Finite Group Databasebeta

← #27182869

Subgroups — #27182869

Order 12928.

482 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692275C211yesmaximal, characteristic, complemented
2C22×#13108064C211yesmaximal, characteristic, complemented
3#20265836C211yesmaximal, characteristic, complemented
4#20265559C211yesmaximal, characteristic, complemented
5#20265416C211yesmaximal, characteristic, complemented
6#20265411C211yesmaximal, characteristic, complemented
7#20265399C211yesmaximal, characteristic, complemented
8#20265396C211yesmaximal, characteristic, complemented
9#20265339C211yesmaximal, characteristic, complemented
10#20265337C211yesmaximal, characteristic, complemented
11#20265316C211yesmaximal, characteristic, complemented
12#20265315C211yesmaximal, characteristic, complemented
13C4×#13108064C211yesmaximal, characteristic, complemented
14D4×#11717579C211yesmaximal, characteristic, complemented
15#548×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C22×Dic202C2211yescharacteristic
18#15692275—24no—
19C2×#13108064—24no—
20C2×#13108064C2211yescharacteristic
21C2×#13108064C2211yescharacteristic
22C2×#13108064C2211yescharacteristic
23C2×#13108064C2211yescharacteristic
24C2×#13108062C2211yescharacteristic
25C2×#13108062C2211yescharacteristic
26C2×#13108061C2211yescharacteristic
27C2×#13108061C2211yescharacteristic
28C2×#13108059C2211yescharacteristic
29C22×#11717579C2211yescharacteristic
30C22×#11717579C2211yescharacteristic
31#15692243C2211yescharacteristic
32#15692241—21no—
33#15692234C2211yescharacteristic
34#15692233—21no—
35#15692229C2211yescharacteristic
36#15692227—21no—
37C4×Dic202C2211yescharacteristic
38#15692215—21no—
39#15692188C2211yescharacteristic
40C4⋊#11717579C2211yescharacteristic
41#15692179C2211yescharacteristic
42#15692179C2211yescharacteristic
43#15692178C2211yescharacteristic
44#15692178C2211yescharacteristic
45#15692176C2211yescharacteristic
46#15692175C2211yescharacteristic
47C22⋊#11717579C2211yescharacteristic
48C22⋊#11717579C2211yescharacteristic
49C4×#11717579C2211yescharacteristic
50C2×#41×C101C2211yescharacteristic
51C22×Q8×C101C2211yescharacteristic
52#124×C101—21no—
53#124×C101C2211yescharacteristic
54C22⋊Q8×C101—21no—
55C22⋊Q8×C101C2211yescharacteristic
56#121×C101—21no—
57#121×C101C2211yescharacteristic
58C4×Q8×C101C2211yescharacteristic
59C4×D4×C101—21no—
60C4×D4×C101C2211yescharacteristic
61#13108074—22no—
62#13108074—41no—
63C2×Dic202—22no—
64C2×Dic202—21no—
65C2×Dic202—41no—
66C2×Dic202C2311yescharacteristic
67C2×Dic202C2311yescharacteristic
68C2×D404C2311yescharacteristic
69C2×C22⋊D101—21no—
70C2×C22⋊D101—21no—
71C22×Dic101—21no—
72C2×C4×D101—21no—
73#13108064—22no—
74#13108064—22no—
75#13108064—22no—
76#13108064D414yescomplemented
77#13108062—22no—
78#13108062—41no—
79#13108061—22no—
80#13108061—41no—
81#13108059—22no—
82C2×#11717579—22no—
83C2×#11717579—22no—
84C2×#11717579C2311yescharacteristic
85C2×#11717579C2311yescharacteristic
86C2×#11717579C2311yescharacteristic
87C2×#11717579C2311yescharacteristic
88C4⋊Dic101C2311yescharacteristic
89#13108053—21no—
90#13108053—21no—
91C22⋊Dic101—21no—
92#13108051—21no—
93#13108051—21no—
94#13108051—21no—
95C4×Dic101—21no—
96#41×C101—22no—
97#41×C101—41no—
98C2×Q8×C101—21no—
99C2×Q8×C101—22no—
100C2×Q8×C101—41no—
101C2×Q8×C101C2311yescharacteristic
102C2×Q8×C101C2311yescharacteristic
103C2×D4×C101—21no—
104C2×D4×C101—21no—
105C2×D4×C101D411yescharacteristic
