Finite Group Databasebeta

← #29779730

Subgroups — #29779730

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519765C211yesmaximal, characteristic
2#21447387C212yesmaximal, complemented
3#21447060C212yesmaximal
4#21446515C212yesmaximal
5#21446508C212yesmaximal, complemented
6#21445831C211yesmaximal, characteristic, complemented
7#21445831C211yesmaximal, characteristic, complemented
8#21445826C211yesmaximal, characteristic
9#21445826C211yesmaximal, characteristic
10#21445672C211yesmaximal, characteristic, complemented
11C61×#2864C211yesmaximal, characteristic, complemented
12C2×#14519765C2211yescharacteristic
13C2×C4×Dic122C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15C2×C4⋊#92418C2211yescharacteristic
16C2×C4⋊#92418C2211yescharacteristic
17C2×#14519700C2212yes—
18C2×#14519700C2212yes—
19C2×#14519697C2212yes—
20C2×#14519697C2212yes—
21C2×#14519688C2212yes—
22#16839668C2211yescharacteristic
23#16839664C2211yescharacteristic
24#16839653—22no—
25#16839628C2211yescharacteristic
26C4×#11758198—22no—
27#16839524C2212yes—
28#16839524C2212yes—
29#16839524C2212yes—
30#16839524C2212yes—
31#16839517C2212yes—
32C61×#389C2212yes—
33C61×#389C2212yes—
34C61×#387—22no—
35C61×#387—22no—
36C61×#382C2211yescharacteristic
37C61×#382C2211yescharacteristic
38C43×C61C2211yescharacteristic
39C22×Dic122C2311yescharacteristic
40C22×D244C2311yescharacteristic
41C22×C4×D61—21no—
42C22×#92418C2312yes—
43C22×#92418C2312yes—
44#14519765—22no—
45#14519765—22no—
46C4×Dic122—22no—
47C4×Dic122—22no—
48C4×D244—22no—
49C4×D244—22no—
50C2×C4⋊Dic61C2311yescharacteristic
51C2×C4⋊Dic61C2311yescharacteristic
52C2×#11758196—21no—
53C2×#11758198—22no—
54C2×#11758198—21no—
55C2×#11758198—22no—
56C2×C4×Dic61—21no—
57C4⋊#92418—22no—
58C4⋊#92418—22no—
59#14519700—24no—
60#14519700—42no—
61#14519697—24no—
62#14519697—42no—
63#14519688—42no—
64#14519684—22no—
65#14519684—22no—
66C2×C4⋊C4×C61—22no—
67C2×C4⋊C4×C61—22no—
68C2×C4⋊C4×C61C2312yes—
69C2×C4⋊C4×C61C2312yes—
70C2×C42×C61D412yes—
71C2×C42×C61D412yes—
72C2×C42×C61C2311yescharacteristic
73C2×C42×C61C2311yescharacteristic
74C2×C42×C61C2311yescharacteristic
75#95×C61—21no—
76#95×C61—21no—
77#95×C61—21no—
78#95×C61—21no—
79#95×C61—22no—
80#95×C61—22no—
81C23×D61—21no—
82C22×C4×C61—22no—
83C22×C4×C61—22no—
84C22×C4×C61C2×D412yes—
85C22×C4×C61C2×D412yes—
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2411yescharacteristic
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C2×D244—21no—
93C2×D244—21no—
94C2×D244—21no—
95C2×D244—21no—
96C2×D244—21no—
97C2×D244#4112yes—
98C2×C4×D61—44no—
99C2×C4×D61—42no—
100C2×Dic122—21no—
101C2×Dic122—21no—
102C2×Dic122—21no—
103C2×Dic122—21no—
104C2×Dic122—21no—
105C2×Dic122#4112yes—
106C4⋊C4×C61—22no—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—22no—
109C4⋊C4×C61—22no—
110C4⋊C4×C61—44no—
111C4⋊C4×C61—44no—
112C42×C61—21no—
113C42×C61—21no—
114C42×C61—21no—
115C42×C61—21no—
116C42×C61—42no—
117C42×C61—42no—
118C42×C61#4112yes—
119C42×C61#4112yes—
120#11758198—28no—
121#11758198—44no—
122#11758198—42no—
123C4⋊Dic61—21no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127#11758196—42no—
