Finite Group Databasebeta

← #27182688

Subgroups — #27182688

Order 12928.

478 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×#15692286C211yesmaximal, characteristic, complemented
2C2×#15692275C211yesmaximal, characteristic, complemented
3#20265832C211yesmaximal, characteristic, complemented
4#20265572C211yesmaximal, characteristic, complemented
5#20265421C211yesmaximal, characteristic, complemented
6#20265416C211yesmaximal, characteristic, complemented
7#20265389C211yesmaximal, characteristic, complemented
8#20265387C211yesmaximal, characteristic, complemented
9#20265339C211yesmaximal, characteristic, complemented
10#20265337C211yesmaximal, characteristic, complemented
11#20265333C211yesmaximal, characteristic, complemented
12#20265320C211yesmaximal, characteristic, complemented
13#20265275C211yesmaximal, characteristic, complemented
14#20265249C211yesmaximal, characteristic, complemented
15#548×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C2×#13108074C2211yescharacteristic
18#15692286—24no—
19#15692275—24no—
20C2×#13108064C2211yescharacteristic
21C2×#13108064C2211yescharacteristic
22C2×#13108064C2211yescharacteristic
23C2×#13108064C2211yescharacteristic
24C2×#13108062C2211yescharacteristic
25C2×#13108062C2211yescharacteristic
26C2×#13108061C2211yescharacteristic
27C2×#13108061C2211yescharacteristic
28C2×#13108059C2211yescharacteristic
29C2×#13108058C2211yescharacteristic
30C22×#11717579C2211yescharacteristic
31#15692255C2211yescharacteristic
32#15692237—21no—
33#15692237—21no—
34#15692234C2211yescharacteristic
35#15692234C2211yescharacteristic
36#15692226—21no—
37C4×D404C2211yescharacteristic
38C4⋊C4×D101—21no—
39#15692188C2211yescharacteristic
40C4⋊#11717579C2211yescharacteristic
41#15692179C2211yescharacteristic
42#15692179C2211yescharacteristic
43#15692178C2211yescharacteristic
44#15692178C2211yescharacteristic
45#15692176C2211yescharacteristic
46#15692175C2211yescharacteristic
47C22⋊#11717579C2211yescharacteristic
48C22⋊#11717579C2211yescharacteristic
49#15692166C2211yescharacteristic
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—22no—
63#13108074—41no—
64#13108074—41no—
65C2×Dic202C2311yescharacteristic
66C2×Dic202C2311yescharacteristic
67C2×D404C2311yescharacteristic
68C2×C22⋊D101—21no—
69C2×C22⋊D101—21no—
70C2×C4×D101—21no—
71C2×C4×D101—21no—
72C2×C4×D101—21no—
73#13108064—22no—
74#13108064—22no—
75#13108064—22no—
76#13108064—41no—
77#13108062—22no—
78#13108062—41no—
79#13108061—22no—
80#13108061D414yescomplemented
81#13108059—22no—
82#13108058—22no—
83C2×#11717579—22no—
84C2×#11717579C2311yescharacteristic
85C2×#11717579C2311yescharacteristic
86C2×#11717579C2311yescharacteristic
87C2×#11717579C2311yescharacteristic
88C4⋊Dic101—21no—
89C4⋊Dic101—21no—
90C4⋊Dic101C2311yescharacteristic
91#13108053—21no—
92#13108053—21no—
93C22⋊Dic101—21no—
94C22⋊Dic101—21no—
95#13108051—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—21no—
133D4×C101—21no—
134D4×C101—41no—
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—
156D404—21no—
157D404C2×D412yes—
158C4×D101—42no—
159C4×D101—42no—
160C4×D101—42no—
161Dic202—21no—
162Dic202—21no—
163Dic202—21no—
164Dic202—21no—
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—21no—
175C22×C101—41no—
176C22×C101—41no—
177C22×C101—41no—
178C22×C101C22×D411yescharacteristic
179C2×D101—22no—
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—41no—
192Dic101—41no—
193Dic101—41no—
194C2×C101—41no—
195C2×C101—41no—
196C2×C101—81no—
197C2×C101D4211yescharacteristic
198C2×C101#57511yescharacteristic
199C2×C101#57811yescharacteristic
200D101—42no—
201#3269—1011nomaximal, Sylow
202C101#326911yescharacteristic, complemented, Sylow
203C2×#137—1011no—
204C2×#135—1011no—
205#548D10111yescharacteristic, complemented
206#547—1011no—
207#497—1011no—
208#468—1011no—
209#467—1011no—
210#463—1011no—
