Finite Group Databasebeta

← #27180583

Subgroups — #27180583

Order 12928.

496 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692275C211yesmaximal, characteristic, complemented
2#20265723C211yesmaximal, characteristic, complemented
3#41×Dic101C211yesmaximal, characteristic, complemented
4#20265401C211yesmaximal, characteristic, complemented
5#20265397C211yesmaximal, characteristic, complemented
6#20265394C211yesmaximal, characteristic, complemented
7#20265390C211yesmaximal, characteristic, complemented
8#20265385C211yesmaximal, characteristic, complemented
9#20265380C211yesmaximal, characteristic, complemented
10#20265375C211yesmaximal, characteristic, complemented
11#20265370C211yesmaximal, characteristic, complemented
12C2×#15692184C211yesmaximal, characteristic, complemented
13#20265174C211yesmaximal, characteristic, complemented
14#20265173C211yesmaximal, characteristic, complemented
15C2×#135×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17#15692275—24no—
18C2×#13108064C2211yescharacteristic
19C2×#13108062C2211yescharacteristic
20C2×#13108061C2211yescharacteristic
21C2×#13108059C2211yescharacteristic
22C22×#11717579C2211yescharacteristic
23#15692253C2211yescharacteristic
24#15692251C2211yescharacteristic
25#15692244C2211yescharacteristic
26#15692242C2211yescharacteristic
27#15692227—21no—
28#15692227—21no—
29Q8×Dic101C2211yescharacteristic
30Q8×Dic101C2211yescharacteristic
31D4×Dic101—21no—
32D4×Dic101—21no—
33D4×Dic101C2211yescharacteristic
34D4×Dic101C2211yescharacteristic
35C4×C22⋊D101—21no—
36C4×C22⋊D101—21no—
37#15692214—21no—
38#15692214C2211yescharacteristic
39C2×C4×Dic101—21no—
40C2×C4×Dic101C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692184C2211yescharacteristic
43#15692184C2211yescharacteristic
44#15692184C2211yescharacteristic
45#15692180C2211yescharacteristic
46#15692180C2211yescharacteristic
47#15692179C2211yescharacteristic
48#15692179C2211yescharacteristic
49#15692177C2211yescharacteristic
50#15692177C2211yescharacteristic
51#15692176C2211yescharacteristic
52#15692176C2211yescharacteristic
53C2×#41×C101C2211yescharacteristic
54C2×#41×C101C2211yescharacteristic
55#135×C101—24no—
56C2×Q16×C101C2211yescharacteristic
57C2×SD16×C101C2211yescharacteristic
58C2×SD16×C101C2211yescharacteristic
59C2×D8×C101C2211yescharacteristic
60C22×C8×C101C2211yescharacteristic
61#13108074—22no—
62#13108074—22no—
63C2×Dic202C2311yescharacteristic
64C2×D404C2311yescharacteristic
65C2×C22⋊D101—21no—
66C2×C22⋊D101—21no—
67C2×C22⋊D101—21no—
68C22×Dic101—21no—
69C22×Dic101—21no—
70C22×Dic101D411yescharacteristic
71C22×Dic101D411yescharacteristic
72C2×C4×D101—21no—
73#13108064—22no—
74#13108062—22no—
75#13108061—22no—
76#13108059—22no—
77C2×#11717579—22no—
78C2×#11717579C2311yescharacteristic
79C2×#11717579C2311yescharacteristic
80C4⋊Dic101—21no—
81C4⋊Dic101—21no—
82C4⋊Dic101C2311yescharacteristic
83C4⋊Dic101C2311yescharacteristic
84#13108053—21no—
85#13108053—21no—
86C22⋊Dic101—21no—
87C22⋊Dic101—21no—
88C22⋊Dic101—21no—
89C22⋊Dic101—21no—
90C22⋊Dic101—21no—
91#13108051—21no—
92#13108051—21no—
93C4×Dic101—21no—
94C4×Dic101—21no—
95C4×Dic101—21no—
96C4×Dic101D411yescharacteristic
97C4×Dic101D411yescharacteristic
98C4×Dic101C2311yescharacteristic
99C4×Dic101C2311yescharacteristic
100#41×C101—22no—
101#41×C101—22no—
102#41×C101—22no—
103#41×C101D414yescomplemented