128C4×Dic61—42no—
129C2×#92418—22no—
130C2×#92418—22no—
131C2×#92418—22no—
132C2×#92418—22no—
133C2×#92418—22no—
134C2×#92418—22no—
135C23×C61C22×D411yescharacteristic
136C22×D61—22no—
137C22×D61—41no—
138C22×D61—41no—
139C22×D61—42no—
140C22×D61—42no—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—22no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—22no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61—42no—
153C2×C4×C61—42no—
154C2×C4×C61—42no—
155C2×C4×C61#12412yes—
156C2×C4×C61#12412yes—
157C2×C4×C61C2×#4111yescharacteristic
158C2×C4×C61C2×#4111yescharacteristic
159C2×C4×C61C2×#4111yescharacteristic
160C2×C4×C61C2×#4111yescharacteristic
161C2×C4×C61H(5,𝔽2)11yescharacteristic
162C2×C4×C612−1+411yescharacteristic
163C2×Dic61—22no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170D244—22no—
171D244—22no—
172D244—41no—
173D244—41no—
174C4×D61—84no—
175Dic122—22no—
176Dic122—22no—
177Dic122—41no—
178Dic122—41no—
179#92418—42no—
180#92418—42no—
181#29128—611nomaximal, Sylow
182C22×C61#52511yescharacteristic
183C22×C61C2×#12411yescharacteristic
184C22×C61#53711yescharacteristic
185C22×C61#57512yes—
186C22×C61#57512yes—
187C2×D61—42no—
188C2×D61—44no—
189C2×D61—81no—
190C2×D61—81no—
191C2×D61—82no—
192C4×C61—22no—
193C4×C61—22no—
194C4×C61—42no—
195C4×C61—42no—
196C4×C61—82no—
197C4×C61—82no—
198C4×C61#55211yescharacteristic
199C4×C61#55211yescharacteristic
200C4×C61#55311yescharacteristic
201C4×C61#55311yescharacteristic
202Dic61—42no—
203Dic61—81no—
204Dic61—81no—
205#2864D6111yescharacteristic, complemented
206C4×#124—611no—
207#2101—612no—
208#2042—612no—
209#1936—612no—
210#1935—612no—
211#1805—611no—
212#1805—611no—
213#1804—611no—
214#1804—611no—
215#1767—611no—
216C2×C61#256211yescharacteristic
217C2×C61#311112yes—
218C2×C61#313511yescharacteristic
219C2×C61#314011yescharacteristic
220C2×C61#325612yes—
221D61—82no—
222C2×#124—611no—
223C2×C4×Q8—611no—
224C2×C4×D4—611no—
225C2×C4⋊C8—611no—
226C2×C4⋊C8—611no—
227C2×#103—612no—
228C2×#103—612no—
229C2×#102—612no—
230C2×#102—612no—
231C2×#97—612no—
232#391—611no—
233#390—611no—
234#389C2×D6112yes—
235#389C2×D6112yes—
236#388—1222no—
237#387—22no—
238#387—22no—
239#382—611no—
240#382C2×D6111yescharacteristic
241#382C2×D6111yescharacteristic
242C4×C22⋊C4—1222no—
243C43C2×D6111yescharacteristic
244#340—612no—
245#340—612no—
246#340—612no—
247#340—612no—
248#336—612no—
249C61#2912811yescharacteristic, complemented, Sylow
250C22×Q8—611no—
251C22×D4—611no—
252C23×C4—1221no—
253C22×C8—612no—
254C22×C8—612no—
255#124—1222no—
256#124—1222no—
257C4×Q8—1222no—
258C4×Q8—1222no—
259C4×D4—1222no—
260C4×D4—1222no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—611no—
264C2×C4⋊C4—611no—
265C2×C4⋊C4—1221no—
266C2×C4⋊C4C22×D6112yes—
267C2×C4⋊C4C22×D6112yes—
268C2×C22⋊C4—1222no—
269C2×C22⋊C4—1221no—
270C2×C22⋊C4—1222no—
271C2×C42—1221no—
272C2×C42C22⋊D6112yes—
273C2×C42C22⋊D6112yes—
274C2×C42C22×D6111yescharacteristic
275C2×C42C22×D6111yescharacteristic
276C2×C42C22×D6111yescharacteristic
277C4⋊C8—1222no—
278C4⋊C8—1222no—
279#103—1224no—
280#103—2442no—
281#102—1224no—
