Finite Group Databasebeta

← #25388

Subgroups — #25388

Order 256.

558 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#466C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic, complemented
3C2×#463C211yesmaximal, characteristic, complemented
4C2×C8×D4C211yesmaximal, characteristic, complemented
5#2540C211yesmaximal, characteristic, complemented
6#2535C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#2000C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1958C211yesmaximal, characteristic, complemented
12#1949C211yesmaximal, characteristic, complemented
13#1893C211yesmaximal, characteristic, complemented
14#1892C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29#466—24no—
30#465—24no—
31#463—24no—
32C8×D4—24no—
33C2×#107C2211yescharacteristic
34C2×#106C2211yescharacteristic
35C2×C4⋊C8C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C4×C8C2211yescharacteristic
47#399—21no—
48#397—21no—
49#392—21no—
50#392—21no—
51#390—21no—
52#390—21no—
53#386—21no—
54#386—21no—
55#382C2211yescharacteristic
56#380C2211yescharacteristic
57#380C2211yescharacteristic
58C4×C4⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4D411yescharacteristic
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C2×SD16—22no—
75C2×SD16—22no—
76C2×SD16—41no—
77C2×D8—22no—
78C2×D8—22no—
79C2×D8—41no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90#124—22no—
91#124—22no—
92C22⋊Q8—22no—
93C22⋊Q8—22no—
94C22⋊Q8—41no—
95C22⋊Q8—41no—
96#121—22no—
97#121—22no—
98#121—22no—
99#121—22no—
100#121—41no—
101#121—41no—
102#121—41no—
103#121—41no—
104C4×D4—22no—
105C4×D4—22no—
106C4×D4—22no—
107C4×D4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4D411yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4D411yescharacteristic
128C2×C22⋊C4D411yescharacteristic
129C2×C42—21no—
130C2×C42C2311yescharacteristic
131#107—22no—
132#106—22no—
133C4⋊C8D414yescomplemented
134#103—22no—
135#103—22no—
136#103—41no—
137#102—22no—
138#102—22no—
139#102—22no—
140#102—41no—
141#102D414yescomplemented
142C22⋊C8—22no—
143C22⋊C8—22no—
144C4×C8—22no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95—21no—
150C24—21no—
151C24—21no—
152C24C2×D411yescharacteristic
153C24C2×D411yescharacteristic
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—81no—
183C2×D4—81no—
184C2×D4—81no—
185C2×D4—81no—
186C2×D4C2×D412yes—
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×C4—21no—
205C22×C4—42no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4C2411yescharacteristic
213SD16—44no—
214D8—44no—
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—44no—
225C2×C8—44no—
226C2×C8C2×D412yes—
227C2×C8C2×D412yes—
228C2×C8C2×D412yes—
229C4⋊C4—22no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4C2×D412yes—
241C4⋊C4#4112yes—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—81no—
254C22⋊C4—81no—
255C42—81no—
256C42C2×D412yes—
257C42#4112yes—
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—22no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23#13512yes—
285C23#13512yes—
286C23C22×D411yescharacteristic
287Q8—41no—
288Q8—41no—
289Q8—41no—
290Q8—41no—
291D4—28no—
292D4—22no—
293D4—22no—
294D4—41no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—41no—
299D4—41no—
300D4—84no—
301D4—84no—
302D4—84no—
303D4—84no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
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×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—84no—
341C2×C4#12112yes—
342C2×C4#12112yes—
343C2×C4C22×D411yescharacteristic
344C2×C4C22×D411yescharacteristic
345C2×C4C2×#4111yescharacteristic
346C2×C4C2×#4111yescharacteristic
347C2×C4H(5,𝔽2)11yescharacteristic
348C2×C4H(5,𝔽2)11yescharacteristic
349C8—22no—
350C8—22no—
351C8—22no—
352C8—41no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22C2×#12111yescharacteristic
382C22D4211yescharacteristic
383C22#54611yescharacteristic
384C22C2×#13511yescharacteristic
385C22C2×#13511yescharacteristic
386C22#57611yescharacteristic
387C22#57711yescharacteristic
388C4—22no—
389C4—22no—
390C4—41no—
391C4—41no—
392C4—42no—
393C4—42no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4#53211yescharacteristic
399C4#54011yescharacteristic
400C4D4211yescharacteristic
401C4#55011yescharacteristic
402C2—42no—
403C2—42no—
404C2—82no—
405C2—161no—
406C2#242811yescharacteristic
407C2#305011yescharacteristic
408C2#306211yescharacteristic
409C2#328011yescharacteristic
410C2#328211yescharacteristic
411C2#329711yescharacteristic
412C2#330111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#466C2semidirect
2. C2×#465C2semidirect
3. C2×#463C2semidirect
4. C2×C8×D4C2semidirect
5. #2540C2semidirect
6. #2535C2semidirect
7. #2036C2semidirect
8. #2034C2semidirect
9. #2000C2semidirect
10. #1999C2semidirect
11. #1958C2semidirect
12. #1949C2semidirect
13. #1893C2semidirect
14. #1892C2semidirect
15. C4×#102C2semidirect
133. C4⋊C8D4semidirect
141. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4662
C2×#4652
C2×#4632
C2×C8×D42
#25402
#25352
#20362
#20342
#20002
#19992
#19582
#19492
#18932
#18922
C4×#1022