Finite Group Databasebeta

← #32910

Subgroups — #32910

Order 256.

747 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×C4×#41C211yesmaximal, characteristic, complemented
2C2×#464C211yesmaximal, characteristic, complemented
3C2×#463C211yesmaximal, characteristic, complemented
4C2×#452C211yesmaximal, characteristic, complemented
5C2×#449C211yesmaximal, characteristic
6#2850C211yesmaximal, characteristic, complemented
7#1880C211yesmaximal, characteristic, complemented
8#1879C211yesmaximal, characteristic, complemented
9#1878C211yesmaximal, characteristic, complemented
10#1877C211yesmaximal, characteristic, complemented
11#1875C211yesmaximal, characteristic, complemented
12#1873C211yesmaximal, characteristic, complemented
13#1805C211yesmaximal, characteristic, complemented
14#1804C211yesmaximal, characteristic, complemented
15#1797C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×D8C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C4×#41—24no—
25C4×#41—24no—
26C2×C4×Q8C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28C2×C4×D4—21no—
29C2×C4×D4—21no—
30C2×C4×D4C2211yescharacteristic
31C2×C4×D4C2211yescharacteristic
32C2×#117—21no—
33C2×#117C2211yescharacteristic
34C22×C42—21no—
35C22×C42C2211yescharacteristic
36#464—24no—
37#463—24no—
38#452—24no—
39#449—24no—
40C2×C4⋊C8C2211yescharacteristic
41C2×C4⋊C8C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×#103C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#102C2211yescharacteristic
48C2×#102C2211yescharacteristic
49C2×#102C2211yescharacteristic
50C2×C22⋊C8C2211yescharacteristic
51C2×C22⋊C8C2211yescharacteristic
52#391C2211yescharacteristic
53#390C2211yescharacteristic
54#388—21no—
55#388—21no—
56#386—21no—
57#386—21no—
58#382C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63#336C2211yescharacteristic
64#336C2211yescharacteristic
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—22no—
70C2×#41D414yes—
71C22×Q8C2311yescharacteristic
72C22×Q8C2311yescharacteristic
73C22×D4—21no—
74C22×D4—21no—
75C22×D4C2311yescharacteristic
76C22×D4C2311yescharacteristic
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4D411yescharacteristic
81C23×C4D411yescharacteristic
82C23×C4C2311yescharacteristic
83C2×Q16—22no—
84C2×Q16—22no—
85C2×Q16—22no—
86C2×SD16—22no—
87C2×SD16—22no—
88C2×SD16—22no—
89C2×SD16—22no—
90C2×SD16—22no—
91C2×SD16—22no—
92C2×D8—22no—
93C2×D8—22no—
94C2×D8—22no—
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99#124—22no—
100#124—22no—
101C22⋊Q8—22no—
102C22⋊Q8—22no—
103#121—22no—
104#121—22no—
105C4×Q8—22no—
106C4×Q8—22no—
107C4×Q8—22no—
108C4×Q8D414yescomplemented
109C4×D4—28no—
110C4×D4—28no—
111C4×D4—22no—
112C4×D4—22no—
113C4×D4—22no—
114C4×D4D414yescomplemented
115#117—28no—
116#117—22no—
117#117—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4C2311yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
123C2×C4⋊C4C2311yescharacteristic
124C2×C4⋊C4C2311yescharacteristic
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C42—28no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—22no—
139C2×C42—22no—
140C2×C42D411yescharacteristic
141C2×C42D411yescharacteristic
142C2×C42C2311yescharacteristic
143C2×C42C2311yescharacteristic
144C4⋊C8—22no—
145C4⋊C8—22no—
146#103—22no—
147#103—22no—
148#103—22no—
149#103—22no—
150#102—22no—
151#102—22no—
152#102—22no—
153#102—22no—
154C22⋊C8—22no—
155C22⋊C8—22no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—21no—
