Finite Group Databasebeta

← #27182713

Subgroups — #27182713

Order 12928.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692285C211yesmaximal, characteristic, complemented
2C2×#15692275C211yesmaximal, characteristic, complemented
3#20265836C211yesmaximal, characteristic, complemented
4#20265564C211yesmaximal, characteristic, complemented
5#20265428C211yesmaximal, characteristic, complemented
6#20265404C211yesmaximal, characteristic, complemented
7#20265399C211yesmaximal, characteristic, complemented
8#20265374C211yesmaximal, characteristic, complemented
9#20265339C211yesmaximal, characteristic, complemented
10#20265322C211yesmaximal, characteristic, complemented
11#20265320C211yesmaximal, characteristic, complemented
12#20265315C211yesmaximal, characteristic, complemented
13#20265282C211yesmaximal, characteristic, complemented
14#20265249C211yesmaximal, characteristic, complemented
15#550×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C22×Dic202C2211yescharacteristic
18#15692285—24no—
19#15692275—24no—
20C2×#13108064C2211yescharacteristic
21C2×#13108064C2211yescharacteristic
22C2×#13108062C2211yescharacteristic
23C2×#13108062C2211yescharacteristic
24C2×#13108062C2211yescharacteristic
25C2×#13108062C2211yescharacteristic
26C2×#13108061C2211yescharacteristic
27C2×#13108059C2211yescharacteristic
28C2×#13108059C2211yescharacteristic
29C2×#13108058C2211yescharacteristic
30C22×#11717579C2211yescharacteristic
31#15692243C2211yescharacteristic
32#15692241—21no—
33#15692234C2211yescharacteristic
34#15692233—21no—
35#15692229C2211yescharacteristic
36#15692227—21no—
37C4×Dic202C2211yescharacteristic
38#15692215—21no—
39#15692186C2211yescharacteristic
40C4⋊#11717579C2211yescharacteristic
41#15692179C2211yescharacteristic
42#15692178C2211yescharacteristic
43#15692178C2211yescharacteristic
44#15692176C2211yescharacteristic
45#15692176C2211yescharacteristic
46#15692175C2211yescharacteristic
47C22⋊#11717579C2211yescharacteristic
48C22⋊#11717579C2211yescharacteristic
49#15692166C2211yescharacteristic
50C2×#41×C101C2211yescharacteristic
51C2×#41×C101C2211yescharacteristic
52#127×C101—21no—
53#127×C101C2211yescharacteristic
54#121×C101—21no—
55#121×C101—21no—
56#121×C101C2211yescharacteristic
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#13108064D414yescomplemented
75#13108062—22no—
76#13108062—22no—
77#13108062—22no—
78#13108062—41no—
79#13108061—22no—
80#13108059—22no—
81#13108059—41no—
82#13108058—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—22no—
98#41×C101—41no—
99#41×C101—41no—
100C2×Q8×C101C2311yescharacteristic
101C2×D4×C101—21no—
102C2×D4×C101—21no—
103C2×D4×C101—21no—
104C2×D4×C101—21no—
105C2×D4×C101—21no—
106C2×D4×C101—21no—
107C2×D4×C101D411yescharacteristic
108C2×D4×C101C2311yescharacteristic
109C2×D4×C101C2311yescharacteristic
110C22×C4×C101—21no—
111C22×C4×C101—21no—
112C22×C4×C101C2311yescharacteristic
113C22×C4×C101C2311yescharacteristic
114C4⋊C4×C101—21no—
115C4⋊C4×C101D411yescharacteristic
116C4⋊C4×C101C2311yescharacteristic
117C22⋊C4×C101—21no—
118C22⋊C4×C101—21no—
119C22⋊C4×C101D411yescharacteristic
