Finite Group Databasebeta

← #27179501

Subgroups — #27179501

Order 12928.

524 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692245C211yesmaximal, characteristic, complemented
2C2×#15692241C211yesmaximal, characteristic
3#20265081C211yesmaximal, characteristic, complemented
4#20265081C211yesmaximal, characteristic, complemented
5#20265077C211yesmaximal, characteristic
6#20265077C211yesmaximal, characteristic
7#20265071C211yesmaximal, characteristic, complemented
8#20265070C211yesmaximal, characteristic, complemented
9#20265069C211yesmaximal, characteristic, complemented
10#20265068C211yesmaximal, characteristic, complemented
11#20265067C211yesmaximal, characteristic, complemented
12#20265066C211yesmaximal, characteristic, complemented
13#20265065C211yesmaximal, characteristic, complemented
14C42×Dic101C211yesmaximal, characteristic, complemented
15#388×C101C211yesmaximal, characteristic, complemented
16C22×Dic202C2211yescharacteristic
17C22×C22⋊D101C2211yescharacteristic
18#15692245—24no—
19#15692241—24no—
20C2×C4⋊Dic101C2211yescharacteristic
21C2×#13108053C2211yescharacteristic
22C2×#13108053C2211yescharacteristic
23C2×#13108053C2211yescharacteristic
24C2×#13108053C2211yescharacteristic
25C2×C22⋊Dic101C2211yescharacteristic
26C2×C22⋊Dic101C2211yescharacteristic
27C2×C22⋊Dic101C2211yescharacteristic
28C2×#13108051C2211yescharacteristic
29C2×#13108051C2211yescharacteristic
30C2×#13108051C2211yescharacteristic
31C2×#13108051C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×C4×Dic101C2211yescharacteristic
35C2×C4×Dic101C2211yescharacteristic
36C2×C4×Dic101C2211yescharacteristic
37C2×C4×Dic101C2211yescharacteristic
38C2×C4×Dic101C2211yescharacteristic
39C2×C4×Dic101C2211yescharacteristic
40#15692162C2211yescharacteristic
41#15692162C2211yescharacteristic
42#15692162C2211yescharacteristic
43#15692162C2211yescharacteristic
44#15692162C2211yescharacteristic
45#15692162C2211yescharacteristic
46C2×C4⋊C4×C101C2211yescharacteristic
47C2×C22⋊C4×C101C2211yescharacteristic
48C2×C22⋊C4×C101C2211yescharacteristic
49C2×C22⋊C4×C101C2211yescharacteristic
50C2×C42×C101C2211yescharacteristic
51#95×C101C2211yescharacteristic
52#95×C101C2211yescharacteristic
53C23×D101C2311yescharacteristic
54C2×Dic202—22no—
55C2×Dic202—22no—
56C2×Dic202—22no—
57C2×C22⋊D101—22no—
58C2×C22⋊D101—22no—
59C2×C22⋊D101—22no—
60C22×Dic101C2311yescharacteristic
61C22×Dic101C2311yescharacteristic
62C22×Dic101C2311yescharacteristic
63C22×Dic101C2311yescharacteristic
64C22×Dic101C2311yescharacteristic
65C22×Dic101C2311yescharacteristic
66C22×Dic101C2311yescharacteristic
67C4⋊Dic101—22no—
68#13108053—22no—
69#13108053—22no—
70#13108053—22no—
71#13108053—22no—
72C22⋊Dic101—22no—
73C22⋊Dic101—22no—
74C22⋊Dic101—22no—
75#13108051—22no—
76#13108051—22no—
77#13108051—22no—
78#13108051—22no—
79#13108051—22no—
80#13108051—22no—
81C4×Dic101—22no—
82C4×Dic101—22no—
83C4×Dic101—22no—
84C4×Dic101—22no—
85C4×Dic101—22no—
86C4×Dic101D414yes—
87C24×C101C2311yescharacteristic
88C22×C4×C101C2311yescharacteristic
89C22×C4×C101C2311yescharacteristic
90C22×C4×C101C2311yescharacteristic
91C22×C4×C101C2311yescharacteristic
92C22×C4×C101C2311yescharacteristic
