Finite Group Databasebeta

← #29779591

Subgroups — #29779591

Order 15616.

866 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454124C211yesmaximal, characteristic, complemented
2#21448306C211yesmaximal, characteristic
3#21447239C211yesmaximal, characteristic, complemented
4#21447232C211yesmaximal, characteristic, complemented
5#21446967C211yesmaximal, characteristic, complemented
6#21446513C212yesmaximal, complemented
7#21446507C212yesmaximal, complemented
8#21445912C212yesmaximal, complemented
9#21445756C212yesmaximal
10#21445580C211yesmaximal, characteristic, complemented
11C61×#2025C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C2×#14519735C2211yescharacteristic
14C2×C42×D61C2211yescharacteristic
15C2×#14519702C2212yes—
16C2×#14519700C2212yes—
17C2×#14519698C2212yes—
18C2×#14519689C2212yes—
19C2×#14519687C2212yes—
20#16839668C2211yescharacteristic
21#16839667C2211yescharacteristic
22#16839665C2211yescharacteristic
23#16839658C2212yes—
24#16839643—22no—
25#16839638—22no—
26#16839633—22no—
27#16839628C2211yescharacteristic
28#16839619C2212yes—
29C22⋊C4×Dic61—22no—
30#16839525C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2211yescharacteristic
34C2×#125×C61C2211yescharacteristic
35C2×#102×C61C2212yes—
36C61×#390C2211yescharacteristic
37C61×#340C2212yes—
38C61×#336C2211yescharacteristic
39C2×#11758224—22no—
40C2×#11758224—24no—
41C2×#11758224—22no—
42C2×#11758224—42no—
43C22×Dic122C2311yescharacteristic
44C22×C22⋊D61—21no—
45C23×Dic61—21no—
46C22×C4×D61D412yes—
47C22×C4×D61C2311yescharacteristic
48C22×#92418C2312yes—
49#14519735—24no—
50C2×C4⋊Dic61C2311yescharacteristic
51C2×C4⋊Dic61C2312yes—
52C2×#11758196—22no—
53C2×C22⋊Dic61—22no—
54C2×#11758198—21no—
55C2×#11758198—22no—
56C2×C4×Dic61—21no—
57C2×C4×Dic61D412yes—
58C2×C4×Dic61C2311yescharacteristic
59C42×D61—42no—
60#14519702—24no—
61#14519700—42no—
62#14519698—24no—
63#14519689—24no—
64#14519687—42no—
65#14519684—22no—
66#14519684—22no—
67#14519684—22no—
68#14519684—22no—
69#14519684—22no—
70C22×D4×C61C2311yescharacteristic
71C22×C8×C61C2312yes—
72#125×C61—42no—
73C2×C4⋊C4×C61—22no—
74C2×C4⋊C4×C61C2311yescharacteristic
75C2×C4⋊C4×C61C2312yes—
76C2×C22⋊C4×C61—22no—
77C2×C42×C61C2311yescharacteristic
78#102×C61—42no—
79C24×C61—21no—
80C23×D61C2×D411yescharacteristic
81C2×D4×C61—22no—
82C2×D4×C61—21no—
83C2×D4×C61—22no—
84C2×D4×C61#4112yes—
85C22×C4×C61—21no—
86C22×C4×C61—22no—
87C22×C4×C61C2×D412yes—
88C22×C4×C61C2411yescharacteristic
89C2×C22⋊D61—24no—
90C2×C22⋊D61—42no—
91C2×C22⋊D61—42no—
92C22×Dic61—24no—
93C22×Dic61—21no—
94C22×Dic61—22no—
95C22×Dic61—21no—
96C22×Dic61—42no—
97C22×Dic61—42no—
98C22×Dic61C2×D412yes—
99C22×Dic61C2×D411yescharacteristic
100#11758224—48no—
101#11758224—48no—
102C2×C4×D61—21no—
103C2×C4×D61—21no—
104C2×C4×D61—22no—
105C2×C4×D61—42no—
106C2×C4×D61—42no—
107C2×C4×D61—42no—
108C2×C4×D61#4114yes—
109C2×Dic122—22no—
110C2×Dic122—22no—
111C2×Dic122—21no—
112C2×Dic122#4112yes—
113C2×C8×C61—22no—
114C2×C8×C61—22no—
115C2×C8×C61—22no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61—22no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—44no—
120C22⋊C4×C61—44no—
121C42×C61—22no—
122C22⋊Dic61—44no—
123#11758198—44no—
124#11758198—42no—
125C4⋊Dic61—22no—
