Finite Group Databasebeta

← #32713

Subgroups — #32713

Order 256.

453 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2616C211yesmaximal, characteristic, complemented
2#2615C211yesmaximal, characteristic
3#2077C211yesmaximal, characteristic, complemented
4#2076C211yesmaximal, characteristic, complemented
5#2075C211yesmaximal, characteristic, complemented
6#2036C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#2034C211yesmaximal, characteristic, complemented
10#2033C211yesmaximal, characteristic, complemented
11#2033C211yesmaximal, characteristic, complemented
12#2000C211yesmaximal, characteristic, complemented
13#2000C211yesmaximal, characteristic, complemented
14#2000C211yesmaximal, characteristic, complemented
15#1780C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39#395—21no—
40#394—21no—
41#394—21no—
42#394—21no—
43#393—21no—
44#392—21no—
45#392—21no—
46#392—21no—
47#380C2211yescharacteristic
48#380C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59C2×#41—41no—
60C2×#41—41no—
61C2×#41—41no—
62C2×#41—41no—
63C2×#41—41no—
64C2×#41—41no—
65C22×Q8D411yescharacteristic
66C22×Q8C2311yescharacteristic
67C22×D4—21no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4D411yescharacteristic
71C22×D4D411yescharacteristic
72C22×D4D411yescharacteristic
73C22×D4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C2×SD16—41no—
78C2×SD16—41no—
79C2×SD16—41no—
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22⋊Q8—41no—
85C22⋊Q8—41no—
86C22⋊Q8—41no—
87C22⋊Q8—41no—
88C22⋊Q8—41no—
89C22⋊Q8—41no—
90#121—41no—
91#121—41no—
92#121—41no—
93#121—41no—
94#121—41no—
95#121—41no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117#107—41no—
118#107—41no—
119#106—41no—
120#103—41no—
121#103—41no—
122#103—41no—
123#102—41no—
124#102—41no—
125#102—41no—
126C22⋊C8—41no—
127C22⋊C8—41no—
128C22⋊C8—41no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24—21no—
139C24C2×D411yescharacteristic
140C24C2×D411yescharacteristic
141C24C2×D411yescharacteristic
142#41—44no—
143#41—44no—
144#41—44no—
145#41—44no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4—41no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4—81no—
176C2×D4—81no—
177C2×D4—81no—
178C2×D4—81no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
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—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2411yescharacteristic
208SD16—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×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—81no—
243C22⋊C4—81no—
244C22⋊C4—81no—
245C22⋊C4—81no—
246C22⋊C4—81no—
247C22⋊C4—81no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23C22×D411yescharacteristic
277Q8—44no—
278Q8—41no—
279Q8—41no—
280Q8—41no—
281Q8—41no—
282D4—44no—
283D4—44no—
284D4—44no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4—84no—
290D4—84no—
291D4—84no—
292C2×C4—21no—
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—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—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4H(5,𝔽2)11yescharacteristic
335C2×C4H(5,𝔽2)11yescharacteristic
336C2×C4H(5,𝔽2)11yescharacteristic
337C2×C4H(5,𝔽2)11yescharacteristic
338C2×C4H(5,𝔽2)11yescharacteristic
339C2×C4H(5,𝔽2)11yescharacteristic
340C8—41no—
341C8—41no—
342C8—41no—
343C8—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22#53411yescharacteristic
373C22#53411yescharacteristic
374C22#53411yescharacteristic
375C22#57611yescharacteristic
376C22#57711yescharacteristic
377C22#57711yescharacteristic
378C22#57811yescharacteristic
379C4—41no—
380C4—41no—
381C4—41no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4#56011yescharacteristic
390C4#56011yescharacteristic
391C4#56011yescharacteristic
392C4H(3,𝔽4)11yescharacteristic
393C2—81no—
394C2—81no—
395C2—81no—
396C2—161no—
397C2#281211yescharacteristic
398C2#319811yescharacteristic
399C2#319811yescharacteristic
400C2#320011yescharacteristic
401C2#320111yescharacteristic
402C2#320111yescharacteristic
403C2#320311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2615C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2616C2semidirect
3. #2077C2semidirect
4. #2076C2semidirect
5. #2075C2semidirect
6. #2036C2semidirect
7. #2036C2semidirect
8. #2035C2semidirect
9. #2034C2semidirect
10. #2033C2semidirect
11. #2033C2semidirect
12. #2000C2semidirect
13. #2000C2semidirect
14. #2000C2semidirect
15. #1780C2semidirect

Maximal subgroups

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

subgroupindex
#26162
#26152
#20772
#20762
#20752
#20362
#20352
#20342
#20332
#20002
#17802