106C2×D4×C101C2311yescharacteristic
107C22×C4×C101—21no—
108C22×C4×C101—21no—
109C22×C4×C101C2311yescharacteristic
110C22×C4×C101C2311yescharacteristic
111C4⋊C4×C101—21no—
112C4⋊C4×C101—21no—
113C4⋊C4×C101D411yescharacteristic
114C4⋊C4×C101C2311yescharacteristic
115C22⋊C4×C101—21no—
116C22⋊C4×C101—21no—
117C22⋊C4×C101—21no—
118C22⋊C4×C101—21no—
119C22⋊C4×C101D411yescharacteristic
120C22⋊C4×C101D411yescharacteristic
121C42×C101—21no—
122C42×C101C2311yescharacteristic
123C23×C101—21no—
124C23×C101C2×D411yescharacteristic
125C23×C101C2×D411yescharacteristic
126C22×D101—21no—
127Q8×C101—21no—
128Q8×C101—21no—
129Q8×C101—21no—
130Q8×C101—42no—
131Q8×C101C2×D412yes—
132D4×C101—22no—
133D4×C101—21no—
134D4×C101—21no—
135D4×C101—42no—
136D4×C101—42no—
137C2×C4×C101—21no—
138C2×C4×C101—21no—
139C2×C4×C101—21no—
140C2×C4×C101—21no—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101—21no—
144C2×C4×C101—42no—
145C2×C4×C101—42no—
146C2×C4×C101C2×D411yescharacteristic
147C2×C4×C101C2×D411yescharacteristic
148C2×C4×C101C2×D411yescharacteristic
149C2×C4×C101C2×D411yescharacteristic
150C2×C4×C101C2411yescharacteristic
151C22⋊D101—42no—
152C22⋊D101—42no—
153C2×Dic101—21no—
154C2×Dic101—21no—
155C2×Dic101—21no—
156C2×Dic101—42no—
157D404—21no—
158D404—21no—
159C4×D101—42no—
160Dic202—21no—
161Dic202—21no—
162Dic202—21no—
163Dic202—42no—
164Dic202C2×D412yes—
165#11717579—21no—
166#11717579—21no—
167#11717579—21no—
168#11717579C2×D412yes—
169C22×C101—21no—
170C22×C101—21no—
171C22×C101—21no—
172C22×C101—21no—
173C22×C101—21no—
174C22×C101—41no—
175C22×C101—41no—
176C22×C101—41no—
177C22×C101#13512yescomplemented
178C22×C101C22×D411yescharacteristic
179C2×D101—41no—
180C2×D101—41no—
181C2×D101—41no—
182C4×C101—21no—
183C4×C101—21no—
184C4×C101—22no—
185C4×C101—21no—
186C4×C101—21no—
187C4×C101—41no—
188C4×C101—41no—
189C4×C101C22×D411yescharacteristic
190C4×C101H(5,𝔽2)11yescharacteristic
191Dic101—22no—
192Dic101—41no—
193Dic101—41no—
194C2×C101—22no—
195C2×C101—41no—
196C2×C101—81no—
197C2×C101D4211yescharacteristic
198C2×C101C2×#13511yescharacteristic
199C2×C101#57811yescharacteristic
200D101—81no—
201#3285—1011nomaximal, Sylow
202C101#328511yescharacteristic, complemented, Sylow
203C2×#135—1011no—
204C22×Q16—1011no—
205#548—1011no—
206#548D10111yescharacteristic, complemented
207#495—1011no—
208#467—1011no—
209#466—1011no—
210#464—1011no—
211#464—1011no—
212#452—1011no—
213#452—1011no—
214#448—1011no—
215#448—1011no—
216C4×Q16—1011no—
217C8×D4—1011no—
218C2×#41—1011no—
219C2×#41C2×D10111yescharacteristic
220C22×Q8—1011no—
221C22×Q8C2×D10111yescharacteristic
222#135—2024no—
223C2×Q16—1011no—
224C2×Q16—1011no—
225C2×Q16—1011no—
226C2×Q16—1011no—
227C2×Q16—2024no—
228C2×SD16—1011no—
229C2×SD16—1011no—
230C2×SD16—1011no—
231C2×SD16—1011no—
232C2×D8—1011no—
233C22×C8—1011no—
234C22×C8—1011no—
235#124—21no—
236#124—1011no—
237#124—2021no—
238#124C2×D10111yescharacteristic
239C22⋊Q8—21no—
240C22⋊Q8—1011no—
241C22⋊Q8—2021no—
242C22⋊Q8C2×D10111yescharacteristic
243#121—21no—
244#121—1011no—
245#121—2021no—
246#121C2×D10111yescharacteristic
247C4×Q8—1011no—
248C4×Q8C2×D10111yescharacteristic
249C4×D4—21no—
250C4×D4—2021no—
251C4×D4C2×D10111yescharacteristic
252#107—1011no—
253C4⋊C8—1011no—
254#103—1011no—
255#103—1011no—
256#103—1011no—
257#103—1011no—
258#102—1011no—
259#102—1011no—
260C22⋊C8—1011no—
261C22⋊C8—1011no—
262C4×C8—1011no—
