Finite Group Databasebeta

← #30619486

Subgroups — #30619486

Order 15616.

666 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461992C212yesmaximal
2#21461991C212yesmaximal
3#21461917C211yesmaximal, characteristic
4#21461916C211yesmaximal, characteristic
5#21461878C212yesmaximal
6#21461877C212yesmaximal
7#21461736C211yesmaximal, characteristic
8#21461615C211yesmaximal, characteristic
9#21461615C211yesmaximal, characteristic
10#21461533C211yesmaximal, characteristic
11#21448390C211yesmaximal, characteristic
12C2×#14519844C2212yes—
13C2×C4⋊#92420C2211yescharacteristic
14C2×C4⋊#92420C2211yescharacteristic
15C2×C4⋊#92420C2212yes—
16C2×#14519832C2212yes—
17C2×#14519832C2212yes—
18C2×C4×#92420C2211yescharacteristic
19C2×C4×#92420C2211yescharacteristic
20#16840628C2211yescharacteristic
21#16840628C2211yescharacteristic
22#16840628C2211yescharacteristic
23#16840628C2211yescharacteristic
24#16840628C2212yes—
25#16840628C2212yes—
26#16840628C2212yes—
27#16840628C2212yes—
28#16840628C2212yes—
29#16840628C2212yes—
30C2×#14519776C411yescharacteristic
31C2×#14519742C2212yes—
32#16839666C412yes—
33#16839660C411yescharacteristic
34#16839657C412yes—
35#16839631C411yescharacteristic
36C4⋊C4×Dic61C2211yescharacteristic
37C4⋊C4×Dic61C2211yescharacteristic
38#16839613C2212yes—
39C42×Dic61C2211yescharacteristic
40C61×#389C411yescharacteristic
41C22×#92420C2311yescharacteristic
42C22×#92420C2311yescharacteristic
43C22×#92420C2312yes—
44C22×#92420C2312yes—
45C22×#92420C2311yescharacteristic
46C22×#92420C2311yescharacteristic
47#14519844—42no—
48C4⋊#92420—22no—
49C4⋊#92420—22no—
50C4⋊#92420—42no—
51#14519832—42no—
52#14519832—42no—
53C4×#92420—22no—
54C4×#92420Q814yescomplemented
55C22×Dic122C2×C412yes—
56#14519776—24no—
57#14519742—42no—
58#14519742—42no—
59C2×C4⋊Dic61C2×C411yescharacteristic
60C2×C4⋊Dic61C2×C411yescharacteristic
61C2×#11758196C2×C412yes—
62C2×#11758196C2×C412yes—
63C2×#11758196C2×C412yes—
64C2×#11758196C2×C412yes—
65C2×C4×Dic61C2×C411yescharacteristic
66C2×C4×Dic61C2×C411yescharacteristic
67C2×C4×Dic61C2×C411yescharacteristic
68C2×C4×Dic61C2312yes—
69C2×C4×Dic61C2312yes—
70C2×C4×Dic61C2311yescharacteristic
71C2×C4×Dic61C2311yescharacteristic
72C2×C4×Dic61C2311yescharacteristic
73#14519684C2×C412yes—
74#14519684C2×C412yes—
75#14519684C2×C412yes—
76C2×C4⋊C4×C61C2×C411yescharacteristic
77C2×C4⋊C4×C61C2×C411yescharacteristic
78C2×C4⋊C4×C61C2×C412yes—
79C2×C42×C61C2×C411yescharacteristic
80#95×C61C2×C412yes—
81C22×C4×C61C22×C412yes—
82C22×C4×C61C22×C411yescharacteristic
83C22×C4×C61C22×C412yes—
84C22×C4×C61C22×C411yescharacteristic
85C22×C4×C61C22×C411yescharacteristic
86C22×Dic61C22×C412yes—
87C22×Dic61C22×C412yes—
88C22×Dic61C22×C411yescharacteristic
89C22×Dic61C22×C411yescharacteristic
90C22×Dic61C22×C411yescharacteristic
91C22×Dic61C2411yescharacteristic
92C2×Dic122—24no—
93C2×Dic122—42no—
94C2×Dic122—42no—
95C2×#92420—21no—
96C2×#92420—21no—
97C2×#92420—21no—
98C2×#92420—21no—
99C2×#92420—21no—
100C2×#92420—21no—
101C2×#92420—22no—
102C2×#92420—22no—
103C2×#92420—22no—
104C2×#92420—22no—
105C2×#92420—21no—
106C2×#92420—21no—
107C2×#92420—21no—
108C2×#92420—21no—
109C2×#92420—22no—
110C2×#92420—22no—
111C2×#92420C2×Q812yes—
