Finite Group Databasebeta

← #20250409

Subgroups — #20250409

Order 6016.

528 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13094004C211yesmaximal, characteristic, complemented
2C2×#13094002C211yesmaximal, characteristic, complemented
3#15689058C211yesmaximal, characteristic, complemented
4#15689058C211yesmaximal, characteristic, complemented
5#15689056C211yesmaximal, characteristic, complemented
6#15689047C211yesmaximal, characteristic, complemented
7#15689045C211yesmaximal, characteristic, complemented
8#15689027C211yesmaximal, characteristic, complemented
9#15689024C211yesmaximal, characteristic
10#15689020C211yesmaximal, characteristic, complemented
11#15689001C211yesmaximal, characteristic
12#15689001C211yesmaximal, characteristic
13#15688997C211yesmaximal, characteristic, complemented
14#15688994C211yesmaximal, characteristic, complemented
15C47×#398C211yesmaximal, characteristic, complemented
16C22×Dic94C2211yescharacteristic
17C22×D188C2211yescharacteristic
18C22×C4×D47C2211yescharacteristic
19C22×C4×D47C2211yescharacteristic
20#13094004—24no—
21#13094002—24no—
22C2×C4⋊Dic47C2211yescharacteristic
23C2×C4⋊Dic47C2211yescharacteristic
24C2×Dic47⋊C4C2211yescharacteristic
25C2×Dic47⋊C4C2211yescharacteristic
26C2×Dic47⋊C4C2211yescharacteristic
27C2×Dic47⋊C4C2211yescharacteristic
28C2×Dic47⋊C4C2211yescharacteristic
29C2×Dic47⋊C4C2211yescharacteristic
30C2×(C2×D47)⋊C4C2211yescharacteristic
31C2×(C2×D47)⋊C4C2211yescharacteristic
32C2×(C2×D47)⋊C4C2211yescharacteristic
33C2×(C2×D47)⋊C4C2211yescharacteristic
34C2×(C2×D47)⋊C4C2211yescharacteristic
35C2×(C2×D47)⋊C4C2211yescharacteristic
36C2×(C2×D47)⋊C4C2211yescharacteristic
37C2×(C2×D47)⋊C4C2211yescharacteristic
38C2×(C2×D47)⋊C4C2211yescharacteristic
39C2×(C2×D47)⋊C4C2211yescharacteristic
40C2×C4×Dic47C2211yescharacteristic
41C47⋊#95C2211yescharacteristic
42C47⋊#95C2211yescharacteristic
43C47⋊#95C2211yescharacteristic
44C47⋊#95C2211yescharacteristic
45C47⋊#95C2211yescharacteristic
46C2×C4⋊C4×C47C2211yescharacteristic
47C2×C4⋊C4×C47C2211yescharacteristic
48C2×C4⋊C4×C47C2211yescharacteristic
49C2×C4⋊C4×C47C2211yescharacteristic
50#95×C47C2211yescharacteristic
51#95×C47C2211yescharacteristic
52#95×C47C2211yescharacteristic
53C23×D47C2311yescharacteristic
54C23×D47C2311yescharacteristic
55C22×C4×C47C2311yescharacteristic
56C22×C4×C47C2311yescharacteristic
57C22×C4×C47C2311yescharacteristic
58C22×C4×C47C2311yescharacteristic
59C22×C4×C47C2311yescharacteristic
60C22×C4×C47C2311yescharacteristic
61C22×C4×C47C2311yescharacteristic
62C22×Dic47C2311yescharacteristic
63C22×Dic47C2311yescharacteristic
64C22×Dic47C2311yescharacteristic
65C22×Dic47C2311yescharacteristic
66C22×Dic47C2311yescharacteristic
67C22×Dic47C2311yescharacteristic
68C2×D188—41no—
69C2×D188—41no—
70C2×D188Q814yescomplemented
71C2×C4×D47—22no—
72C2×C4×D47—22no—
73C2×C4×D47—41no—
74C2×C4×D47—41no—
75C2×C4×D47—41no—
76C2×C4×D47—41no—
77C2×Dic94—41no—
78C2×Dic94—41no—
79C2×Dic94D414yescomplemented
80C4⋊C4×C47—22no—
81C4⋊C4×C47—22no—
82C4⋊C4×C47—41no—
83C4⋊C4×C47—41no—
84(C2×D47)⋊C4—22no—
85(C2×D47)⋊C4—22no—
86(C2×D47)⋊C4—22no—
