Finite Group Databasebeta

← #29573870

Subgroups — #29573870

Order 15616.

476 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839818C211yesmaximal, characteristic
2#21445796C211yesmaximal, characteristic
3C4×#14519689C211yesmaximal, characteristic
4C2×#16839514C211yesmaximal, characteristic, complemented
5C2×#16839513C211yesmaximal, characteristic, complemented
6#21443653C211yesmaximal, characteristic, complemented
7C61×#1277C211yesmaximal, characteristic, complemented
8C2×C42×D61C2211yescharacteristic
9#16839818—22no—
10#16839818—22no—
11C2×C4⋊#92418C2211yescharacteristic
12C2×C4⋊#92418C2211yescharacteristic
13C2×#14519689—21no—
14C2×#14519689—21no—
15C2×#14519689C411yescharacteristic
16C2×#14519689C411yescharacteristic
17C2×#14519687—21no—
18C2×C4×#92418C2211yescharacteristic
19#16839516—21no—
20#16839516C411yescharacteristic
21#16839516C411yescharacteristic
22#16839514—22no—
23#16839513—22no—
24C2×C4⋊C8×C61C2211yescharacteristic
25C2×C4⋊C8×C61C2211yescharacteristic
26C2×C4×C8×C61C2211yescharacteristic
27C22×C4×D61C2×C411yescharacteristic
28C22×C4×D61D411yescharacteristic
29C22×C4×D61Q811yescharacteristic
30C22×#92418—21no—
31C22×#92418—21no—
32C22×#92418C2×C411yescharacteristic
33C22×#92418C2×C411yescharacteristic
34C2×C4×Dic61C2×C411yescharacteristic
35C2×C4×Dic61D411yescharacteristic
36C2×C4×Dic61Q811yescharacteristic
37C42×D61—21no—
38C42×D61—21no—
39C42×D61—21no—
40C42×D61—21no—
41C4⋊#92418—21no—
42C4⋊#92418—21no—
43C4⋊#92418—21no—
44C4⋊#92418D412yes—
45#14519689—24no—
46#14519689—22no—
47#14519689—22no—
48#14519689—42no—
49#14519687—24no—
50C4×#92418—21no—
51C4×#92418—21no—
52C22×C8×C61—21no—
53C22×C8×C61—21no—
54C22×C8×C61C2×C411yescharacteristic
55C22×C8×C61C2×C411yescharacteristic
56C2×C42×C61C2311yescharacteristic
57C4⋊C8×C61—21no—
58C4⋊C8×C61—21no—
59C4⋊C8×C61—21no—
60C4⋊C8×C61D412yes—
61C4×C8×C61—21no—
62C4×C8×C61—21no—
63C23×D61C4⋊C411yescharacteristic
64C22×C4×C61C22×C411yescharacteristic
65C22×C4×C61C2×D411yescharacteristic
66C22×C4×C61C2×Q811yescharacteristic
67C22×Dic61C4⋊C411yescharacteristic
68C22×Dic61C4⋊C411yescharacteristic
69C22×Dic61C4⋊C411yescharacteristic
70C2×C4×D61—22no—
71C2×C4×D61—21no—
72C2×C4×D61—21no—
73C2×C4×D61—21no—
74C2×C4×D61—21no—
75C2×C4×D61—21no—
76C2×C4×D61—21no—
77C2×C4×D61—21no—
78C2×C4×D61—21no—
79C2×C4×D61—21no—
80C2×C4×D61—21no—
81C2×C4×D61—21no—
82C2×C4×D61—21no—
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—41no—
86C2×C8×C61—21no—
87C2×C8×C61—21no—
88C2×C8×C61—21no—
89C2×C8×C61—22no—
90C2×C8×C61—21no—
91C2×C8×C61—21no—
92C2×C8×C61—41no—
93C2×C8×C61—41no—
94C2×C8×C61—41no—
95C2×C8×C61—41no—
96C2×C8×C61—41no—
97C2×C8×C61M1612yes—
98C42×C61C2×D411yescharacteristic
99C42×C61C2×D411yescharacteristic
100C42×C61#4111yescharacteristic
101C42×C61#4111yescharacteristic
102C4×Dic61—21no—
103C4×Dic61—21no—
104C4×Dic61—21no—
105C4×Dic61—21no—
106C4×Dic61—21no—
107C4×Dic61—21no—
108C2×#92418—21no—
109C2×#92418—21no—
110C2×#92418—21no—
111C2×#92418—22no—
112C2×#92418—21no—
113C2×#92418—21no—
114C2×#92418—41no—
115C2×#92418—41no—
116C2×#92418—41no—
117C2×#92418—41no—
118C2×#92418—41no—
119C2×#92418M1612yes—
120C23×C61C2×C4⋊C411yescharacteristic
121C22×D61—21no—
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—21no—
125C22×D61—21no—
126C22×D61—21no—
127C22×D61#10812yes—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61C4×D411yescharacteristic
131C2×C4×C61C4×D411yescharacteristic
132C2×C4×C61#12011yescharacteristic