112C2×#92420C2×Q812yes—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—22no—
115C4⋊C4×C61—42no—
116C42×C61—22no—
117C4⋊Dic61—22no—
118C4⋊Dic61—22no—
119#11758196—24no—
120#11758196—42no—
121#11758196—42no—
122#11758196—42no—
123C4×Dic61—22no—
124C4×Dic61—22no—
125C4×Dic61—22no—
126C4×Dic61—22no—
127C4×Dic61—21no—
128C4×Dic61—21no—
129C4×Dic61—21no—
130C4×Dic61—21no—
131C4×Dic61—41no—
132C4×Dic61—41no—
133C4×Dic61M1614yes—
134C4×Dic61C2×Q812yes—
135C4×Dic61#4112yes—
136C23×C61C23×C411yescharacteristic
137C2×C4×C61—21no—
138C2×C4×C61—22no—
139C2×C4×C61—22no—
140C2×C4×C61—21no—
141C2×C4×C61—22no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—22no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—22no—
148C2×C4×C61—22no—
149C2×C4×C61C4×Q812yes—
150C2×C4×C61C2×M1612yes—
151C2×C4×C61#13112yes—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—21no—
155C2×Dic61—21no—
156C2×Dic61—21no—
157C2×Dic61—21no—
158C2×Dic61—22no—
159C2×Dic61—22no—
160C2×Dic61—22no—
161C2×Dic61—22no—
162C2×Dic61—22no—
163C2×Dic61—22no—
164C2×Dic61C4×Q812yes—
165C2×Dic61C2×M1612yes—
166C2×Dic61C2×M1612yes—
167C2×Dic61C22×Q811yescharacteristic
168C2×Dic61C2×#4111yescharacteristic
169C2×Dic61H(5,𝔽2)11yescharacteristic
170C2×Dic61H(5,𝔽2)11yescharacteristic
171C2×Dic612−1+411yescharacteristic
172C2×Dic612−1+411yescharacteristic
173Dic122—48no—
174#92420—22no—
175#92420—22no—
176#92420—41no—
177#92420—41no—
178#92420—42no—
179#92420—42no—
180#22311—611nomaximal, Sylow
181C22×C61C2×C4×Q811yescharacteristic
182C22×C61#51811yescharacteristic
183C22×C61#51911yescharacteristic
184C22×C61C22×M1611yescharacteristic
185C22×C61C2×#13111yescharacteristic
186C22×C61#56811yescharacteristic
187C22×C61#56811yescharacteristic
188C4×C61—42no—
189C4×C61—41no—
190C4×C61—42no—
191C4×C61—41no—
192C4×C61#44614yes—
193Dic61—22no—
194Dic61—22no—
195Dic61—22no—
196Dic61—42no—
197Dic61—42no—
198Dic61#54211yescharacteristic
199Dic61#54211yescharacteristic
200Dic61#54411yescharacteristic
201Dic61#54411yescharacteristic
202C4×C4⋊Q8—611no—
203#1985—612no—
204#1985—612no—
205#1939—611no—
206#1939—611no—
207#1917—612no—
208#1917—612no—
209#1831—611no—
210#1770—611no—
211#1770—611no—
212#1744—611no—
213C2×C61#233111yescharacteristic
214C2×C61C2×#44611yescharacteristic
215C2×C61#296111yescharacteristic
216C2×C61#297511yescharacteristic
217C2×C61#298111yescharacteristic
218C2×C61#298311yescharacteristic
219C2×C61#298511yescharacteristic
220C2×C4⋊Q8—611no—
221C2×C4×Q8—612no—
222C2×C4⋊C8—611no—
223C2×C4⋊C8—611no—
224C2×C4⋊C8—612no—
225C2×C4⋊C8—612no—
226C2×#97—612no—
227C2×#97—612no—
228C2×C4×C8—611no—
229C2×C4×C8—611no—
230#391—612no—
231#389—612no—
232#389—611no—
233#389#692311yescharacteristic
234#384—611no—
235C4×C4⋊C4—611no—
236C4×C4⋊C4—611no—
237C4×C4⋊C4—612no—
238C43—611no—
239#336—612no—
240#336—612no—
241#336—611no—
242#336—611no—
243#336—612no—
244#336—612no—
245#336—612no—
246#336—612no—
247#336—611no—
248#336—611no—
249C61#2231111yescharacteristic, complemented, Sylow
250C22×Q8—612no—
251C22×C8—611no—
252C22×C8—611no—
253C22×C8—611no—
254C22×C8—611no—
255C22×C8—612no—
256C22×C8—612no—
257C4⋊Q8—1224no—
258C4×Q8—2442no—
259C4×Q8—2442no—
260C2×C4⋊C4—612no—
261C2×C4⋊C4—611no—
262C2×C4⋊C4—611no—
263C2×C4⋊C4—612no—
264C2×C4⋊C4—612no—
