Finite Group Databasebeta

← #22039

Subgroups — #22039

Order 256.

514 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#436C211yesmaximal, characteristic, complemented
2C2×#408C211yesmaximal, characteristic, complemented
3#2388C211yesmaximal, characteristic
4#1955C211yesmaximal, characteristic, complemented
5#1954C211yesmaximal, characteristic, complemented
6#1924C211yesmaximal, characteristic, complemented
7#1923C211yesmaximal, characteristic, complemented
8#1923C211yesmaximal, characteristic, complemented
9#1923C211yesmaximal, characteristic, complemented
10#1870C211yesmaximal, characteristic, complemented
11#1870C211yesmaximal, characteristic, complemented
12#1869C211yesmaximal, characteristic, complemented
13#1850C211yesmaximal, characteristic, complemented
14#1780C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23#436—24no—
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27#408—24no—
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×#97C2211yescharacteristic
42C2×C4×C8C2211yescharacteristic
43C2×C4×C8C2211yescharacteristic
44#395C411yescharacteristic
45#394C411yescharacteristic
46#394C411yescharacteristic
47#394C411yescharacteristic
48#393C411yescharacteristic
49#392C411yescharacteristic
50#392C411yescharacteristic
51#392C411yescharacteristic
52C4×C22⋊C4C2211yescharacteristic
53C4×C22⋊C4C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—41no—
64C2×#41—41no—
65C2×#41—41no—
66C2×#41—41no—
67C22×Q8C2×C411yescharacteristic
68C22×D4C2×C411yescharacteristic
69C22×D4C2×C411yescharacteristic
70C22×D4C2×C411yescharacteristic
71C22×D4C2×C411yescharacteristic
72C22×D4C2×C411yescharacteristic
73C22×D4C2×C411yescharacteristic
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C23×C4C2311yescharacteristic
78C2×M16—22no—
79C2×M16—41no—
80C2×M16—41no—
81C2×M16—41no—
82C2×M16—41no—
83C2×M16D414yescomplemented
84C22×C8—22no—
85C22×C8—41no—
86C22×C8—41no—
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—41no—
98C4×D4—41no—
99C4×D4—41no—
100C4×D4—41no—
101C2×C4⋊C4C2×C411yescharacteristic
102C2×C4⋊C4C2×C411yescharacteristic
103C2×C4⋊C4C2×C411yescharacteristic
104C2×C22⋊C4C2×C411yescharacteristic
105C2×C22⋊C4C2×C411yescharacteristic
106C2×C22⋊C4C2×C411yescharacteristic
107C2×C22⋊C4C2×C411yescharacteristic
108C2×C22⋊C4C2×C411yescharacteristic
109C2×C22⋊C4C2×C411yescharacteristic
110C2×C22⋊C4C2×C411yescharacteristic
111C2×C22⋊C4C2×C411yescharacteristic
112C2×C22⋊C4C2×C411yescharacteristic
113C2×C22⋊C4C2×C411yescharacteristic
114C2×C22⋊C4C2×C411yescharacteristic
115C2×C22⋊C4C2×C411yescharacteristic
116C2×C42C2311yescharacteristic
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119C4⋊C8—22no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C22⋊C8—22no—
123C22⋊C8—22no—
124C22⋊C8—22no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135#97—41no—
136C4×C8—22no—
137C4×C8—41no—
138#95C2×C411yescharacteristic
139#95C2×C411yescharacteristic
140#95C2×C411yescharacteristic
141#95C2×C411yescharacteristic
142#95C2×C411yescharacteristic
143#95C2×C411yescharacteristic
144C24C22×C411yescharacteristic
145C24C22×C411yescharacteristic
146C24C22×C411yescharacteristic
147C24C22×C411yescharacteristic
148#41—44no—
149#41—44no—
150#41—44no—
151#41—44no—
152C2×Q8—22no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—22no—
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—41no—
171C2×D4—41no—
172C2×D4—41no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
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×C4C22×C411yescharacteristic
196C22×C4C22×C411yescharacteristic
197C22×C4C22×C411yescharacteristic
198C22×C4C22×C411yescharacteristic
199C22×C4C22×C411yescharacteristic
200C22×C4C22×C411yescharacteristic
201C22×C4C22×C411yescharacteristic
202C22×C4C22×C411yescharacteristic
203C22×C4C22×C411yescharacteristic
204C22×C4C22×C411yescharacteristic
205C22×C4C2×D412yes—
206C22×C4#4112yes—
207C22×C4C2411yescharacteristic
208M16—44no—
209M16—44no—
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—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—44no—
233C2×C8C2×D412yes—
234C2×C8C2×D412yes—
235C4⋊C4—22no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C22⋊C4—22no—
239C22⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—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—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23C4×D412yes—
284C23C23×C411yescharacteristic
285Q8—44no—
286D4—44no—
287D4—44no—
288D4—44no—
289D4—44no—
290D4—44no—
291D4—44no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
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—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
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×C4—41no—
335C2×C4C4×D412yes—
336C2×C4#13112yes—
337C2×C4#13112yes—
338C2×C4C22×D411yescharacteristic
339C2×C4C2×#4111yescharacteristic
340C2×C4H(5,𝔽2)11yescharacteristic
341C2×C4H(5,𝔽2)11yescharacteristic
342C2×C4H(5,𝔽2)11yescharacteristic
343C2×C4H(5,𝔽2)11yescharacteristic
344C8—22no—
345C8—22no—
346C8—41no—
347C8—41no—
348C8—41no—
349C8—41no—
350C8—41no—
351C8—41no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22C2×C4×D411yescharacteristic
381C22#51811yescharacteristic
382C22#51811yescharacteristic
383C22C2×#13111yescharacteristic
384C22C2×#13111yescharacteristic
385C22#56811yescharacteristic
386C22#56811yescharacteristic
387C4—22no—
388C4—22no—
389C4—22no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4#53411yescharacteristic
398C4#53511yescharacteristic
399C4#53811yescharacteristic
400C4#53811yescharacteristic
401C2—42no—
402C2—81no—
403C2—81no—
404C2—81no—
405C2#232111yescharacteristic
406C2#293311yescharacteristic
407C2#293511yescharacteristic
408C2#297611yescharacteristic
409C2#297811yescharacteristic
410C2#297911yescharacteristic
411C2#298111yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C24
commutator, socleC23C23×C4
prime radical#2388C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#436C2semidirect
2. C2×#408C2semidirect
4. #1955C2semidirect
5. #1954C2semidirect
6. #1924C2semidirect
7. #1923C2semidirect
8. #1923C2semidirect
9. #1923C2semidirect
10. #1870C2semidirect
11. #1870C2semidirect
12. #1869C2semidirect
13. #1850C2semidirect
14. #1780C2semidirect
15. #1753C2semidirect
83. C2×M16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4362
C2×#4082
#23882
#19552
#19542
#19242
#19232
#18702
#18692
#18502
#17802
#17532