104C2×Q8×C101C2311yescharacteristic
105C2×Q8×C101C2311yescharacteristic
106C2×D4×C101—21no—
107C2×D4×C101—21no—
108C2×D4×C101C2311yescharacteristic
109C2×D4×C101C2311yescharacteristic
110C22×C4×C101—21no—
111C22×C4×C101—21no—
112C22×C4×C101C2311yescharacteristic
113Q16×C101—22no—
114SD16×C101—22no—
115SD16×C101—22no—
116D8×C101—22no—
117C2×C8×C101—22no—
118C2×C8×C101C2311yescharacteristic
119C2×C8×C101C2311yescharacteristic
120C23×C101—21no—
121C23×C101—21no—
122C23×C101C2×D411yescharacteristic
123C22×D101—21no—
124Q8×C101—21no—
125Q8×C101—21no—
126Q8×C101—21no—
127Q8×C101C2×D412yes—
128D4×C101—24no—
129D4×C101—21no—
130D4×C101—21no—
131D4×C101—21no—
132D4×C101—42no—
133D4×C101C2×D412yes—
134C2×C4×C101—21no—
135C2×C4×C101—24no—
136C2×C4×C101—21no—
137C2×C4×C101—42no—
138C2×C4×C101C2×D411yescharacteristic
139C2×C4×C101C2×D412yes—
140C2×C4×C101#4112yes—
141C2×C4×C101C2411yescharacteristic
142C22⋊D101—42no—
143C22⋊D101—42no—
144C22⋊D101—42no—
145C2×Dic101—22no—
146C2×Dic101—21no—
147C2×Dic101—21no—
148C2×Dic101—21no—
149C2×Dic101—41no—
150C2×Dic101—42no—
151C2×Dic101—42no—
152C2×Dic101C2×D411yescharacteristic
153C2×Dic101C2×D411yescharacteristic
154C2×Dic101C2×D411yescharacteristic
155C2×Dic101C2×D411yescharacteristic
156D404—21no—
157D404—21no—
158C4×D101—42no—
159Dic202—21no—
160Dic202—21no—
161C8×C101—21no—
162C8×C101—21no—
163#11717579—21no—
164#11717579—21no—
165C22×C101—21no—
166C22×C101—21no—
167C22×C101—22no—
168C22×C101—41no—
169C22×C101—41no—
170C22×C101—41no—
171C22×C101—41no—
172C22×C101—41no—
173C22×C101#12112yes—
174C22×C101C22×D411yescharacteristic
175C2×D101—41no—
176C2×D101—41no—
177C2×D101—41no—
178C4×C101—22no—
179C4×C101—41no—
180C4×C101#12112yes—
181C4×C101C22×D411yescharacteristic
182C4×C101C2×#4111yescharacteristic
183Dic101—21no—
184Dic101—21no—
185Dic101—21no—
186Dic101—41no—
187Dic101—41no—
188Dic101—41no—
189Dic101#13512yescomplemented
190C2×C101—22no—
191C2×C101—42no—
192C2×C101—81no—
193C2×C101C2×#12111yescharacteristic
194C2×C101C2×#13511yescharacteristic
195C2×C101#57511yescharacteristic
196D101—81no—
197#3035—1011nomaximal, Sylow
198C101#303511yescharacteristic, complemented, Sylow
199C2×#135—1011no—
200C2×#135D10111yescharacteristic, complemented
201#532—1011no—
202C4×#41—1011no—
203#464—1011no—
204#464—1011no—
205#463—1011no—
206#463—1011no—
207#462—1011no—
208#461—1011no—
209#460—1011no—
210#459—1011no—
211C2×C4⋊C8—1011no—
212#416—1011no—
213#416—1011no—
214C2×#41—1011no—
215C2×#41C2×D10111yescharacteristic
216C2×#41C2×D10111yescharacteristic
217#135—24no—
218#135—2024no—
219C2×Q16—1011no—
220C2×Q16C2×D10111yescharacteristic
221C2×SD16—1011no—
222C2×SD16—1011no—
223C2×SD16C2×D10111yescharacteristic
224C2×SD16C2×D10111yescharacteristic
225C2×D8—1011no—
226C2×D8C2×D10111yescharacteristic
227C22×C8—1011no—
228C22×C8C2×D10111yescharacteristic
229C4⋊Q8—1011no—
230#127—1011no—
231#124—1011no—
232#124—1011no—
233#121—2021no—
234#121—2021no—
235C4×Q8—1011no—
236C4×Q8—1011no—
237C4×D4—1011no—
238C4×D4—1011no—
239C4×D4—2021no—
240C4×D4—2021no—
241C4×D4—2021no—
242C4×D4—2021no—
243#117—1011no—
244#117—2021no—
245C2×C42—1011no—
246C2×C42—2021no—
247C4⋊C8—1011no—
248C4⋊C8—1011no—
249C4⋊C8—1011no—
250C4⋊C8—1011no—
251#103—1011no—
252#103—1011no—
253#103—1011no—
254#103—1011no—
255#102—1011no—
256#102—1011no—
257#102—1011no—
258#102—1011no—
259#41—22no—