265C2×C4⋊C4—612no—
266C2×C4⋊C4C2×#692311yescharacteristic
267C2×C4⋊C4C2×#692311yescharacteristic
268C2×C4⋊C4C2×#692312yes—
269C2×C42—611no—
270C2×C42—611no—
271C2×C42—611no—
272C2×C42—612no—
273C2×C42—612no—
274C2×C42—611no—
275C2×C42—611no—
276C2×C42—611no—
277C2×C42C2×#692311yescharacteristic
278C4⋊C8—1222no—
279C4⋊C8—1222no—
280C4⋊C8—2442no—
281C4⋊C8—2442no—
282#97—2442no—
283#97—2442no—
284C4×C8—614no—
285C4×C8—1222no—
286#95—612no—
287#95—612no—
288#95—612no—
289#95C2×#692312yes—
290C2×Q8—1224no—
291C2×Q8—2442no—
292C2×Q8—2442no—
293C22×C4—611no—
294C22×C4—611no—
295C22×C4—611no—
296C22×C4—611no—
297C22×C4—612no—
298C22×C4—612no—
299C22×C4C22×#692311yescharacteristic
300C22×C4C22×#692312yes—
301C22×C4C22×#692312yes—
302C22×C4C22×#692311yescharacteristic
303C22×C4C22×#692311yescharacteristic
304C2×C8—612no—
305C2×C8—612no—
306C2×C8—1221no—
307C2×C8—1221no—
308C2×C8—1221no—
309C2×C8—1221no—
310C2×C8—1221no—
311C2×C8—1221no—
312C2×C8—1222no—
313C2×C8—1222no—
314C2×C8—1222no—
315C2×C8—1222no—
316C2×C8—1221no—
317C2×C8—1221no—
318C2×C8—1221no—
319C2×C8—1221no—
320C2×C8—1222no—
321C2×C8—1222no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—42no—
325C4⋊C4—1224no—
326C4⋊C4—1222no—
327C4⋊C4—1222no—
328C4⋊C4—2442no—
329C4⋊C4—2442no—
330C4⋊C4—2442no—
331C42—22no—
332C42—612no—
333C42—612no—
334C42—614no—
335C42—1222no—
336C42—1222no—
337C42—1222no—
338C42—1222no—
339C42—1221no—
340C42—1221no—
341C42—1221no—
342C42—1221no—
343C42—2441no—
344C42—2441no—
345C23C23×#692311yescharacteristic
346Q8—2448no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—611no—
360C2×C4—611no—
361C2×C4—611no—
362C2×C4—611no—
363C2×C4—611no—
364C2×C4—611no—
365C2×C4—612no—
366C2×C4—612no—
367C2×C4—612no—
368C2×C4—1221no—
369C2×C4—1221no—
370C2×C4—1221no—
371C2×C4—1221no—
372C2×C4—1221no—
373C2×C4—1221no—
374C2×C4—1222no—
375C2×C4—1222no—
376C2×C4—1222no—
377C2×C4—1222no—
378C2×C4—1222no—
379C2×C4—1222no—
380C2×C4Q8×#692312yes—
381C2×C4D61⋊M1612yes—
382C2×C4#1451986412yes—
383C8—1222no—
384C8—1222no—
385C8—2441no—
386C8—2441no—
387C8—2442no—
388C8—2442no—
389C22C2×Q8×#692311yescharacteristic
390C22#1684082911yescharacteristic
391C22#1684083011yescharacteristic
392C22C2×D61⋊M1611yescharacteristic
393C22C2×#1451986411yescharacteristic
394C22#1684084111yescharacteristic
395C22#1684084111yescharacteristic
396C4—42no—
397C4—41no—
398C4—42no—
399C4—41no—
400C4—611no—
401C4—611no—
402C4—611no—
403C4—611no—
404C4—1222no—
405C4—1222no—
406C4—1222no—
407C4—2442no—
408C4—2442no—
409C4#1684081414yes—
410C2#2146231411yescharacteristic
411C2C2×#1684081411yescharacteristic
412C2#2146261111yescharacteristic
413C2#2146268211yescharacteristic
414C2#2146270911yescharacteristic
415C2#2146271611yescharacteristic
416C2#2146272311yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socle, prime radicalC23×C61C23×C4
FittingC61×#389C4
cosocleC22×Dic61C24
hypercenter#389#6923
hypocenterC61#22311
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
54. C4×#92420Q8semidirect
249. C61#22311semidirect

Maximal subgroups

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

subgroupindex
#214619922
#214619912
#214619172
#214619162
#214618782
#214618772
#214617362
#214616152
#214615332
#214483902
#2231161