126C4⋊Dic61—22no—
127C4⋊Dic61—22no—
128#11758196—44no—
129C4×Dic61—22no—
130C4×Dic61—42no—
131C4×Dic61—42no—
132C2×#92418—22no—
133C2×#92418—22no—
134C2×#92418—22no—
135C23×C61—22no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61—42no—
139C23×C61—42no—
140C23×C61C22×D411yescharacteristic
141C22×D61—21no—
142C22×D61—21no—
143C22×D61—22no—
144C22×D61—22no—
145C22×D61#12412yes—
146D4×C61—24no—
147D4×C61—42no—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61—41no—
152C2×C4×C61—42no—
153C2×C4×C61—42no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61C2×#4111yescharacteristic
157C2×C4×C61C2×#4112yes—
158C2×C4×C61C2×#4111yescharacteristic
159C2×C4×C61H(5,𝔽2)12yes—
160C22⋊D61—48no—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—22no—
164C2×Dic61—22no—
165C2×Dic61—22no—
166C2×Dic61—22no—
167C2×Dic61—48no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—42no—
171C2×Dic61—41no—
172C2×Dic61—42no—
173C2×Dic61—42no—
174C2×Dic61—42no—
175C2×Dic61#12412yes—
176C4×D61—24no—
177C4×D61—24no—
178C4×D61—48no—
179Dic122—24no—
180Dic122—42no—
181C8×C61—42no—
182#92418—42no—
183#29114—611nomaximal, Sylow
184C22×C61—42no—
185C22×C61—44no—
186C22×C61—81no—
187C22×C61—81no—
188C22×C61—82no—
189C22×C61#52511yescharacteristic
190C22×C61C2×#12411yescharacteristic
191C22×C61#53411yescharacteristic
192C22×C61#57512yes—
193C22×C61#57512yes—
194C2×D61—22no—
195C2×D61—22no—
196C2×D61—22no—
197C2×D61—42no—
198C2×D61—42no—
199C4×C61—42no—
200C4×C61—81no—
201C4×C61—82no—
202C4×C61#55112yes—
203C4×C61#55112yes—
204Dic61—22no—
205Dic61—42no—
206Dic61—42no—
207Dic61—82no—
208Dic61—81no—
209#2851—611no—
210#2297—611no—
211#2074—611no—
212#2073—611no—
213#2026—611no—
214#2025D6111yescharacteristic, complemented
215#1936—612no—
216#1935—612no—
217#1825—612no—
218#1788—612no—
219#1749—611no—
220C2×C61—82no—
221C2×C61#256111yescharacteristic
222C2×C61#311112yes—
223C2×C61#313511yescharacteristic
224C2×C61#314211yescharacteristic
225C2×C61#319712yes—
226D61—42no—
227C22×#41—611no—
228C2×#125C2×D6111yescharacteristic
229C2×#117—611no—
230C22×C42—611no—
231C2×#103—612no—
232C2×#103—612no—
233C2×#102—612no—
234C2×#102C2×D6112yes—
235C2×C22⋊C8—612no—
236C2×#97—612no—
237#391—611no—
238#391—611no—
239#390—611no—
240#390C2×D6111yescharacteristic
241#389—612no—
242#387—1222no—
243#386—1222no—
244#385—1222no—
245#382—611no—
246#380—612no—
247C4×C22⋊C4—1222no—
248#340—612no—
249#340—612no—
250#340—612no—
251#340C2×D6112yes—
252#336—611no—
253#336C2×D6111yescharacteristic
254C61#2911411yescharacteristic, complemented, Sylow
255C2×#41—1224no—
256C2×#41—1222no—
257C2×#41—1222no—
258C2×#41—2442no—
259C22×Q8—611no—
260C22×D4—1221no—
261C22×D4C22×D6111yescharacteristic
262C23×C4—611no—
263C23×C4—612no—
264C23×C4—1221no—
265C22×C8—612no—
266C22×C8C22×D6112yes—
267#125—42no—
268#117—1224no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—611no—
271C2×C4⋊C4—612no—
272C2×C4⋊C4—1222no—
273C2×C4⋊C4C22×D6111yescharacteristic
274C2×C4⋊C4C22×D6112yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—1221no—
277C2×C22⋊C4—1222no—
278C2×C22⋊C4—1222no—
279C2×C42—611no—
280C2×C42—612no—
281C2×C42—1221no—
282C2×C42—2442no—
283C2×C42C22×D6111yescharacteristic
284#103—1224no—
285#103—2442no—
286#102—42no—
