Finite Group Databasebeta

← #22847

Subgroups — #22847

Order 256.

589 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#435C211yesmaximal, characteristic, complemented
2C2×C8×D4C211yesmaximal, characteristic, complemented
3C2×#408C211yesmaximal, characteristic, complemented
4C2×#408C211yesmaximal, characteristic, complemented
5#2388C211yesmaximal, characteristic
6#1954C211yesmaximal, characteristic, complemented
7#1924C211yesmaximal, characteristic, complemented
8#1923C211yesmaximal, characteristic, complemented
9#1870C211yesmaximal, characteristic, complemented
10#1869C211yesmaximal, characteristic, complemented
11#1869C211yesmaximal, characteristic, complemented
12#1851C211yesmaximal, characteristic, complemented
13#1850C211yesmaximal, characteristic, complemented
14#1753C211yesmaximal, characteristic, complemented
15C8×C22⋊C4C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C23×C8C2211yescharacteristic
21C23×C8C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25#435—24no—
26C8×D4—24no—
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30#408—24no—
31#408—24no—
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×C22⋊C8C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×#97C2211yescharacteristic
44C2×C4×C8C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic
46#395C411yescharacteristic
47#394C411yescharacteristic
48#394C411yescharacteristic
49#394C411yescharacteristic
50#393C411yescharacteristic
51#392C411yescharacteristic
52#392C411yescharacteristic
53#392C411yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55C4×C22⋊C4C2211yescharacteristic
56C4×C22⋊C4C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—41no—
68C2×#41—41no—
69C22×Q8C2×C411yescharacteristic
70C22×D4C2×C411yescharacteristic
71C22×D4C2×C411yescharacteristic
72C22×D4C2×C411yescharacteristic
73C22×D4C2×C411yescharacteristic
74C22×D4C2×C411yescharacteristic
75C22×D4C2×C411yescharacteristic
76C23×C4C2311yescharacteristic
77C23×C4C2311yescharacteristic
78C23×C4C2311yescharacteristic
79C23×C4C2311yescharacteristic
80C2×M16—22no—
81C2×M16—22no—
82C2×M16—22no—
83C2×M16—22no—
84C2×M16—41no—
85C2×M16—41no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—41no—
93C22×C8—41no—
94C22×C8—41no—
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99C22×C8C2311yescharacteristic
100C22×C8C2311yescharacteristic
101C22×C8C2311yescharacteristic
102C22×C8C2311yescharacteristic
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—22no—
106C4×D4—22no—
107C4×D4—41no—
108C4×D4—41no—
109C2×C4⋊C4C2×C411yescharacteristic
110C2×C4⋊C4C2×C411yescharacteristic
111C2×C4⋊C4C2×C411yescharacteristic
112C2×C22⋊C4C2×C411yescharacteristic
113C2×C22⋊C4C2×C411yescharacteristic
114C2×C22⋊C4C2×C411yescharacteristic
115C2×C22⋊C4C2×C411yescharacteristic
116C2×C22⋊C4C2×C411yescharacteristic
117C2×C22⋊C4C2×C411yescharacteristic
118C2×C22⋊C4C2×C411yescharacteristic
119C2×C22⋊C4C2×C411yescharacteristic
120C2×C22⋊C4C2×C411yescharacteristic
121C2×C22⋊C4C2×C411yescharacteristic
122C2×C22⋊C4C2×C411yescharacteristic
123C2×C22⋊C4C2×C411yescharacteristic
124C2×C42C2311yescharacteristic
125C2×C42C2311yescharacteristic
126C2×C42C2311yescharacteristic
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—41no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138C22⋊C8—41no—
139C22⋊C8D414yescomplemented
140C22⋊C8D414yescomplemented
141#97—22no—
142C4×C8—22no—
143C4×C8—41no—
144#95C2×C411yescharacteristic
145#95C2×C411yescharacteristic
146#95C2×C411yescharacteristic
147#95C2×C411yescharacteristic
148#95C2×C411yescharacteristic
149#95C2×C411yescharacteristic
150C24C22×C411yescharacteristic
151C24C22×C411yescharacteristic
152C24C22×C411yescharacteristic
153C24C22×C411yescharacteristic
154#41—44no—
155#41—44no—
156#41—44no—
157#41—44no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8—41no—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—22no—
169C2×D4—22no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—41no—
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×C4C22×C411yescharacteristic
200C22×C4C22×C411yescharacteristic
201C22×C4C22×C411yescharacteristic
202C22×C4C22×C411yescharacteristic
203C22×C4C22×C411yescharacteristic
204C22×C4C22×C411yescharacteristic
205C22×C4C22×C411yescharacteristic
206C22×C4C22×C411yescharacteristic
207C22×C4C22×C411yescharacteristic
208C22×C4C22×C411yescharacteristic
209C22×C4C2×D412yes—
210C22×C4C2×D412yes—
211C22×C4#4112yes—
212C22×C4#4112yes—
213C22×C4C2411yescharacteristic
214M16—44no—
215M16—44no—
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—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—44no—
237C2×C8—44no—
238C2×C8—44no—
239C2×C8C2×D412yes—
240C2×C8C2×D412yes—
241C2×C8C2×D412yes—
242C2×C8C2×D412yes—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—41no—
246C22⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—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—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23C4×D412yes—
290C23C4×D412yes—
291C23#13112yes—
292C23C23×C411yescharacteristic
293Q8—44no—
294D4—28no—
295D4—44no—
296D4—44no—
297D4—44no—
298D4—44no—
299D4—44no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4C4×D412yes—
343C2×C4C4×D412yes—
344C2×C4#13112yes—
345C2×C4#13112yes—
346C2×C4C22×D411yescharacteristic
347C2×C4C22×D411yescharacteristic
348C2×C4C2×#4111yescharacteristic
349C2×C4C2×#4111yescharacteristic
350C2×C4H(5,𝔽2)11yescharacteristic
351C2×C4H(5,𝔽2)11yescharacteristic
352C8—22no—
353C8—22no—
354C8—22no—
355C8—22no—
356C8—41no—
357C8—41no—
358C8—41no—
359C8—41no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22C2×C4×D411yescharacteristic
389C22C2×C4×D411yescharacteristic
390C22#51811yescharacteristic
391C22C2×#13111yescharacteristic
392C22C2×#13111yescharacteristic
393C22C2×#13111yescharacteristic
394C22#56811yescharacteristic
395C4—22no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4D4211yescharacteristic
406C4#54611yescharacteristic
407C4#54611yescharacteristic
408C4#55111yescharacteristic
409C2—42no—
410C2—42no—
411C2—42no—
412C2—81no—
413C2#235811yescharacteristic
414C2#292911yescharacteristic
415C2#293111yescharacteristic
416C2#293311yescharacteristic
417C2#293511yescharacteristic
418C2#297711yescharacteristic
419C2#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×#435C2semidirect
2. C2×C8×D4C2semidirect
3. C2×#408C2semidirect
4. C2×#408C2semidirect
6. #1954C2semidirect
7. #1924C2semidirect
8. #1923C2semidirect
9. #1870C2semidirect
10. #1869C2semidirect
11. #1869C2semidirect
12. #1851C2semidirect
13. #1850C2semidirect
14. #1753C2semidirect
15. C8×C22⋊C4C2semidirect
139. C22⋊C8D4semidirect
140. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4352
C2×C8×D42
C2×#4082
#23882
#19542
#19242
#19232
#18702
#18692
#18512
#18502
#17532
C8×C22⋊C42