⋊Q8D414yes—
94#124—22no—
95#124—22no—
96#124—22no—
97#124—41no—
98C4×Q8—22no—
99C4×Q8—22no—
100C4×Q8—22no—
101C4×Q8—22no—
102C4×Q8—41no—
103C4×Q8—41no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4D411yescharacteristic
112C2×C4⋊C4D411yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
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×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42C2311yescharacteristic
127C2×C42C2311yescharacteristic
128C2×C42C2311yescharacteristic
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132#103—22no—
133#103—22no—
134#103—22no—
135#103—22no—
136#103—41no—
137#103D414yes—
138#102—22no—
139#102—41no—
140#97—22no—
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—
151C2×Q8—21no—
152C2×Q8—21no—
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—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8C2×D412yes—
173C2×Q8C2×D412yes—
174C2×Q8C2×D412yes—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2411yescharacteristic
195Q16—44no—
196Q16—44no—
197Q16—44no—
198Q16—44no—
199SD16—44no—
200SD16—44no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8C2×D412yes—
212C2×C8#4112yes—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—22no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4C2×D412yes—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—42no—
239C42—42no—
240C42—42no—
241C42—42no—
242C42C2×D412yes—
243C42#4112yes—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23C22×D411yescharacteristic
252Q8—22no—
253Q8—22no—
254Q8—22no—
255Q8—22no—
256Q8—22no—
257Q8—22no—
258Q8—41no—
259Q8—41no—
260Q8—41no—
261Q8—41no—
262Q8—41no—
263Q8—41no—
264Q8—41no—
265Q8—41no—
266Q8—41no—
267Q8—41no—
268D4—41no—
269D4—41no—
270D4—41no—
271D4—41no—
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—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
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—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4#12112yes—
309C2×C4#12112yes—
310C2×C4#13712yes—
311C2×C4C22×D411yescharacteristic
312C2×C4C22×D411yescharacteristic
313C2×C4C2×#4111yescharacteristic
314C2×C4C2×#4111yescharacteristic
315C2×C4H(5,𝔽2)11yescharacteristic
316C2×C4H(5,𝔽2)11yescharacteristic
317C8—22no—
318C8—22no—
319C8—41no—
320C8—41no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22C2×#12111yescharacteristic
329C22D4211yescharacteristic
330C22#55011yescharacteristic
331C22C2×#13711yescharacteristic
332C22#57511yescharacteristic
333C22#57811yescharacteristic
334C22#57811yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—41no—
342C4—41no—
343C4—42no—
344C4—41no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4#53211yescharacteristic
349C4#54011yescharacteristic
350C4D4211yescharacteristic
351C4#54611yescharacteristic
352C2—161no—
353C2#242811yescharacteristic
354C2#303711yescharacteristic
355C2#304411yescharacteristic
356C2#326911yescharacteristic
357C2#327011yescharacteristic
358C2#339811yescharacteristic
359C2#340211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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
3. C2×#462C2semidirect
4. C2×#462C2semidirect
6. #2365C2semidirect
7. #2056C2semidirect
9. #2016C2semidirect
12. #1940C2semidirect
14. #1878C2semidirect
15. #1763C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4972
C2×#4642
C2×#4622
#25452
#23652
#20562
#20552
#20162
#20152
#19832
#19402
#18802
#18782
#17632