263#41—22no—
264#41—41no—
265#41—4041no—
266#41—2022no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—1011no—
271C2×Q8—1011no—
272C2×Q8—4041no—
273C2×Q8—2022no—
274C2×Q8—2021no—
275C2×Q8C22×D10111yescharacteristic
276C2×Q8C22×D10111yescharacteristic
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—1011no—
280C2×D4—2021no—
281C2×D4—2021no—
282C2×D4C22⋊D10111yescharacteristic
283C2×D4C22×D10111yescharacteristic
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—2021no—
287C22×C4—2021no—
288C22×C4C22×D10111yescharacteristic
289C22×C4C22×D10111yescharacteristic
290Q16—1014no—
291Q16—2022no—
292Q16—2022no—
293Q16—2022no—
294SD16—4041no—
295SD16—4041no—
296SD16—2022no—
297SD16—2022no—
298D8—2022no—
299C2×C8—1011no—
300C2×C8—1011no—
301C2×C8—1011no—
302C2×C8—1011no—
303C2×C8—2022no—
304C2×C8—2022no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—1011no—
308C4⋊C4—2021no—
309C4⋊C4—2021no—
310C4⋊C4C22⋊D10111yescharacteristic
311C4⋊C4C22×D10111yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—2021no—
317C22⋊C4—2021no—
318C22⋊C4—2021no—
319C22⋊C4—2021no—
320C22⋊C4C22⋊D10111yescharacteristic
321C22⋊C4C22⋊D10111yescharacteristic
322C42—21no—
323C42—2021no—
324C42C22×D10111yescharacteristic
325C23—21no—
326C23—2021no—
327C23C2×C22⋊D10111yescharacteristic
328C23C2×C22⋊D10111yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—1012no—
334Q8—4042no—
335Q8—2021no—
336Q8—2021no—
337Q8—2021no—
338Q8D4×D10112yes—
339D4—22no—
340D4—21no—
341D4—21no—
342D4—42no—
343D4—42no—
344D4—4042no—
345D4—4042no—
346D4—2021no—
347D4—2021no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—4042no—
358C2×C4—4042no—
359C2×C4—2021no—
360C2×C4—2021no—
361C2×C4—2021no—
362C2×C4C2×C22⋊D10111yescharacteristic
363C2×C4C2×C22⋊D10111yescharacteristic
364C2×C4C2×C22⋊D10111yescharacteristic
365C2×C4C2×C22⋊D10111yescharacteristic
366C2×C4C23×D10111yescharacteristic
367C8—1012no—
368C8—2021no—
369C8—2021no—
370C8—2021no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—4041no—
380C22—4041no—
381C22—4041no—
382C22#1569227512yescomplemented
383C22C22×C22⋊D10111yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—22no—
388C4—21no—
389C4—41no—
390C4—41no—
391C4—4041no—
392C4—4041no—
393C4—2022no—
394C4C2×D4×D10111yescharacteristic
395C4#1569230011yescharacteristic
396C2—22no—
397C2—41no—
398C2—81no—
399C2—8081no—
400C2D4×C22⋊D10111yescharacteristic
401C2C2×#1569227511yescharacteristic
402C2#2026599211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#548×C101C2
socleC22×C101C22×D4
hypercenter#548D101
hypocenterC101#3285
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#15692275C2semidirect
2. C22×#13108064C2semidirect
3. #20265836C2semidirect
4. #20265559C2semidirect
5. #20265416C2semidirect
6. #20265411C2semidirect
7. #20265399C2semidirect
8. #20265396C2semidirect
9. #20265339C2semidirect
10. #20265337C2semidirect
11. #20265316C2semidirect
12. #20265315C2semidirect
13. C4×#13108064C2semidirect
14. D4×#11717579C2semidirect
15. #548×C101C2semidirect
76. #13108064D4semidirect
177. C22×C101#135semidirect
202. C101#3285semidirect
206. #548D101semidirect
382. C22#15692275semidirect

Maximal subgroups

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

subgroupindex
C2×#156922752
C22×#131080642
#202658362
#202655592
#202654162
#202654112
#202653992
#202653962
#202653392
#202653372
#202653162
#202653152
C4×#131080642
D4×#117175792
#548×C1012
#3285101