87(C2×D47)⋊C4—22no—
88(C2×D47)⋊C4—22no—
89(C2×D47)⋊C4—22no—
90(C2×D47)⋊C4—41no—
91(C2×D47)⋊C4—41no—
92(C2×D47)⋊C4—41no—
93(C2×D47)⋊C4—41no—
94C4⋊Dic47—22no—
95C4⋊Dic47—22no—
96Dic47⋊C4—22no—
97Dic47⋊C4—22no—
98Dic47⋊C4—22no—
99Dic47⋊C4—41no—
100Dic47⋊C4—41no—
101Dic47⋊C4—41no—
102C4×Dic47—41no—
103C23×C47C2411yescharacteristic
104C22×D47—21no—
105C22×D47—21no—
106C22×D47—21no—
107C22×D47—21no—
108C22×D47—21no—
109C22×D47—21no—
110C22×D47—21no—
111C22×D47—21no—
112C22×D47—21no—
113C22×D47—21no—
114C22×D47—21no—
115C22×D47—21no—
116C22×D47C2×Q812yes—
117C22×D47C2×Q812yes—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—21no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47C2×D412yes—
138C2×C4×C47C2×Q812yes—
139C2×Dic47—21no—
140C2×Dic47—21no—
141C2×Dic47—21no—
142C2×Dic47—21no—
143C2×Dic47—21no—
144C2×Dic47—21no—
145C2×Dic47—21no—
146C2×Dic47—21no—
147C2×Dic47—21no—
148C2×Dic47—21no—
149C2×Dic47—21no—
150C2×Dic47—21no—
151C2×Dic47—21no—
152C2×Dic47—21no—
153C2×Dic47—21no—
154C2×Dic47C2×D412yes—
155C2×Dic47C2×D412yes—
156C2×Dic47#4112yes—
157D188—44no—
158C4×D47—44no—
159C4×D47—44no—
160Dic94—44no—
161C22×C47C22×D411yescharacteristic
162C22×C47C22×Q811yescharacteristic
163C22×C47C2×#4111yescharacteristic
164C22×C47H(5,𝔽2)11yescharacteristic
165C22×C47H(5,𝔽2)11yescharacteristic
166C22×C47H(5,𝔽2)11yescharacteristic
167C22×C472−1+411yescharacteristic
168C2×D47—22no—
169C2×D47—22no—
170C2×D47—22no—
171C2×D47—22no—
172C2×D47—22no—
173C2×D47—22no—
174C2×D47—41no—
175C2×D47—41no—
176C2×D47—41no—
177C2×D47—41no—
178C2×D47—41no—
179C2×D47—41no—
180C2×D47—41no—
181C2×D47—41no—
182C4×C47—22no—
183C4×C47—22no—
184C4×C47—41no—
185C4×C47—41no—
186C4×C47—41no—
187C4×C47—41no—
188C4×C47—41no—
189Dic47—22no—
190Dic47—22no—
191Dic47—22no—
192Dic47—41no—
193Dic47—41no—
194Dic47—41no—
195#2732—471nomaximal, Sylow
196C2×C47#53411yescharacteristic
197C2×C47#53911yescharacteristic
198C2×C47#54311yescharacteristic
199C2×C47D4×Q811yescharacteristic
200C2×C47#55011yescharacteristic
201C2×C47#55411yescharacteristic
202C2×C47#56211yescharacteristic
203D47—42no—
204D47—42no—
205C2×C22⋊Q8—471no—
206C2×C22⋊Q8—471no—
207#399—471no—
208#399—471no—
209#398—471no—
210#398D4711yescharacteristic, complemented
211#396—471no—
212#396—471no—
213#391—471no—
214#390—471no—
215#389—471no—
216#386—471no—
217#386—471no—
218#385—471no—
219#385—471no—
220C47#273211yescharacteristic, complemented, Sylow
221C22×Q8—471no—
222C22×D4—471no—
223C23×C4—471no—
224C23×C4—471no—
225C22⋊Q8—944no—
226C22⋊Q8—944no—
227C2×C4⋊C4—471no—
228C2×C4⋊C4—471no—
229C2×C4⋊C4—471no—
230C2×C4⋊C4—471no—
231C2×C4⋊C4—471no—
232C2×C4⋊C4—471no—
233C2×C4⋊C4—471no—
234C2×C4⋊C4—471no—
235C2×C4⋊C4C2×D4711yescharacteristic
236C2×C4⋊C4C2×D4711yescharacteristic
237C2×C4⋊C4C2×D4711yescharacteristic
238C2×C4⋊C4C2×D4711yescharacteristic
239C2×C22⋊C4—471no—
240C2×C22⋊C4—471no—
241C2×C22⋊C4—471no—
242C2×C22⋊C4—471no—
243C2×C22⋊C4—471no—
244C2×C22⋊C4—471no—
245C2×C22⋊C4—471no—
246C2×C22⋊C4—471no—
247C2×C22⋊C4—471no—
248C2×C22⋊C4—471no—
249C2×C42—471no—
250#95—471no—
251#95—471no—
252#95—471no—
253#95—471no—
254#95—471no—
