Finite Group Databasebeta

← #21519

Subgroups — #21519

Order 256.

451 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3035C211yesmaximal, characteristic
2C2×#444C211yesmaximal, characteristic, complemented
3C2×#443C211yesmaximal, characteristic, complemented
4#2932C211yesmaximal, characteristic, complemented
5#2875C211yesmaximal, characteristic, complemented
6#1980C211yesmaximal, characteristic, complemented
7#1950C211yesmaximal, characteristic, complemented
8#1929C211yesmaximal, characteristic, complemented
9#1929C211yesmaximal, characteristic, complemented
10#1876C211yesmaximal, characteristic, complemented
11#1876C211yesmaximal, characteristic, complemented
12#1806C211yesmaximal, characteristic, complemented
13#1771C211yesmaximal, characteristic, complemented
14#1765C211yesmaximal, characteristic, complemented
15#1764C211yesmaximal, characteristic, complemented
16C2×#135C2211yescharacteristic
17C2×#135C2211yescharacteristic
18C2×#131C2211yescharacteristic
19C2×#131C2211yescharacteristic
20C2×#131C2211yescharacteristic
21C2×#131C2211yescharacteristic
22#532C2211yescharacteristic
23C4×#41C2211yescharacteristic
24#464C411yescharacteristic, complemented
25#464C411yescharacteristic, complemented
26#463C411yescharacteristic, complemented
27#463C411yescharacteristic, complemented
28#462C411yescharacteristic, complemented
29#461C411yescharacteristic, complemented
30#460C411yescharacteristic, complemented
31#459C411yescharacteristic, complemented
32#444—24no—
33#443—24no—
34#435—21no—
35#435—21no—
36#435—21no—
37#435—21no—
38C2×#108C2211yescharacteristic
39C2×#108C2211yescharacteristic
40#423C2211yescharacteristic
41C2×C4⋊C8C2211yescharacteristic
42C2×#104C2211yescharacteristic
43C2×#104C2211yescharacteristic
44C2×#104C2211yescharacteristic
45C2×#104C2211yescharacteristic
46C2×#104C2211yescharacteristic
47C2×#104C2211yescharacteristic
48#416C2211yescharacteristic
49#416C2211yescharacteristic
50#408C2211yescharacteristic
51#408C2211yescharacteristic
52#405—21no—
53#405—21no—
54#405C2211yescharacteristic
55#405C2211yescharacteristic
56C4×M16—21no—
57C4×M16C2211yescharacteristic
58C2×#97—21no—
59C2×#97C2211yescharacteristic
60C2×#97C2211yescharacteristic
61C2×C4×C8C2211yescharacteristic
62C4×C4⋊C4C2211yescharacteristic
63#341C2211yescharacteristic
64#341C2211yescharacteristic
65#339C2211yescharacteristic
66#339C2211yescharacteristic
67#336C2211yescharacteristic
68#336C2211yescharacteristic
69C2×#41C2311yescharacteristic
70C2×#41C2311yescharacteristic
71C2×#41C2311yescharacteristic
72#135—22no—
73#135—22no—
74#135—22no—
75#135—22no—
76C2×Q16C2×C411yescharacteristic
77C2×Q16C2×C411yescharacteristic
78C2×SD16C2×C411yescharacteristic
79C2×SD16C2×C411yescharacteristic
80C2×SD16C2×C411yescharacteristic
81C2×SD16C2×C411yescharacteristic
82C2×D8C2×C411yescharacteristic
83C2×D8C2×C411yescharacteristic
84#131—22no—
85#131—22no—
86#131—22no—
87#131—22no—
88#131—22no—
89#131—22no—
90#131—22no—
91#131D414yescomplemented
92C2×M16—21no—
93C2×M16—21no—
94C2×M16—21no—
95C2×M16—21no—
96C2×M16C2311yescharacteristic
97C2×M16C2311yescharacteristic
98C2×M16C2311yescharacteristic
99C2×M16C2311yescharacteristic
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8C2311yescharacteristic
105C22×C8C2311yescharacteristic
106C22×C8C2311yescharacteristic
107C22×C8C2311yescharacteristic
108C4⋊Q8C2×C411yescharacteristic
109#127C2×C411yescharacteristic
110#124C2×C411yescharacteristic
111#124C2×C411yescharacteristic
112#121—21no—
113#121—21no—
114C4×Q8C2×C411yescharacteristic
115C4×Q8C2×C411yescharacteristic
116C4×D4—21no—
117C4×D4—21no—
118C4×D4—21no—
119C4×D4—21no—
120C4×D4C2×C411yescharacteristic
121C4×D4C2×C411yescharacteristic
122#117—21no—
123#117C2311yescharacteristic
124C2×C4⋊C4—21no—
125C2×C42—21no—
126C2×C42C2311yescharacteristic
127C2×C42C2311yescharacteristic
128C2×C42C2311yescharacteristic
129#108—22no—
130#108—22no—
131C4⋊C8—21no—
132C4⋊C8—21no—
133C4⋊C8C2×C411yescharacteristic
134C4⋊C8C2×C411yescharacteristic
135C4⋊C8C2×C411yescharacteristic
136C4⋊C8C2×C411yescharacteristic
137#104—22no—
138#104—22no—
139#104—22no—
140#104—22no—
141#104—22no—
142#104—22no—
143#103C2×C411yescharacteristic
144#103C2×C411yescharacteristic
145#103C2×C411yescharacteristic
146#103C2×C411yescharacteristic
147#102C2×C411yescharacteristic