211#462—1011no—
212#452—1011no—
213#452—1011no—
214#451—1011no—
215#449—1011no—
216#440—1011no—
217#435—1011no—
218C2×#41—1011no—
219C2×#41—1011no—
220C2×#41C2×D10111yescharacteristic
221C22×Q8C2×D10111yescharacteristic
222#137—2024no—
223#135—2024no—
224C2×Q16—1011no—
225C2×Q16—1011no—
226C2×Q16—1011no—
227C2×Q16—1011no—
228C2×SD16—1011no—
229C2×SD16—1011no—
230C2×SD16—1011no—
231C2×SD16—1011no—
232C2×D8—1011no—
233C2×M16—1011no—
234C22×C8—1011no—
235C4⋊Q8—1011no—
236#124—21no—
237#124C2×D10111yescharacteristic
238#123—2021no—
239#123—2021no—
240C22⋊Q8—21no—
241C22⋊Q8—1011no—
242C22⋊Q8—1011no—
243C22⋊Q8C2×D10111yescharacteristic
244#121—21no—
245#121—2021no—
246#121C2×D10111yescharacteristic
247C4×Q8C2×D10111yescharacteristic
248C4×D4—21no—
249C4×D4—1011no—
250C4×D4C2×D10111yescharacteristic
251C2×C4⋊C4—2021no—
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—
262#97—1011no—
263#41—22no—
264#41—41no—
265#41—4041no—
266#41—4041no—
267#41—2022no—
268#41—2022no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—1011no—
273C2×Q8—1011no—
274C2×Q8C22×D10111yescharacteristic
275C2×Q8C22×D10111yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—1011no—
279C2×D4—2021no—
280C2×D4—2021no—
281C2×D4C22⋊D10111yescharacteristic
282C2×D4C22×D10111yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—2021no—
286C22×C4—2021no—
287C22×C4—2021no—
288C22×C4C22×D10111yescharacteristic
289C22×C4C22×D10111yescharacteristic
290Q16—4041no—
291Q16—2022no—
292Q16—2022no—
293Q16—2022no—
294SD16—1014no—
295SD16—4041no—
296SD16—2022no—
297SD16—2022no—
298D8—2022no—
299M16—2022no—
300C2×C8—1011no—
301C2×C8—1011no—
302C2×C8—1011no—
303C2×C8—1011no—
304C2×C8—2022no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—1011no—
308C4⋊C4—2021no—
309C4⋊C4—2021no—
310C4⋊C4—2021no—
311C4⋊C4—2021no—
312C4⋊C4C22⋊D10111yescharacteristic
313C4⋊C4C22×D10111yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—2021no—
319C22⋊C4—2021no—
320C22⋊C4—2021no—
321C22⋊C4C22⋊D10111yescharacteristic
322C22⋊C4C22⋊D10111yescharacteristic
323C42—21no—
324C42C22×D10111yescharacteristic
325C23—21no—
326C23—2021no—
327C23C2×C22⋊D10111yescharacteristic
328C23C2×C22⋊D10111yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—2021no—
334Q8—2021no—
335Q8—2021no—
336Q8—2021no—
337Q8D4×D10112yes—
338D4—21no—
339D4—21no—
340D4—41no—
341D4—42no—
342D4—42no—
343D4—1012no—
344D4—4042no—
345D4—4042no—
346D4—2021no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—4042no—
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—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—4041no—
381C22—4041no—
382C22—2022no—
383C22C22×C22⋊D10111yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—22no—
388C4—21no—
389C4—41no—
390C4—41no—
391C4—4041no—
392C4—4041no—
393C4—4041no—
394C4C2×D4×D10111yescharacteristic
395C4#1569230011yescharacteristic
396C2—41no—
397C2—41no—
398C2—81no—
399C2—4042no—
400C2D4×C22⋊D10111yescharacteristic
401C2#2026597511yescharacteristic
402C2#2026599211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#548×C101C2
socleC22×C101C22×D4
hypercenter#548D101
hypocenterC101#3269
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×#15692286C2semidirect
2. C2×#15692275C2semidirect
3. #20265832C2semidirect
4. #20265572C2semidirect
5. #20265421C2semidirect
6. #20265416C2semidirect
7. #20265389C2semidirect
8. #20265387C2semidirect
9. #20265339C2semidirect
10. #20265337C2semidirect
11. #20265333C2semidirect
12. #20265320C2semidirect
13. #20265275C2semidirect
14. #20265249C2semidirect
15. #548×C101C2semidirect
80. #13108061D4semidirect
202. C101#3269semidirect
205. #548D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922862
C2×#156922752
#202658322
#202655722
#202654212
#202654162
#202653892
#202653872
#202653392
#202653372
#202653332
#202653202
#202652752
#202652492
#548×C1012
#3269101