Finite Group Databasebeta

← #25381

Subgroups — #25381

Order 256.

506 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#497C211yesmaximal, characteristic, complemented
2C2×#464C211yesmaximal, characteristic, complemented
3C2×#463C211yesmaximal, characteristic
4C2×#463C211yesmaximal, characteristic
5#2547C211yesmaximal, characteristic, complemented
6#2367C211yesmaximal, characteristic, complemented
7#2056C211yesmaximal, characteristic, complemented
8#2055C211yesmaximal, characteristic, complemented
9#2015C211yesmaximal, characteristic, complemented
10#2012C211yesmaximal, characteristic
11#1981C211yesmaximal, characteristic, complemented
12#1940C211yesmaximal, characteristic, complemented
13#1878C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×D8C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23C2×#124C2211yescharacteristic
24C2×#124C2211yescharacteristic
25C2×#124C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29#497—24no—
30#464—24no—
31#463—24no—
32#463—24no—
33C2×C4⋊C8C2211yescharacteristic
34C2×C4⋊C8C2211yescharacteristic
35C2×C4⋊C8C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#97C2211yescharacteristic
45#397—21no—
46#397—21no—
47#391C2211yescharacteristic
48#391C2211yescharacteristic
49#391C2211yescharacteristic
50#390C2211yescharacteristic
51#388—21no—
52#388—21no—
53#385—21no—
54#380—21no—
55#380—21no—
56C4×C4⋊C4C2211yescharacteristic
57C4×C4⋊C4C2211yescharacteristic
58C4×C22⋊C4—21no—
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C23×C4—21no—
70C2×Q16—22no—
71C2×Q16—22no—
72C2×Q16—22no—
73C2×Q16—22no—
74C2×Q16—41no—
75C2×Q16—41no—
76C2×SD16—22no—
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16—22no—
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—41no—
83C2×SD16—41no—
84C2×SD16—41no—
85C2×D8—22no—
86C2×D8—22no—
87C2×D8—41no—
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C4⋊Q8—22no—
93C4⋊Q8—22no—
94#124—22no—
95#124—22no—
96#124—22no—
97#124—41no—
98#124—41no—
99#124D414yes—
100C4×Q8—22no—
101C4×Q8—41no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—41no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4D411yescharacteristic
110C2×C4⋊C4D411yescharacteristic
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⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—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—41no—
136#103—41no—
137#102—22no—
138#102—22no—
139#102D414yes—
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—
151C24—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×Q8C2×D412yes—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4C2×D412yes—
181C2×D4C2×D412yes—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—81no—
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4C2411yescharacteristic
201Q16—44no—
202Q16—44no—
203SD16—44no—
204SD16—44no—
205SD16—44no—
206D8—44no—
207C2×C8—21no—
208C2×C8—21no—
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×C8#4112yes—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4C2×D412yes—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—81no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—42no—
245C42—81no—
246C42C2×D412yes—
247C42#4112yes—
248C23—22no—
249C23—22no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23C22×D411yescharacteristic
263Q8—22no—
264Q8—22no—
265Q8—41no—
266Q8—41no—
267Q8—41no—
268Q8—41no—
269Q8—41no—
270Q8—41no—
271Q8—41no—
272Q8—41no—
273Q8—41no—
274Q8—41no—
275D4—22no—
276D4—22no—
277D4—22no—
278D4—22no—
279D4—41no—
280D4—41no—
281D4—41no—
282D4—41no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
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—84no—
318C2×C4—84no—
319C2×C4#12112yes—
320C2×C4#12112yes—
321C2×C4C22×D411yescharacteristic
322C2×C4C22×D411yescharacteristic
323C2×C4C2×#4111yescharacteristic
324C2×C4C2×#4111yescharacteristic
325C2×C4H(5,𝔽2)11yescharacteristic
326C2×C4H(5,𝔽2)11yescharacteristic
327C8—22no—
328C8—22no—
329C8—41no—
330C8—41no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22C2×#12111yescharacteristic
346C22D4211yescharacteristic
347C22#55011yescharacteristic
348C22#57511yescharacteristic
349C22#57511yescharacteristic
350C22#57711yescharacteristic
351C22#57811yescharacteristic
352C4—22no—
353C4—22no—
354C4—42no—
355C4—41no—
356C4—41no—
357C4—42no—
358C4—41no—
359C4—41no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4#53211yescharacteristic
365C4#54011yescharacteristic
366C4D4211yescharacteristic
367C4#54611yescharacteristic
368C2—82no—
369C2—82no—
370C2#242811yescharacteristic
371C2#304011yescharacteristic
372C2#304611yescharacteristic
373C2#326711yescharacteristic
374C2#326911yescharacteristic
375C2#340111yescharacteristic
376C2#340211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
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×#497C2semidirect
2. C2×#464C2semidirect
5. #2547C2semidirect
6. #2367C2semidirect
7. #2056C2semidirect
8. #2055C2semidirect
9. #2015C2semidirect
11. #1981C2semidirect
12. #1940C2semidirect
13. #1878C2semidirect
14. #1877C2semidirect
15. #1762C2semidirect

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×#4632
#25472
#23672
#20562
#20552
#20152
#20122
#19812
#19402
#18782
#18772
#17622