120C22⋊C4×C101D411yescharacteristic
121C42×C101—21no—
122C42×C101C2311yescharacteristic
123C23×C101—21no—
124C23×C101—21no—
125C23×C101C2×D411yescharacteristic
126C23×C101C2×D411yescharacteristic
127C22×D101—21no—
128Q8×C101—21no—
129Q8×C101C2×D412yes—
130D4×C101—21no—
131D4×C101—21no—
132D4×C101—21no—
133D4×C101—21no—
134D4×C101—41no—
135D4×C101—42no—
136D4×C101—42no—
137D4×C101—42no—
138D4×C101—81no—
139D4×C101—81no—
140D4×C101—81no—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101—21no—
144C2×C4×C101—21no—
145C2×C4×C101—21no—
146C2×C4×C101—21no—
147C2×C4×C101—42no—
148C2×C4×C101—42no—
149C2×C4×C101C2×D411yescharacteristic
150C2×C4×C101C2×D411yescharacteristic
151C2×C4×C101C2×D411yescharacteristic
152C2×C4×C101C2×D411yescharacteristic
153C2×C4×C101C2411yescharacteristic
154C22⋊D101—42no—
155C22⋊D101—42no—
156C2×Dic101—21no—
157C2×Dic101—21no—
158C2×Dic101—21no—
159C2×Dic101—42no—
160D404—21no—
161D404—21no—
162C4×D101—42no—
163Dic202—21no—
164Dic202—21no—
165Dic202—21no—
166Dic202—42no—
167Dic202C2×D412yes—
168#11717579—21no—
169#11717579—21no—
170#11717579—21no—
171#11717579C2×D412yes—
172C22×C101—21no—
173C22×C101—21no—
174C22×C101—21no—
175C22×C101—21no—
176C22×C101—21no—
177C22×C101—21no—
178C22×C101—41no—
179C22×C101—41no—
180C22×C101—41no—
181C22×C101—41no—
182C22×C101—41no—
183C22×C101—41no—
184C22×C101C22×D411yescharacteristic
185C2×D101—41no—
186C2×D101—41no—
187C2×D101—41no—
188C4×C101—21no—
189C4×C101—21no—
190C4×C101—22no—
191C4×C101—21no—
192C4×C101—21no—
193C4×C101—41no—
194C4×C101C22×D411yescharacteristic
195C4×C101H(5,𝔽2)11yescharacteristic
196Dic101—22no—
197Dic101—41no—
198Dic101—41no—
199C2×C101—41no—
200C2×C101—41no—
201C2×C101—81no—
202C2×C101—81no—
203C2×C101D4211yescharacteristic
204C2×C101#57511yescharacteristic
205C2×C101#57711yescharacteristic
206D101—81no—
207#3271—1011nomaximal, Sylow
208C101#327111yescharacteristic, complemented, Sylow
209C2×#137—1011no—
210C2×#135—1011no—
211#550D10111yescharacteristic, complemented
212#548—1011no—
213#496—1011no—
214#469—1011no—
215#465—1011no—
216#464—1011no—
217#460—1011no—
218#452—1011no—
219#449—1011no—
220#449—1011no—
221#448—1011no—
222#441—1011no—
223#435—1011no—
224C2×#41—1011no—
225C2×#41C2×D10111yescharacteristic
226C2×#41C2×D10111yescharacteristic
227C22×Q8—1011no—
228#137—2024no—
229#135—2024no—
230C2×Q16—1011no—
231C2×Q16—1011no—
232C2×SD16—1011no—
233C2×SD16—1011no—
234C2×SD16—1011no—
235C2×SD16—1011no—
236C2×SD16—1011no—
237C2×D8—1011no—
238C2×D8—1011no—
239C2×M16—1011no—
240C22×C8—1011no—
241#127—21no—
242#127C2×D10111yescharacteristic
243#124—1011no—
244#124—2021no—
245C22⋊Q8—1011no—
246C22⋊Q8—2021no—
247#121—21no—
248#121—21no—
249#121—1011no—
250#121—2021no—
251#121C2×D10111yescharacteristic
252#121C2×D10111yescharacteristic
253C4×Q8—1011no—
254C4×Q8C2×D10111yescharacteristic
255C4×D4—21no—
256C4×D4—2021no—
257C4×D4C2×D10111yescharacteristic
258#106—1011no—
259C4⋊C8—1011no—
260#103—1011no—
261#103—1011no—
262#103—1011no—
263#102—1011no—
264#102—1011no—
265#102—1011no—
266C22⋊C8—1011no—
267C22⋊C8—1011no—
268#97—1011no—
269#41—22no—
270#41—22no—
271#41—41no—
