Finite Group Databasebeta

← #18518

Subgroups — #18518

Order 256.

484 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#41C211yesmaximal, characteristic, complemented
2#3167C211yesmaximal, characteristic, complemented
3#3103C211yesmaximal, characteristic, complemented
4#3035C211yesmaximal, characteristic
5#1653C211yesmaximal, characteristic, complemented
6#1651C211yesmaximal, characteristic, complemented
7#1645C211yesmaximal, characteristic, complemented
8#1643C211yesmaximal, characteristic, complemented
9#1627C211yesmaximal, characteristic, complemented
10#1626C211yesmaximal, characteristic, complemented
11#1623C211yesmaximal, characteristic, complemented
12#1622C211yesmaximal, characteristic, complemented
13#1542C211yesmaximal, characteristic, complemented
14#1476C211yesmaximal, characteristic, complemented
15C2×Q8⋊C8C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17Q82C2211yescharacteristic
18#557—21no—
19#557C2211yescharacteristic
20#554—22no—
21#554—21no—
22#554—21no—
23#554C2211yescharacteristic
24D4×Q8—21no—
25D4×Q8C2211yescharacteristic
26#533—22no—
27#533—21no—
28#533—21no—
29#533C2211yescharacteristic
30#532C2211yescharacteristic
31C4×#41—21no—
32C4×#41C2211yescharacteristic
33C2×C4×Q8—21no—
34C2×C4×Q8C2211yescharacteristic
35#498C2211yescharacteristic
36#492C2211yescharacteristic
37#487C2211yescharacteristic
38#486C2211yescharacteristic
39#481—21no—
40#475C2211yescharacteristic
41#475C2211yescharacteristic
42#465—22no—
43#464—21no—
44#463—21no—
45#462C2211yescharacteristic
46#461C2211yescharacteristic
47#460C2211yescharacteristic
48#460C2211yescharacteristic
49#460C2211yescharacteristic
50#459C2211yescharacteristic
51#448—21no—
52C4×SD16—21no—
53C4×SD16—21no—
54C4×SD16—22no—
55#431C2211yescharacteristic
56C2×C4⋊C8C2211yescharacteristic
57#416—21no—
58C2×C4×C8C2211yescharacteristic
59#333C2211yescharacteristic
60#333C2211yescharacteristic
61#332C2211yescharacteristic
62#332C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64Q8⋊C8C2211yescharacteristic
65Q8⋊C8C2211yescharacteristic
66Q8⋊C8C2211yescharacteristic
67Q8⋊C8C2211yescharacteristic
68Q8⋊C8C2211yescharacteristic
69D4⋊C8C2211yescharacteristic
70D4⋊C8C2211yescharacteristic
71C2×#41—21no—
72C2×#41D411yescharacteristic
73C22×Q8—21no—
74C22×Q8D411yescharacteristic
75#135—44no—
76C2×Q16—21no—
77C2×SD16—21no—
78C2×SD16—21no—
79C2×SD16—21no—
80C2×SD16—22no—
81C2×SD16—21no—
82C2×D8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C4⋊Q8—22no—
86C4⋊Q8—21no—
87C4⋊Q8—22no—
88C4⋊Q8—21no—
89C4⋊Q8—21no—
90C4⋊Q8—21no—
91C4⋊Q8C2311yescharacteristic
92C4⋊Q8C2311yescharacteristic
93C4⋊Q8C2311yescharacteristic
94#127C2311yescharacteristic
95#125—22no—
96#125—22no—
97#125—22no—
98#125—21no—
99#125—21no—
100#125—21no—
101#124—21no—
102C22⋊Q8—22no—
103C22⋊Q8—22no—
104C22⋊Q8—22no—
105C22⋊Q8—22no—
106C22⋊Q8—22no—
107C22⋊Q8—21no—
108C22⋊Q8—22no—
109C22⋊Q8—21no—
110C22⋊Q8—21no—
111C22⋊Q8—22no—
112C22⋊Q8D412yes—
113#121—21no—
114#121—21no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×Q8—21no—
119C4×Q8—21no—
120C4×Q8—21no—
121C4×Q8—21no—
122C4×Q8—21no—
123C4×Q8—21no—
124C4×Q8—21no—
125C4×Q8D411yescharacteristic
126C4×Q8D411yescharacteristic
127C4×Q8D411yescharacteristic
128C4×Q8D412yes—
129C4×Q8C2311yescharacteristic
130C4×Q8C2311yescharacteristic
131C4×Q8C2311yescharacteristic
132C4×D4—22no—
133C4×D4—22no—
134C4×D4—21no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4—21no—
138C4×D4—21no—
139C4×D4—21no—
140C4×D4D411yescharacteristic
141C4×D4C2311yescharacteristic
142#117—22no—
143#117—21no—
144#117—21no—
145#117D411yescharacteristic
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4D411yescharacteristic
150C2×C42—22no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42C2311yescharacteristic
154#106—22no—
