Finite Group Databasebeta

← #30741

Subgroups — #30741

Order 256.

480 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#492C211yesmaximal, characteristic
2C2×#461C211yesmaximal, characteristic, complemented
3#2546C211yesmaximal, characteristic, complemented
4#2445C211yesmaximal, characteristic
5#2055C211yesmaximal, characteristic, complemented
6#2043C211yesmaximal, characteristic, complemented
7#2014C211yesmaximal, characteristic
8#2013C211yesmaximal, characteristic, complemented
9#2013C211yesmaximal, characteristic, complemented
10#2013C211yesmaximal, characteristic, complemented
11#1933C211yesmaximal, characteristic, complemented
12#1916C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1880C211yesmaximal, characteristic, complemented
15#1877C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×SD16C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23C2×C4⋊Q8C2211yescharacteristic
24C2×C4⋊Q8C2211yescharacteristic
25C2×#127C2211yescharacteristic
26C2×#123—21no—
27C2×C4×Q8C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29#492—24no—
30#461—24no—
31C2×C4⋊C8C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic
41#398—21no—
42#398—21no—
43#394—21no—
44#394—21no—
45#391C2211yescharacteristic
46#391C2211yescharacteristic
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#390C2211yescharacteristic
50#389—21no—
51#389—21no—
52#389C2211yescharacteristic
53#386—21no—
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4—21no—
65C22×D4—21no—
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C2×SD16—22no—
70C2×SD16—22no—
71C2×SD16—22no—
72C2×SD16—22no—
73C2×SD16—22no—
74C2×SD16—22no—
75C2×SD16—41no—
76C2×SD16—41no—
77C2×SD16—41no—
78C2×SD16—41no—
79C2×SD16—41no—
80C2×SD16—41no—
81C2×SD16—41no—
82C2×SD16—41no—
83C2×SD16—41no—
84C2×SD16—41no—
85C2×SD16—41no—
86C2×SD16D414yes—
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C4⋊Q8—22no—
92C4⋊Q8—22no—
93C4⋊Q8—41no—
94C4⋊Q8—41no—
95C4⋊Q8—41no—
96C4⋊Q8—41no—
97#127—22no—
98#127—41no—
99#123—44no—
100C4×Q8—41no—
101C4×Q8—41no—
102C4×D4—22no—
103C4×D4—41no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4D411yescharacteristic
113C2×C4⋊C4D411yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
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×C42—21no—
125C2×C42C2311yescharacteristic
126C2×C42C2311yescharacteristic
127C2×C42C2311yescharacteristic
128C4⋊C8—22no—
129#103—22no—
130#103—22no—
131#103—22no—
132#103—41no—
133#103—41no—
134#103—41no—
135#102—41no—
136#102—41no—
137C4×C8—22no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95D411yescharacteristic
144#95D411yescharacteristic
145C24—21no—
146C24—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8C2×D412yes—
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—21no—
175C2×D4—21no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—81no—
181C2×D4—81no—
182C2×D4C2×D412yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—81no—
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2411yescharacteristic
199SD16—44no—
200SD16—44no—
201SD16—44no—
202SD16—44no—
203SD16—44no—
204SD16—44no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8C2×D412yes—
216C2×C8#4112yes—
217C4⋊C4—22no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4SD1614yescomplemented
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—81no—
238C22⋊C4—81no—
239C22⋊C4—81no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—42no—
247C23—22no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23C22×D411yescharacteristic
262Q8—22no—
263Q8—22no—
264Q8—41no—
265Q8—41no—
266Q8—41no—
267Q8—41no—
268Q8—41no—
269Q8—41no—
270Q8—41no—
271Q8—41no—
272Q8—41no—
273Q8—41no—
274D4—22no—
275D4—22no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—41no—
280D4—41no—
281D4—41no—
282D4—84no—
283D4—84no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—84no—
317C2×C4C2×SD1612yes—
318C2×C4C2×SD1612yes—
319C2×C4C2×SD1612yes—
320C2×C4#13712yes—
321C2×C4C22×D411yescharacteristic
322C2×C4C2×#4111yescharacteristic
323C2×C4H(5,𝔽2)11yescharacteristic
324C2×C4H(5,𝔽2)11yescharacteristic
325C2×C4H(5,𝔽2)11yescharacteristic
326C2×C4H(5,𝔽2)11yescharacteristic
327C8—22no—
328C8—22no—
329C8—41no—
330C8—41no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22#53511yescharacteristic
346C22D4211yescharacteristic
347C22#55011yescharacteristic
348C22C22×SD1611yescharacteristic
349C22C2×#13711yescharacteristic
350C22#57711yescharacteristic
351C22#57711yescharacteristic
352C4—22no—
353C4—22no—
354C4—22no—
355C4—22no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4#53411yescharacteristic
365C4#54011yescharacteristic
366C4#54611yescharacteristic
367C4#56011yescharacteristic
368C2—82no—
369C2—161no—
370C2#267311yescharacteristic
371C2#320611yescharacteristic
372C2#322111yescharacteristic
373C2#326611yescharacteristic
374C2D4×SD1611yescharacteristic
375C2#339211yescharacteristic
376C2#340011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#461C2semidirect
3. #2546C2semidirect
5. #2055C2semidirect
6. #2043C2semidirect
8. #2013C2semidirect
9. #2013C2semidirect
10. #2013C2semidirect
11. #1933C2semidirect
12. #1916C2semidirect
13. #1880C2semidirect
14. #1880C2semidirect
15. #1877C2semidirect
232. C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4922
C2×#4612
#25462
#24452
#20552
#20432
#20142
#20132
#19332
#19162
#18802
#18772