260#41—22no—
261#41—22no—
262#41—2022no—
263#41—2022no—
264#41C22⋊D10114yescomplemented
265C2×Q8—1011no—
266C2×Q8C22×D10111yescharacteristic
267C2×Q8C22×D10111yescharacteristic
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—1011no—
271C2×D4—2021no—
272C2×D4—2021no—
273C2×D4—2021no—
274C2×D4C22×D10111yescharacteristic
275C2×D4C22×D10111yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—1011no—
279C22×C4—1011no—
280C22×C4—2021no—
281C22×C4—2021no—
282C22×C4—2021no—
283C22×C4C22×D10111yescharacteristic
284Q16—22no—
285Q16—2022no—
286SD16—22no—
287SD16—22no—
288SD16—2022no—
289SD16—2022no—
290D8—22no—
291D8—2022no—
292C2×C8—22no—
293C2×C8—1011no—
294C2×C8—1011no—
295C2×C8—2022no—
296C2×C8C22×D10111yescharacteristic
297C2×C8C22×D10111yescharacteristic
298C4⋊C4—1011no—
299C4⋊C4—1011no—
300C4⋊C4—2021no—
301C4⋊C4—2021no—
302C4⋊C4—2021no—
303C4⋊C4—2021no—
304C22⋊C4—2021no—
305C22⋊C4—2021no—
306C22⋊C4—2021no—
307C22⋊C4—2021no—
308C22⋊C4—2021no—
309C22⋊C4—2021no—
310C22⋊C4—2021no—
311C42—1011no—
312C42—1011no—
313C42—1011no—
314C42—1011no—
315C42—2021no—
316C42—2021no—
317C42—2021no—
318C23—21no—
319C23—21no—
320C23—2021no—
321C23D4×D10111yescharacteristic
322Q8—21no—
323Q8—21no—
324Q8—21no—
325Q8—2021no—
326Q8—2021no—
327Q8C2×C22⋊D10112yes—
328D4—24no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—42no—
333D4—4042no—
334D4—4042no—
335D4—4042no—
336D4—2021no—
337D4—2021no—
338D4C2×C22⋊D10112yes—
339C2×C4—21no—
340C2×C4—24no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—1011no—
344C2×C4—1011no—
345C2×C4—1011no—
346C2×C4—1011no—
347C2×C4—4042no—
348C2×C4—4042no—
349C2×C4—4042no—
350C2×C4—4041no—
351C2×C4—2022no—
352C2×C4—2021no—
353C2×C4—2021no—
354C2×C4—2021no—
355C2×C4D4×D10111yescharacteristic
356C2×C4C2×C22⋊D10112yes—
357C2×C4#1310807312yes—
358C2×C4C23×D10111yescharacteristic
359C8—21no—
360C8—21no—
361C8—2021no—
362C8—2021no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—4041no—
372C22—4041no—
373C22—4041no—
374C22#1569222712yes—
375C22C2×D4×D10111yescharacteristic
376C4—22no—
377C4—41no—
378C4—1012no—
379C4—4041no—
380C4—4041no—
381C4—4041no—
382C4—2021no—
383C4—2021no—
384C4—2021no—
385C4#1569222712yes—
386C4C22×C22⋊D10111yescharacteristic
387C4C2×#1310807311yescharacteristic
388C2—22no—
389C2—42no—
390C2—81no—
391C2—8081no—
392C2C2×#1569222711yescharacteristic
393C2#135×D10111yescharacteristic
394C2#2026597411yescharacteristic
subgroupG/H
centerC2×C4D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#135×C101C2
socleC22×C101C22×D4
hypercenterC2×#135D101
hypocenterC101#3035
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#15692275C2semidirect
2. #20265723C2semidirect
3. #41×Dic101C2semidirect
4. #20265401C2semidirect
5. #20265397C2semidirect
6. #20265394C2semidirect
7. #20265390C2semidirect
8. #20265385C2semidirect
9. #20265380C2semidirect
10. #20265375C2semidirect
11. #20265370C2semidirect
12. C2×#15692184C2semidirect
13. #20265174C2semidirect
14. #20265173C2semidirect
15. C2×#135×C101C2semidirect
103. #41×C101D4semidirect
189. Dic101#135semidirect
198. C101#3035semidirect
200. C2×#135D101semidirect
264. #41C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922752
#202657232
#41×Dic1012
#202654012
#202653972
#202653942
#202653902
#202653852
#202653802
#202653752
#202653702
C2×#156921842
#202651742
#202651732
C2×#135×C1012
#3035101