133C2×C4×C61C22⋊Q811yescharacteristic
134C2×C4×C61C22⋊Q811yescharacteristic
135C2×C4×C61#12311yescharacteristic
136C2×C4×C61C2×M1611yescharacteristic
137C2×C4×C61C2×M1611yescharacteristic
138C2×C4×C61#13111yescharacteristic
139C2×C4×C61#13111yescharacteristic
140C2×C4×C61#13511yescharacteristic
141C2×C4×C61#13511yescharacteristic
142C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C61#13711yescharacteristic
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—21no—
148C2×Dic61—21no—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—21no—
152C2×Dic61—21no—
153C4×D61—42no—
154C4×D61—42no—
155C4×D61—41no—
156C4×D61—41no—
157C4×D61—41no—
158C4×D61—41no—
159C4×D61—41no—
160C4×D61—41no—
161C4×D61—41no—
162C4×D61—41no—
163C8×C61—22no—
164C8×C61—41no—
165C8×C61—42no—
166C8×C61—81no—
167#92418—22no—
168#92418—41no—
169#92418—42no—
170#92418—81no—
171#8834—611nomaximal, Sylow
172C22×C61#38511yescharacteristic
173C22×C61#42311yescharacteristic
174C22×C61#42411yescharacteristic
175C22×C61#42711yescharacteristic
176C22×C61#42811yescharacteristic
177C22×C61C2×#10811yescharacteristic
178C22×C61#43011yescharacteristic
179C2×D61—22no—
180C2×D61—22no—
181C2×D61—22no—
182C2×D61—41no—
183C2×D61—41no—
184C2×D61—41no—
185C2×D61—41no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61#43511yescharacteristic
189C4×C61#43511yescharacteristic
190C4×C61#43611yescharacteristic
191C4×C61#43611yescharacteristic
192C4×C61#44911yescharacteristic
193C4×C61#45211yescharacteristic
194C4×C61#48211yescharacteristic
195C4×C61#48511yescharacteristic
196Dic61—41no—
197Dic61—41no—
198Dic61—41no—
199C2×#432—611no—
200#1797—611no—
201C4×C22⋊C8—611no—
202C2×#335—611no—
203C2×#334—611no—
204#1277—611no—
205#1277D6111yescharacteristic, complemented
206C2×C61#156811yescharacteristic
207C2×C61#156911yescharacteristic
208C2×C61#157211yescharacteristic
209C2×C61#157311yescharacteristic
210C2×C61#185111yescharacteristic
211C2×C61#185411yescharacteristic
212C2×C61#185511yescharacteristic
213D61—42no—
214C22×C42—611no—
215#432—1222no—
216#432—1222no—
217C2×C4⋊C8—611no—
218C2×C4⋊C8—611no—
219C2×C4⋊C8C2×D6111yescharacteristic
220C2×C4⋊C8C2×D6111yescharacteristic
221C2×C22⋊C8—611no—
222C2×C22⋊C8—611no—
223C2×C22⋊C8—1221no—
224C2×C22⋊C8—1221no—
225C2×#97—1221no—
226C2×C4×C8—611no—
227C2×C4×C8C2×D6111yescharacteristic
228#336—611no—
229#336—611no—
230#336—1221no—
231#335—1222no—
232#334—1222no—
233C61#883411yescharacteristic, complemented, Sylow
234C23×C4—611no—
235C23×C4—611no—
236C23×C4—611no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—611no—
240C22×C8—611no—
241C22×C8—1221no—
242C22×C8—1221no—
243C22×C8C4×D6111yescharacteristic
244C22×C8C4×D6111yescharacteristic
245C2×C42—611no—
246C2×C42—611no—
247C2×C42—611no—
248C2×C42—1221no—
249C2×C42—1221no—
250C2×C42—1221no—
251C2×C42—1221no—
252C2×C42C22×D6111yescharacteristic
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—612no—
257C4⋊C8—1221no—
258C4⋊C8—1221no—
259C4⋊C8—1221no—
260C4⋊C8D24412yes—
261C22⋊C8—1222no—
262C22⋊C8—1224no—
263C22⋊C8—1222no—
264C22⋊C8—2442no—
265#97—1224no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—1221no—
269C4×C8—1221no—
270C24—611no—
271C22×C4—611no—
272C22×C4—611no—
273C22×C4—611no—
274C22×C4—1222no—
275C22×C4—1221no—
276C22×C4—1221no—
277C22×C4—1221no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—1221no—
281C22×C4—1221no—
282C22×C4—1221no—
283C22×C4—1221no—
284C22×C4—1221no—