272#41—41no—
273#41—4041no—
274#41—2022no—
275C2×Q8—1011no—
276C2×Q8—1011no—
277C2×Q8—4041no—
278C2×Q8—2022no—
279C2×Q8—2021no—
280C2×Q8C22×D10111yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—1011no—
288C2×D4—2021no—
289C2×D4—2021no—
290C2×D4C22⋊D10111yescharacteristic
291C2×D4C22×D10111yescharacteristic
292C2×D4C22×D10111yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—2021no—
296C22×C4—2021no—
297C22×C4C22×D10111yescharacteristic
298C22×C4C22×D10111yescharacteristic
299Q16—1014no—
300Q16—2022no—
301SD16—4041no—
302SD16—2022no—
303SD16—2022no—
304SD16—2022no—
305SD16—2022no—
306D8—4041no—
307D8—2022no—
308M16—2022no—
309C2×C8—1011no—
310C2×C8—1011no—
311C2×C8—1011no—
312C2×C8—1011no—
313C2×C8—2022no—
314C4⋊C4—21no—
315C4⋊C4—1011no—
316C4⋊C4—2021no—
317C4⋊C4—2021no—
318C4⋊C4C22⋊D10111yescharacteristic
319C4⋊C4C22×D10111yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—2021no—
323C22⋊C4—2021no—
324C22⋊C4—2021no—
325C22⋊C4—2021no—
326C22⋊C4C22⋊D10111yescharacteristic
327C22⋊C4C22⋊D10111yescharacteristic
328C42—21no—
329C42—2021no—
330C42C22×D10111yescharacteristic
331C23—21no—
332C23—21no—
333C23—2021no—
334C23C2×C22⋊D10111yescharacteristic
335C23C2×C22⋊D10111yescharacteristic
336Q8—21no—
337Q8—1012no—
338Q8—4042no—
339Q8—2021no—
340Q8—2021no—
341Q8—2021no—
342Q8D4×D10112yes—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—4042no—
355D4—4042no—
356D4—2021no—
357D4—2021no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—4042no—
367C2×C4—4042no—
368C2×C4—2021no—
369C2×C4—2021no—
370C2×C4—2021no—
371C2×C4C2×C22⋊D10111yescharacteristic
372C2×C4C2×C22⋊D10111yescharacteristic
373C2×C4C2×C22⋊D10111yescharacteristic
374C2×C4C2×C22⋊D10111yescharacteristic
375C2×C4C23×D10111yescharacteristic
376C8—1012no—
377C8—2021no—
378C8—2021no—
379C8—2021no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—4041no—
393C22—4041no—
394C22—4041no—
395C22C22×C22⋊D10111yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—22no—
400C4—21no—
401C4—41no—
402C4—4041no—
403C4—4041no—
404C4—2022no—
405C4C2×D4×D10111yescharacteristic
406C4#1569230011yescharacteristic
407C2—41no—
408C2—41no—
409C2—81no—
410C2—81no—
411C2—8081no—
412C2D4×C22⋊D10111yescharacteristic
413C2#2026597511yescharacteristic
414C2#2026598811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#550×C101C2
socleC22×C101C22×D4
hypercenter#550D101
hypocenterC101#3271
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#15692285C2semidirect
2. C2×#15692275C2semidirect
3. #20265836C2semidirect
4. #20265564C2semidirect
5. #20265428C2semidirect
6. #20265404C2semidirect
7. #20265399C2semidirect
8. #20265374C2semidirect
9. #20265339C2semidirect
10. #20265322C2semidirect
11. #20265320C2semidirect
12. #20265315C2semidirect
13. #20265282C2semidirect
14. #20265249C2semidirect
15. #550×C101C2semidirect
74. #13108064D4semidirect
208. C101#3271semidirect
211. #550D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922852
C2×#156922752
#202658362
#202655642
#202654282
#202654042
#202653992
#202653742
#202653392
#202653222
#202653202
#202653152
#202652822
#202652492
#550×C1012
#3271101