93C22×C4×C101C2311yescharacteristic
94C4⋊C4×C101—22no—
95C22⋊C4×C101—22no—
96C22⋊C4×C101—22no—
97C22⋊C4×C101—22no—
98C42×C101—22no—
99C23×C101—21no—
100C23×C101—21no—
101C23×C101—21no—
102C23×C101—21no—
103C23×C101—21no—
104C23×C101—21no—
105C23×C101—21no—
106C23×C101C2411yescharacteristic
107C22×D101—21no—
108C22×D101—21no—
109C22×D101—21no—
110C22×D101—21no—
111C22×D101—21no—
112C22×D101—21no—
113C22×D101—21no—
114C2×C4×C101—21no—
115C2×C4×C101—21no—
116C2×C4×C101—21no—
117C2×C4×C101—21no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—21no—
128C2×C4×C101—21no—
129C2×C4×C101C2×D412yes—
130C2×C4×C101#4112yes—
131C2×C4×C101#4112yes—
132C22⋊D101—44no—
133C2×Dic101—21no—
134C2×Dic101—21no—
135C2×Dic101—21no—
136C2×Dic101—21no—
137C2×Dic101—21no—
138C2×Dic101—21no—
139C2×Dic101—21no—
140C2×Dic101—21no—
141C2×Dic101—21no—
142C2×Dic101—21no—
143C2×Dic101—21no—
144C2×Dic101—21no—
145C2×Dic101—21no—
146C2×Dic101—21no—
147C2×Dic101—21no—
148C2×Dic101C2×D412yes—
149C2×Dic101C2×D412yes—
150C2×Dic101#4112yes—
151C2×Dic101#4112yes—
152C2×Dic101#4112yes—
153C2×Dic101#4112yes—
154Dic202—44no—
155C22×C101—41no—
156C22×C101—41no—
157C22×C101—41no—
158C22×C101—41no—
159C22×C101—41no—
160C22×C101—41no—
161C22×C101—41no—
162C22×C101C22×D411yescharacteristic
163C22×C101C2×#4111yescharacteristic
164C22×C101C2×#4111yescharacteristic
165C22×C101C2×#4111yescharacteristic
166C22×C101C2×#4111yescharacteristic
167C22×C101C2×#4111yescharacteristic
168C22×C101C2×#4111yescharacteristic
169C2×D101—41no—
170C2×D101—41no—
171C2×D101—41no—
172C2×D101—41no—
173C2×D101—41no—
174C2×D101—41no—
175C2×D101—41no—
176C4×C101—22no—
177C4×C101—22no—
178C4×C101—22no—
179C4×C101—41no—
180C4×C101—41no—
181C4×C101—41no—
182Dic101—22no—
183Dic101—22no—
184Dic101—22no—
185Dic101—41no—
186Dic101—41no—
187Dic101—41no—
188Dic101#12414yes—
189C2×C101—81no—
190C2×C101C2×#12411yescharacteristic
191C2×C101#52911yescharacteristic
192C2×C101#52911yescharacteristic
193C2×C101#52911yescharacteristic
194C2×C101#52911yescharacteristic
195C2×C101#53211yescharacteristic
196C2×C101#53211yescharacteristic
197D101—81no—
198#2861—1011nomaximal, Sylow
199C101#286111yescharacteristic, complemented, Sylow
200C2×#125—1011no—
201C2×#124—1011no—
202#391—1011no—
203#391—1011no—
204#390—1011no—
205#390—1011no—
206#388—1011no—
207#388—1011no—
208#388—1011no—
209#388—1011no—
210#388—1011no—
211#388—1011no—
212#388—1011no—
213#388D10111yescharacteristic, complemented
214C43—1011no—
215C22×Q8—1011no—
216C22×D4—1011no—
217#125—2024no—
218#124—2024no—
219C2×C4⋊C4—1011no—
220C2×C4⋊C4—1011no—
221C2×C4⋊C4—1011no—
222C2×C4⋊C4—1011no—
223C2×C4⋊C4—1011no—
224C2×C4⋊C4C2×D10111yescharacteristic
225C2×C22⋊C4—1011no—
226C2×C22⋊C4—1011no—
227C2×C22⋊C4—1011no—
228C2×C22⋊C4—1011no—
229C2×C22⋊C4—1011no—
230C2×C22⋊C4—1011no—
231C2×C22⋊C4—1011no—
232C2×C22⋊C4—1011no—
233C2×C22⋊C4—1011no—
234C2×C22⋊C4C2×D10111yescharacteristic
235C2×C22⋊C4C2×D10111yescharacteristic
236C2×C22⋊C4C2×D10111yescharacteristic
237C2×C42—1011no—
238C2×C42—1011no—
239C2×C42—1011no—
240C2×C42—1011no—
241C2×C42—1011no—
242C2×C42—1011no—
243C2×C42C2×D10111yescharacteristic
244#95—1011no—
245#95—1011no—
246#95—1011no—
247#95—1011no—
248#95—1011no—