287#102—1224no—
288C22⋊C8—1224no—
289#97—2442no—
290#95—1222no—
291#95—1222no—
292#95—1222no—
293#95—1222no—
294#95—1222no—
295C24—21no—
296C24—611no—
297#41—2448no—
298#41—2448no—
299C2×Q8—612no—
300C2×Q8—1221no—
301C2×Q8—1222no—
302C2×Q8—1222no—
303C2×D4—22no—
304C2×D4—21no—
305C2×D4—22no—
306C2×D4—1224no—
307C2×D4—2442no—
308C2×D4—2442no—
309C2×D4#1175822612yes—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—611no—
313C22×C4—614no—
314C22×C4—612no—
315C22×C4—1221no—
316C22×C4—1221no—
317C22×C4—1222no—
318C22×C4—1224no—
319C22×C4—1221no—
320C22×C4—1222no—
321C22×C4—1221no—
322C22×C4—2442no—
323C22×C4—2442no—
324C22×C4—2442no—
325C22×C4—2442no—
326C22×C4—2442no—
327C22×C4D4×D6112yes—
328C22×C4C23×D6111yescharacteristic
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—1222no—
333C2×C8—1222no—
334C2×C8—1222no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—44no—
339C4⋊C4—1222no—
340C4⋊C4—1222no—
341C4⋊C4—1222no—
342C4⋊C4—2444no—
343C22⋊C4—44no—
344C22⋊C4—2442no—
345C22⋊C4—2444no—
346C22⋊C4—2444no—
347C42—22no—
348C42—1222no—
349C42—2442no—
350C42—2442no—
351C23—22no—
352C23—41no—
353C23—42no—
354C23—42no—
355C23—41no—
356C23—612no—
357C23—1221no—
358C23—1222no—
359C23—1222no—
360C23—1221no—
361C23C2×D4×D6111yescharacteristic
362Q8—1224no—
363Q8—2442no—
364D4—24no—
365D4—42no—
366D4—2448no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—612no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1224no—
379C2×C4—1224no—
380C2×C4—1222no—
381C2×C4—1222no—
382C2×C4—1222no—
383C2×C4—1222no—
384C2×C4—2448no—
385C2×C4—2448no—
386C2×C4—2442no—
387C2×C4—2441no—
388C2×C4—2442no—
389C2×C4—2442no—
390C2×C4—2442no—
391C2×C4—2441no—
392C2×C4—2441no—
393C2×C4#41×D6112yes—
394C2×C4#41×D6111yescharacteristic
395C2×C4C2×#1175822611yescharacteristic
396C2×C4#1451982212yes—
397C8—42no—
398C8—2442no—
399C22—44no—
400C22—42no—
401C22—82no—
402C22—81no—
403C22—81no—
404C22—1222no—
405C22—1222no—
406C22—1222no—
407C22—2442no—
408C22—2442no—
409C22#1684027511yescharacteristic
410C22#124×D6111yescharacteristic
411C22#1684032211yescharacteristic
412C22#1684055812yes—
413C22#1684055912yes—
414C4—42no—
415C4—82no—
416C4—81no—
417C4—1222no—
418C4—4882no—
419C4—4881no—
420C4—2442no—
421C4—2442no—
422C4#1684043912yes—
423C4#1684044212yes—
424C2—82no—
425C2—2442no—
426C2#2145097311yescharacteristic
427C2#2145593312yes—
428C2#2145616311yescharacteristic
429C2#2145621611yescharacteristic
430C2#2145665812yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2025C2
socleC23×C61C22×D4
prime radicalC22×#11758224C22
hypercenter#2025D61
hypocenterC61#29114
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2025

Complements

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

NH ≅ G/Ntype
1. #21454124C2semidirect
3. #21447239C2semidirect
4. #21447232C2semidirect
5. #21446967C2semidirect
6. #21446513C2semidirect
7. #21446507C2semidirect
8. #21445912C2semidirect
10. #21445580C2semidirect
11. C61×#2025C2semidirect
214. #2025D61semidirect
254. C61#29114semidirect

Maximal subgroups

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

subgroupindex
#214541242
#214483062
#214472392
#214472322
#214469672
#214465132
#214465072
#214459122
#214457562
#214455802
C61×#20252
#2911461