160C24—21no—
161C24—21no—
162C24C2×D411yescharacteristic
163#41—28no—
164#41—28no—
165#41—28no—
166#41—28no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8C2×D412yes—
177C2×Q8C2×D412yes—
178C2×Q8#4112yes—
179C2×D4—24no—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4C2×D412yes—
195C2×D4C2×D412yes—
196C2×D4#4112yes—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—24no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4C2×D411yescharacteristic
223C22×C4C2×D411yescharacteristic
224C22×C4C2×D411yescharacteristic
225C22×C4C2×D411yescharacteristic
226C22×C4C2×D412yes—
227C22×C4C2×D411yescharacteristic
228C22×C4#4112yes—
229C22×C4C2411yescharacteristic
230Q16—44no—
231SD16—44no—
232SD16—44no—
233D8—44no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—42no—
255C4⋊C4C2×D412yes—
256C4⋊C4C2×D412yes—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—24no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C42—24no—
267C42—24no—
268C42—22no—
269C42—24no—
270C42—22no—
271C42—21no—
272C42—21no—
273C42C2×D412yes—
274C42C2×D412yes—
275C23—21no—
276C23—22no—
277C23—22no—
278C23—21no—
279C23—22no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23#12112yes—
296C23C22×D411yescharacteristic
297Q8—22no—
298Q8—22no—
299Q8—22no—
300Q8—41no—
301Q8—41no—
302Q8—41no—
303Q8—41no—
304Q8#12114yes—
305D4—216no—
306D4—22no—
307D4—22no—
308D4—22no—
309D4—41no—
310D4—41no—
311D4—41no—
312D4—41no—
313D4—84no—
314D4#12114yes—
315C2×C4—216no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—44no—
337C2×C4—48no—
338C2×C4—44no—
339C2×C4—48no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4#12012yes—
354C2×C4#12012yes—
355C2×C4#12112yes—
356C2×C4#13512yes—
357C2×C4C22×D411yescharacteristic
358C2×C4C22×D411yescharacteristic
359C2×C4C22×D411yescharacteristic
360C2×C4C2×#4111yescharacteristic
361C2×C4C2×#4111yescharacteristic
362C2×C4C2×#4111yescharacteristic
363C8—41no—
364C8—41no—
365C8—41no—
366C8—41no—
367C22—22no—
368C22—24no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—42no—
373C22—41no—
374C22—42no—
375C22—41no—
376C22—42no—
377C22—41no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22C2×#12011yescharacteristic
389C22C2×#12111yescharacteristic
390C22#52511yescharacteristic
391C22C2×#13511yescharacteristic
392C22#57511yescharacteristic
393C22#57511yescharacteristic
394C22#57511yescharacteristic
395C4—22no—
396C4—24no—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—42no—
401C4—41no—
402C4—42no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4C2×#12111yescharacteristic
407C4C2×#12111yescharacteristic
408C4#52511yescharacteristic
409C4#52511yescharacteristic
410C2—42no—
411C2—44no—
412C2—161no—
413C2#284711yescharacteristic
414C2#300211yescharacteristic
415C2#300511yescharacteristic
416C2#303511yescharacteristic
417C2#304011yescharacteristic
418C2#310211yescharacteristic
419C2#311111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C4×#41C2semidirect
2. C2×#464C2semidirect
3. C2×#463C2semidirect
4. C2×#452C2semidirect
6. #2850C2semidirect
7. #1880C2semidirect
8. #1879C2semidirect
9. #1878C2semidirect
10. #1877C2semidirect
11. #1875C2semidirect
12. #1873C2semidirect
13. #1805C2semidirect
14. #1804C2semidirect
15. #1797C2semidirect
108. C4×Q8D4semidirect
114. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#412
C2×#4642
C2×#4632
C2×#4522
C2×#4492
#28502
#18802
#18792
#18782
#18772
#18752
#18732
#18052
#18042
#17972