249#95—1011no—
250#95C2×D10111yescharacteristic
251#95C2×D10111yescharacteristic
252C24—1011no—
253C24C22×D10111yescharacteristic
254C2×Q8—2022no—
255C2×Q8—2022no—
256C2×Q8—2022no—
257C2×D4—2022no—
258C2×D4—2022no—
259C2×D4—2022no—
260C22×C4—1011no—
261C22×C4—1011no—
262C22×C4—1011no—
263C22×C4—1011no—
264C22×C4—1011no—
265C22×C4—1011no—
266C22×C4—1011no—
267C22×C4C22×D10111yescharacteristic
268C22×C4C22×D10111yescharacteristic
269C22×C4C22×D10111yescharacteristic
270C22×C4C22×D10111yescharacteristic
271C22×C4C22×D10111yescharacteristic
272C22×C4C22×D10111yescharacteristic
273C4⋊C4—22no—
274C4⋊C4—2022no—
275C4⋊C4—2022no—
276C4⋊C4—2022no—
277C4⋊C4—2022no—
278C4⋊C4—2022no—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—2022no—
283C22⋊C4—2022no—
284C22⋊C4—2022no—
285C22⋊C4—2022no—
286C22⋊C4—2022no—
287C22⋊C4—2022no—
288C22⋊C4—2022no—
289C22⋊C4—2022no—
290C22⋊C4—2022no—
291C42—22no—
292C42—1014no—
293C42—2022no—
294C42—2022no—
295C42—2022no—
296C42—2022no—
297C42—2022no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—2021no—
306C23—2021no—
307C23—2021no—
308C23—2021no—
309C23—2021no—
310C23—2021no—
311C23—2021no—
312C23C23×D10111yescharacteristic
313Q8—4044no—
314D4—4044no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—1012no—
331C2×C4—1012no—
332C2×C4—1012no—
333C2×C4—1012no—
334C2×C4—1012no—
335C2×C4—1012no—
336C2×C4—2021no—
337C2×C4—2021no—
338C2×C4—2021no—
339C2×C4—2021no—
340C2×C4—2021no—
341C2×C4—2021no—
342C2×C4—2021no—
343C2×C4—2021no—
344C2×C4—2021no—
345C2×C4—2021no—
346C2×C4—2021no—
347C2×C4—2021no—
348C2×C4—2021no—
349C2×C4—2021no—
350C2×C4—2021no—
351C2×C4D4×D10112yes—
352C2×C4#1310807312yes—
353C2×C4#1310807412yes—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—4041no—
362C22—4041no—
363C22—4041no—
364C22—4041no—
365C22—4041no—
366C22—4041no—
367C22—4041no—
368C22C2×D4×D10111yescharacteristic
369C22#41×D10111yescharacteristic
370C22#41×D10111yescharacteristic
371C22#41×D10111yescharacteristic
372C22#41×D10111yescharacteristic
373C22C2×#1310807311yescharacteristic
374C22C2×#1310807411yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—1014no—
382C4—4041no—
383C4—4041no—
384C4—4041no—
385C4—2022no—
386C4—2022no—
387C4—2022no—
388C2—81no—
389C2—8081no—
390C2#124×D10111yescharacteristic
391C2#2026570411yescharacteristic
392C2#2026570511yescharacteristic
393C2#2026570711yescharacteristic
394C2#2026571011yescharacteristic
395C2#2026572211yescharacteristic
396C2#2026572411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#388×C101C2
prime radicalC22×C22⋊D101C22
hypercenter#388D101
hypocenterC101#2861
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#15692245C2semidirect
3. #20265081C2semidirect
4. #20265081C2semidirect
7. #20265071C2semidirect
8. #20265070C2semidirect
9. #20265069C2semidirect
10. #20265068C2semidirect
11. #20265067C2semidirect
12. #20265066C2semidirect
13. #20265065C2semidirect
14. C42×Dic101C2semidirect
15. #388×C101C2semidirect
199. C101#2861semidirect
213. #388D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922452
C2×#156922412
#202650812
#202650772
#202650712
#202650702
#202650692
#202650682
#202650672
#202650662
#202650652
C42×Dic1012
#388×C1012
#2861101