282#102—2442no—
283#97—2442no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—22no—
289#95—22no—
290#95—1222no—
291#95—1222no—
292C24—1221no—
293C2×Q8—612no—
294C2×Q8—1221no—
295C2×Q8—1221no—
296C2×Q8—1221no—
297C2×Q8—1221no—
298C2×Q8—1221no—
299C2×D4—612no—
300C2×D4—1221no—
301C2×D4—1221no—
302C2×D4—1221no—
303C2×D4—1221no—
304C2×D4—1221no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—1221no—
308C22×C4—1221no—
309C22×C4—1221no—
310C22×C4—2444no—
311C22×C4—2442no—
312C22×C4C2×C22⋊D6112yes—
313C22×C4C2×C22⋊D6112yes—
314C22×C4C2×C22⋊D6111yescharacteristic
315C22×C4C2×C22⋊D6111yescharacteristic
316C22×C4C23×D6111yescharacteristic
317C2×C8—1222no—
318C2×C8—1222no—
319C2×C8—1222no—
320C2×C8—1222no—
321C2×C8—1222no—
322C2×C8—1222no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—1221no—
330C4⋊C4—1221no—
331C4⋊C4—1221no—
332C4⋊C4—1221no—
333C4⋊C4—2442no—
334C22⋊C4—1228no—
335C22⋊C4—2444no—
336C22⋊C4—2442no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—42no—
342C42—42no—
343C42—2442no—
344C42#1175822612yes—
345C42#1175822612yes—
346C23—1222no—
347C23—2441no—
348C23—2442no—
349C23—2442no—
350C23—2441no—
351C23C22×C22⋊D6111yescharacteristic
352Q8—1222no—
353Q8—1222no—
354Q8—2441no—
355Q8—2441no—
356D4—1222no—
357D4—1222no—
358D4—2441no—
359D4—2441no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—1222no—
375C2×C4—4884no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2442no—
379C2×C4—2442no—
380C2×C4—2441no—
381C2×C4—2441no—
382C2×C4#1451976612yes—
383C2×C4#1451976612yes—
384C2×C4#41×D6111yescharacteristic
385C2×C4#41×D6111yescharacteristic
386C2×C4C2×#1175822611yescharacteristic
387C2×C4C2×#1175822611yescharacteristic
388C2×C4#1451982311yescharacteristic
389C2×C4#1451982411yescharacteristic
390C8—2442no—
391C8—2442no—
392C22—4881no—
393C22—4882no—
394C22—4881no—
395C22—2444no—
396C22—2442no—
397C22#1684026911yescharacteristic
398C22C2×#1451976611yescharacteristic
399C22#1684034011yescharacteristic
400C22#1684056012yes—
401C22#1684056012yes—
402C4—22no—
403C4—22no—
404C4—42no—
405C4—42no—
406C4—82no—
407C4—82no—
408C4—4881no—
409C4—4881no—
410C4—2442no—
411C4#1684045011yescharacteristic
412C4#1684045011yescharacteristic
413C4#1684046511yescharacteristic
414C4#1684046511yescharacteristic
415C2—4882no—
416C2#2145098611yescharacteristic
417C2#2145592212yes—
418C2#2145616711yescharacteristic
419C2#2145620811yescharacteristic
420C2#2145721512yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2864C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2864D61
hypocenterC61#29128
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C43 ◁ #2864

Complements

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

NH ≅ G/Ntype
2. #21447387C2semidirect
5. #21446508C2semidirect
6. #21445831C2semidirect
7. #21445831C2semidirect
10. #21445672C2semidirect
11. C61×#2864C2semidirect
205. #2864D61semidirect
249. C61#29128semidirect

Maximal subgroups

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

subgroupindex
C4×#145197652
#214473872
#214470602
#214465152
#214465082
#214458312
#214458262
#214456722
C61×#28642
#2912861