Finite Group Databasebeta

← #23193

Subgroups — #23193

Order 256.

563 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#496C211yesmaximal, characteristic
2C2×#442C211yesmaximal, characteristic, complemented
3C2×C4×SD16C211yesmaximal, characteristic, complemented
4#2367C211yesmaximal, characteristic, complemented
5#2366C211yesmaximal, characteristic, complemented
6#1983C211yesmaximal, characteristic, complemented
7#1982C211yesmaximal, characteristic, complemented
8#1973C211yesmaximal, characteristic, complemented
9#1970C211yesmaximal, characteristic, complemented
10#1880C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic, complemented
13#1877C211yesmaximal, characteristic, complemented
14#1776C211yesmaximal, characteristic, complemented
15#1771C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C2×#127C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4—21no—
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27#496C418yescomplemented
28#442—24no—
29C4×SD16—24no—
30C2×#107C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#97C2211yescharacteristic
42C2×#97C2211yescharacteristic
43C2×C4×C8C2211yescharacteristic
44#391C2211yescharacteristic
45#390—21no—
46#390C2211yescharacteristic
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#388—21no—
50#388—21no—
51#386—21no—
52#386—21no—
53#385—21no—
54C4×C4⋊C4C2211yescharacteristic
55C4×C4⋊C4C2211yescharacteristic
56C4×C4⋊C4C2211yescharacteristic
57C4×C22⋊C4—21no—
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×D4—21no—
64C22×D4—21no—
65C22×D4C2311yescharacteristic
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C2×SD16—22no—
72C2×SD16—22no—
73C2×SD16—41no—
74C2×SD16—41no—
75C2×SD16C2×C414yescomplemented
76C2×SD16C2×C414yescomplemented
77C2×D8—22no—
78C2×D8—22no—
79C2×D8—41no—
80C2×D8—41no—
81C2×D8C2×C414yescomplemented
82C2×D8C2×C414yescomplemented
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87#127—22no—
88#127C2×C414yescomplemented
89#124—22no—
90#124C2×C414yescomplemented
91C4×Q8—22no—
92C4×Q8—41no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—41no—
97C4×D4—41no—
98C4×D4—41no—
99C4×D4—44no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4D411yescharacteristic
102C2×C4⋊C4D411yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42C2311yescharacteristic
122C2×C42C2311yescharacteristic
123C2×C42C2311yescharacteristic
124#107—22no—
125#106D414yescomplemented
126C4⋊C8—22no—
127#103—22no—
128#103—41no—
129#102—22no—
130#102—22no—
131#102—22no—
132#102—41no—
133#102—41no—
134#102—41no—
135#97—22no—
136#97C2×C414yescomplemented
137C4×C8—22no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95D411yescharacteristic
143#95D411yescharacteristic
144C24—21no—
145C24—21no—
146C24—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8C22×C412yes—
153C2×D4—24no—
154C2×D4—24no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4C22×C412yes—
175C2×D4C22×C412yes—
176C2×D4C22×C412yes—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2411yescharacteristic
198SD16—28no—
199SD16—44no—
200D8—28no—
201D8—44no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8C22×C412yes—
211C2×C8C22×C412yes—
212C2×C8C2×D412yes—
213C2×C8#4112yes—
214C4⋊C4—22no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C4⋊C4C2×D412yes—
224C4⋊C4C2×D412yes—
225C22⋊C4—24no—
226C22⋊C4—24no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—81no—
234C22⋊C4—81no—
235C22⋊C4—81no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—42no—
241C42—42no—
242C42—81no—
243C42C22×C412yes—
244C42#4112yes—
245C23—22no—
246C23—22no—
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—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23C22×D411yescharacteristic
267Q8—22no—
268Q8—22no—
269Q8—41no—
270Q8—41no—
271D4—22no—
272D4—22no—
273D4—22no—
274D4—22no—
275D4—22no—
276D4—22no—
277D4—48no—
278D4—41no—
279D4—41no—
280D4—41no—
281D4—41no—
282D4—41no—
283D4—41no—
284D4—84no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
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—84no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4C4×D412yes—
319C2×C4C4×D412yes—
320C2×C4AGL(1,ℤ/8)12yes—
321C2×C4C23×C411yescharacteristic
322C2×C4C22×D411yescharacteristic
323C2×C4C2×#4111yescharacteristic
324C2×C4C2×#4111yescharacteristic
325C2×C4H(5,𝔽2)11yescharacteristic
326C2×C4H(5,𝔽2)11yescharacteristic
327C8—22no—
328C8—41no—
329C8—41no—
330C8C4×D414yescomplemented
331C22—42no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22C2×C4×D411yescharacteristic
353C22D4211yescharacteristic
354C22#55011yescharacteristic
355C22C2×AGL(1,ℤ/8)11yescharacteristic
356C22#57511yescharacteristic
357C22#57611yescharacteristic
358C22#57711yescharacteristic
359C4—22no—
360C4—22no—
361C4—22no—
362C4—42no—
363C4—41no—
364C4—42no—
365C4—41no—
366C4—41no—
367C4—42no—
368C4—81no—
369C4—81no—
370C4C2×C4×D411yescharacteristic
371C4#51811yescharacteristic
372C4#55011yescharacteristic
373C4#55911yescharacteristic
374C2—82no—
375C2—82no—
376C2—82no—
377C2#236611yescharacteristic
378C2C4×AGL(1,ℤ/8)11yescharacteristic
379C2#294611yescharacteristic
380C2#326511yescharacteristic
381C2#326811yescharacteristic
382C2#339911yescharacteristic
383C2#340111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#496C2
LCSG ▷ C2×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×#442C2semidirect
3. C2×C4×SD16C2semidirect
4. #2367C2semidirect
5. #2366C2semidirect
6. #1983C2semidirect
7. #1982C2semidirect
8. #1973C2semidirect
9. #1970C2semidirect
10. #1880C2semidirect
11. #1879C2semidirect
12. #1879C2semidirect
13. #1877C2semidirect
14. #1776C2semidirect
15. #1771C2semidirect
27. #496C4semidirect
75. C2×SD16C2×C4semidirect
76. C2×SD16C2×C4semidirect
81. C2×D8C2×C4semidirect
82. C2×D8C2×C4semidirect
88. #127C2×C4semidirect
90. #124C2×C4semidirect
125. #106D4semidirect
136. #97C2×C4semidirect
330. C8C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4962
C2×#4422
C2×C4×SD162
#23672
#23662
#19832
#19822
#19732
#19702
#18802
#18792
#18772
#17762
#17712