155#106—21no—
156#106—21no—
157C4⋊C8—21no—
158C4⋊C8C2311yescharacteristic
159C4⋊C8C2311yescharacteristic
160C4⋊C8C2311yescharacteristic
161C4⋊C8C2311yescharacteristic
162#103—21no—
163#103—21no—
164#103—22no—
165#103—21no—
166#103—21no—
167#102—21no—
168#102—21no—
169#102—21no—
170#102—21no—
171#102—22no—
172C22⋊C8—21no—
173C4×C8—21no—
174C4×C8—21no—
175C4×C8C2311yescharacteristic
176C4×C8C2311yescharacteristic
177#41—22no—
178#41—42no—
179#41—42no—
180#41SD1614yescomplemented
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—22no—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8C2×D411yescharacteristic
191C2×Q8C2×D411yescharacteristic
192C2×Q8C2×D411yescharacteristic
193C2×Q8C2×D412yes—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—41no—
199C2×D4C2×D411yescharacteristic
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—41no—
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211Q16—42no—
212SD16—44no—
213SD16—42no—
214SD16—42no—
215SD16—42no—
216SD16—42no—
217D8—42no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—42no—
225C2×C8—42no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—21no—
235C4⋊C4—22no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—22no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4C2×D411yescharacteristic
249C4⋊C4C2×D411yescharacteristic
250C4⋊C4C2×D411yescharacteristic
251C4⋊C4C2×D411yescharacteristic
252C4⋊C4C2×D412yes—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C42—22no—
262C42—22no—
263C42—21no—
264C42—21no—
265C42—22no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42C2×D411yescharacteristic
272C42C2×D411yescharacteristic
273C42C2×D411yescharacteristic
274C42C2411yescharacteristic
275C23—21no—
276C23—41no—
277C23#12011yescharacteristic
278Q8—21no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—42no—
283Q8—42no—
284Q8—44no—
285Q8—41no—
286Q8—41no—
287Q8—41no—
288Q8—41no—
289Q8—82no—
290Q8C2×SD1612yes—
291Q8#13512yescomplemented
292D4—24no—
293D4—21no—
294D4—42no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—41no—
299D4—82no—
300D4C2×SD1612yes—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—24no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4#12011yescharacteristic
324C2×C4#12011yescharacteristic
325C2×C4#12011yescharacteristic
326C2×C4C2×SD1612yes—
327C2×C4#13712yes—
328C2×C4C22×D411yescharacteristic
329C2×C4C22×D411yescharacteristic
330C2×C4C22×D411yescharacteristic
331C8—22no—
332C8—41no—
333C8—41no—
334C8—41no—
335C8—41no—
336C8—41no—
337C22—21no—
338C22—21no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22#44812yes—
346C22C2×#12011yescharacteristic
347C4—22no—
348C4—21no—
349C4—22no—
350C4—21no—
351C4—21no—
352C4—41no—
353C4—42no—
354C4—41no—
355C4—41no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—81no—
360C4#44812yes—
361C4C22×SD1611yescharacteristic
362C4C2×#13511yescharacteristic
363C4C2×#13711yescharacteristic
364C4#57511yescharacteristic
365C2—22no—
366C2—42no—
367C2—161no—
368C2C2×#44811yescharacteristic
369C2#300211yescharacteristic
370C2#301811yescharacteristic
subgroupG/H
centerC2×C4#120
commutator, Frattini, cosocleC42C24
socleC22C2×#120
prime radical#3035C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. Q8×#41C2semidirect
2. #3167C2semidirect
3. #3103C2semidirect
5. #1653C2semidirect
6. #1651C2semidirect
7. #1645C2semidirect
8. #1643C2semidirect
9. #1627C2semidirect
10. #1626C2semidirect
11. #1623C2semidirect
12. #1622C2semidirect
13. #1542C2semidirect
14. #1476C2semidirect
15. C2×Q8⋊C8C2semidirect
180. #41SD16semidirect
291. Q8#135semidirect

Maximal subgroups

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

subgroupindex
Q8×#412
#31672
#31032
#30352
#16532
#16512
#16452
#16432
#16272
#16262
#16232
#16222
#15422
#14762
C2×Q8⋊C82