285C22×C4—1221no—
286C22×C4—1221no—
287C22×C4—1221no—
288C22×C4—1221no—
289C22×C4—2441no—
290C22×C4C2×C4×D6111yescharacteristic
291C22×C4D4×D6111yescharacteristic
292C22×C4Q8×D6111yescharacteristic
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—22no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—612no—
305C2×C8—1221no—
306C2×C8—1221no—
307C2×C8—1221no—
308C2×C8—1222no—
309C2×C8—1221no—
310C2×C8—1221no—
311C2×C8—2441no—
312C2×C8—2441no—
313C2×C8—2441no—
314C2×C8—2441no—
315C2×C8—2441no—
316C2×C8#1175818912yes—
317C42—1221no—
318C42—1221no—
319C42—1221no—
320C42—1221no—
321C42—1221no—
322C42—1221no—
323C42C2×D24411yescharacteristic
324C42#1175822211yescharacteristic
325C42D4×D6111yescharacteristic
326C42#1175822411yescharacteristic
327C23—612no—
328C23—1221no—
329C23—1221no—
330C23—1221no—
331C23—1221no—
332C23—1221no—
333C23—1221no—
334C23C4⋊C4×D6111yescharacteristic
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—1221no—
338C2×C4—1221no—
339C2×C4—1221no—
340C2×C4—1221no—
341C2×C4—1221no—
342C2×C4—1221no—
343C2×C4—1221no—
344C2×C4—1221no—
345C2×C4—1221no—
346C2×C4—2442no—
347C2×C4—2442no—
348C2×C4—2441no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4—2441no—
352C2×C4—2441no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2441no—
356C2×C4#1451973711yescharacteristic
357C2×C4C4×D24411yescharacteristic
358C2×C4#1451974711yescharacteristic
359C2×C4#1451975511yescharacteristic
360C2×C4#1451975711yescharacteristic
361C2×C4#1451975911yescharacteristic
362C2×C4M16×D6111yescharacteristic
363C2×C4C2×#1175818911yescharacteristic
364C2×C4#1451978311yescharacteristic
365C2×C4#1451978511yescharacteristic
366C2×C4#1451979811yescharacteristic
367C2×C4#1451979811yescharacteristic
368C2×C4#1451980111yescharacteristic
369C2×C4#1451980911yescharacteristic
370C8—22no—
371C8—41no—
372C8—42no—
373C8—81no—
374C8—1222no—
375C8—4881no—
376C8—2441no—
377C8—2442no—
378C22—1222no—
379C22—1222no—
380C22—1222no—
381C22—2441no—
382C22—2441no—
383C22—2441no—
384C22—2441no—
385C22#1683963611yescharacteristic
386C22#1683978411yescharacteristic
387C22#1683978711yescharacteristic
388C22#1683979711yescharacteristic
389C22#1683980311yescharacteristic
390C22#108×D6111yescharacteristic
391C22#1683980911yescharacteristic
392C4—21no—
393C4—21no—
394C4—2441no—
395C4—2441no—
396C4—2441no—
397C4#1683983011yescharacteristic
398C4#1683983511yescharacteristic
399C4#1683983711yescharacteristic
400C4#1683984111yescharacteristic
401C4#1683990811yescharacteristic
402C4#1683992711yescharacteristic
403C4#1684008511yescharacteristic
404C4#1684010111yescharacteristic
405C2—2442no—
406C2#2144457911yescharacteristic
407C2#2144458511yescharacteristic
408C2#2144459611yescharacteristic
409C2#2144460311yescharacteristic
410C2#2144600311yescharacteristic
411C2#2144601911yescharacteristic
412C2#2144602511yescharacteristic
subgroupG/H
centerC22×C4D4×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1277C2
socleC23×C61C2×C4⋊C4
prime radicalC23×D61C4⋊C4
cosocleC2×C42×C61C23
hypercenter#1277D61
hypocenterC61#8834
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
4. C2×#16839514C2semidirect
5. C2×#16839513C2semidirect
6. #21443653C2semidirect
7. C61×#1277C2semidirect
205. #1277D61semidirect
233. C61#8834semidirect

Maximal subgroups

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

subgroupindex
C2×#168398182
#214457962
C4×#145196892
C2×#168395142
C2×#168395132
#214436532
C61×#12772
#883461