148#102C2×C411yescharacteristic
149#102C2×C411yescharacteristic
150#102C2×C411yescharacteristic
151C22⋊C8—21no—
152C22⋊C8—21no—
153C22⋊C8—21no—
154C22⋊C8—21no—
155#97—21no—
156#97—21no—
157#97—22no—
158#97—21no—
159#97D411yescharacteristic
160#97D411yescharacteristic
161#97D411yescharacteristic
162#97D411yescharacteristic
163C4×C8—22no—
164C4×C8—21no—
165C4×C8—21no—
166C4×C8—21no—
167#95—21no—
168#41—21no—
169#41—21no—
170#41—21no—
171#41—21no—
172#41—21no—
173#41—21no—
174#41—21no—
175#41—21no—
176#41—21no—
177#41—21no—
178#41C2×D412yes—
179#41#4112yes—
180C2×Q8C22×C411yescharacteristic
181C2×Q8C22×C411yescharacteristic
182C2×Q8C22×C411yescharacteristic
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4C22×C411yescharacteristic
189C2×D4C22×C411yescharacteristic
190C2×D4C22×C411yescharacteristic
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4#4111yescharacteristic
199C22×C4#4111yescharacteristic
200C22×C4C2411yescharacteristic
201Q16—22no—
202Q16—22no—
203SD16—22no—
204SD16—22no—
205SD16—22no—
206SD16—22no—
207D8—22no—
208D8—22no—
209M16—24no—
210M16—21no—
211M16—21no—
212M16—24no—
213M16—21no—
214M16—21no—
215M16—21no—
216M16—21no—
217M16—42no—
218M16—42no—
219M16C2×D412yes—
220M16#4112yes—
221C2×C8—24no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—24no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8C22×C411yescharacteristic
236C2×C8C22×C411yescharacteristic
237C2×C8C22×C411yescharacteristic
238C2×C8C22×C411yescharacteristic
239C2×C8C2×D411yescharacteristic
240C2×C8C2×D411yescharacteristic
241C2×C8C2×D412yes—
242C2×C8C2×D411yescharacteristic
243C2×C8C2×D411yescharacteristic
244C2×C8#4112yes—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—42no—
250C4⋊C4C22×C411yescharacteristic
251C4⋊C4C22×C411yescharacteristic
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42C22×C411yescharacteristic
267C42C22×C411yescharacteristic
268C42C2×D411yescharacteristic
269C42C2×D411yescharacteristic
270C23—21no—
271C23—21no—
272C23—21no—
273C23C2×#4111yescharacteristic
274Q8—21no—
275Q8—21no—
276Q8—21no—
277Q8—21no—
278Q8—21no—
279Q8C4×D412yes—
280D4—22no—
281D4—22no—
282D4—21no—
283D4—21no—
284D4—21no—
285D4—21no—
286D4—21no—
287D4—41no—
288D4—41no—
289D4—41no—
290D4—41no—
291D4—81no—
292D4—81no—
293D4C4×D412yes—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
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—81no—
314C2×C4—81no—
315C2×C4C4×D411yescharacteristic
316C2×C4C4×D411yescharacteristic
317C2×C4C4×D411yescharacteristic
318C2×C4C4×D411yescharacteristic
319C2×C4C23×C411yescharacteristic
320C2×C4C22×D411yescharacteristic
321C2×C4C22×D411yescharacteristic
322C2×C4C2×#4111yescharacteristic
323C2×C4C2×#4111yescharacteristic
324C2×C4C2×#4111yescharacteristic
325C8—21no—
326C8—21no—
327C8—21no—
328C8—21no—
329C8—22no—
330C8—21no—
331C8—22no—
332C8—21no—
333C8—41no—
334C8—41no—
335C8#12112yes—
336C8#12112yes—
337C22—21no—
338C22—21no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22C2×C4×D411yescharacteristic
349C22#52911yescharacteristic
350C22#53211yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—22no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—81no—
362C4C2×C4×D411yescharacteristic
363C4C4×#4111yescharacteristic
364C4C2×#12111yescharacteristic
365C4#52511yescharacteristic
366C2—21no—
367C2—21no—
368C2—42no—
369C2—81no—
370C2—81no—
371C2C4×#12111yescharacteristic
372C2#295611yescharacteristic
373C2#295711yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC22C2×C4×D4
prime radical#3035C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#97 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#444C2semidirect
3. C2×#443C2semidirect
4. #2932C2semidirect
5. #2875C2semidirect
6. #1980C2semidirect
7. #1950C2semidirect
8. #1929C2semidirect
9. #1929C2semidirect
10. #1876C2semidirect
11. #1876C2semidirect
12. #1806C2semidirect
13. #1771C2semidirect
14. #1765C2semidirect
15. #1764C2semidirect
24. #464C4semidirect
25. #464C4semidirect
26. #463C4semidirect
27. #463C4semidirect
28. #462C4semidirect
29. #461C4semidirect
30. #460C4semidirect
31. #459C4semidirect
91. #131D4semidirect

Maximal subgroups

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

subgroupindex
#30352
C2×#4442
C2×#4432
#29322
#28752
#19802
#19502
#19292
#18762
#18062
#17712
#17652
#17642