255#95C2×D4711yescharacteristic
256#95C2×D4711yescharacteristic
257#95C2×D4711yescharacteristic
258C24—471no—
259C24—471no—
260C2×Q8—474no—
261C2×Q8—1881no—
262C2×Q8—1881no—
263C2×D4—474no—
264C2×D4—1881no—
265C2×D4—1881no—
266C22×C4—471no—
267C22×C4—471no—
268C22×C4—471no—
269C22×C4—471no—
270C22×C4—471no—
271C22×C4—471no—
272C22×C4—942no—
273C22×C4—942no—
274C22×C4—1881no—
275C22×C4—1881no—
276C22×C4—1881no—
277C22×C4—1881no—
278C22×C4C22×D4711yescharacteristic
279C22×C4C22×D4711yescharacteristic
280C22×C4C22×D4711yescharacteristic
281C22×C4C22×D4711yescharacteristic
282C22×C4C22×D4711yescharacteristic
283C22×C4C22×D4711yescharacteristic
284C22×C4C22×D4711yescharacteristic
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—942no—
290C4⋊C4—942no—
291C4⋊C4—942no—
292C4⋊C4—942no—
293C4⋊C4—942no—
294C4⋊C4—1881no—
295C4⋊C4—1881no—
296C4⋊C4—1881no—
297C22⋊C4—942no—
298C22⋊C4—942no—
299C22⋊C4—942no—
300C22⋊C4—942no—
301C22⋊C4—942no—
302C22⋊C4—942no—
303C22⋊C4—1881no—
304C22⋊C4—1881no—
305C22⋊C4—1881no—
306C22⋊C4—1881no—
307C42—1881no—
308C23—472no—
309C23—472no—
310C23—941no—
311C23—941no—
312C23—941no—
313C23—941no—
314C23—941no—
315C23—941no—
316C23—941no—
317C23—941no—
318C23—941no—
319C23—941no—
320C23—941no—
321C23—941no—
322C23C23×D4711yescharacteristic
323Q8—1884no—
324D4—1884no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—472no—
345C2×C4—472no—
346C2×C4—472no—
347C2×C4—941no—
348C2×C4—941no—
349C2×C4—941no—
350C2×C4—941no—
351C2×C4—941no—
352C2×C4—941no—
353C2×C4—941no—
354C2×C4—941no—
355C2×C4—941no—
356C2×C4—941no—
357C2×C4—941no—
358C2×C4—941no—
359C2×C4—941no—
360C2×C4—941no—
361C2×C4—941no—
362C2×C4—1884no—
363C2×C4—1884no—
364C2×C4D4×D4712yes—
365C2×C4Q8×D4712yes—
366C22—942no—
367C22—942no—
368C22—942no—
369C22—942no—
370C22—942no—
371C22—942no—
372C22—1881no—
373C22—1881no—
374C22—1881no—
375C22—1881no—
376C22—1881no—
377C22—1881no—
378C22—1881no—
379C22—1881no—
380C22C2×D4×D4711yescharacteristic
381C22C2×Q8×D4711yescharacteristic
382C22#41×D4711yescharacteristic
383C22#1309406911yescharacteristic
384C22#1309407011yescharacteristic
385C22#1309407011yescharacteristic
386C22#1309407211yescharacteristic
387C4—22no—
388C4—22no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—942no—
395C4—942no—
396C4—942no—
397C4—1881no—
398C4—1881no—
399C4—1881no—
400C2—1882no—
401C2—1882no—
402C2#1568968411yescharacteristic
403C2#1568972111yescharacteristic
404C2#1568974811yescharacteristic
405C2#1568978811yescharacteristic
406C2#1568979111yescharacteristic
407C2#1568982911yescharacteristic
408C2#1568986911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC47×#398C2
hypercenter#398D47
hypocenterC47#2732
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#13094004C2semidirect
2. C2×#13094002C2semidirect
3. #15689058C2semidirect
4. #15689058C2semidirect
5. #15689056C2semidirect
6. #15689047C2semidirect
7. #15689045C2semidirect
8. #15689027C2semidirect
10. #15689020C2semidirect
13. #15688997C2semidirect
14. #15688994C2semidirect
15. C47×#398C2semidirect
70. C2×D188Q8semidirect
79. C2×Dic94D4semidirect
210. #398D47semidirect
220. C47#2732semidirect