Finite Group Databasebeta

← #29716286

Subgroups — #29716286

Order 15616.

1014 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455620C212yesmaximal, complemented
2#21448531C211yesmaximal, characteristic
3#21447161C212yesmaximal, complemented
4#21446977C212yesmaximal, complemented
5#21446801C212yesmaximal, complemented
6#21446207C212yesmaximal
7#21446029C212yesmaximal
8#21445811C211yesmaximal, characteristic
9C61×#3501C211yesmaximal, characteristic
10C2×#14519764C2211yescharacteristic
11C2×C4×Dic122C2211yescharacteristic
12#16840066—24no—
13#16840054—22no—
14#16840044—22no—
15#16839805C2214yes—
16#16839786C2212yes—
17#16839770C2214yes—
18C2×#14519694C2212yes—
19#16839667C2211yescharacteristic
20#16839665C2211yescharacteristic
21#16839650—22no—
22#16839635—22no—
23#16839631C2211yescharacteristic
24#16839608C2211yescharacteristic
25#16839531C2212yes—
26#16839521C2214yes—
27#16839521C2214yes—
28C61×#554—22no—
29C61×#554—24no—
30D4×Q8×C61—22no—
31C61×#544C2212yes—
32C61×#543—22no—
33C61×#533C2212yes—
34C2×C22⋊Q8×C61—22no—
35C61×#518C2212yes—
36C2×C4×D4×C61C2211yescharacteristic
37C22×Dic122C2311yescharacteristic
38C2×#11758194C2314yes—
39#14519764—22no—
40#14519764Q814yescomplemented
41C4×Dic122—22no—
42C4×Dic122—22no—
43C2×C4⋊Dic61C2311yescharacteristic
44C2×#11758196—21no—
45C2×#11758196—22no—
46C2×C22⋊Dic61—22no—
47C2×C22⋊Dic61—22no—
48C2×C4×Dic61—21no—
49C2×C4×Dic61C2311yescharacteristic
50C2×C4×Dic61C2311yescharacteristic
51#14519710—24no—
52#14519708—24no—
53C4⋊#92418—22no—
54C4⋊#92418—22no—
55#14519700—24no—
56#14519700—24no—
57#14519694—24no—
58#14519684—21no—
59#14519684—21no—
60#14519684—22no—
61C22×Q8×C61—22no—
62C22×D4×C61C2311yescharacteristic
63C23×C4×C61D412yes—
64C4⋊Q8×C61—24no—
65C4⋊Q8×C61D412yes—
66C4⋊Q8×C61D412yes—
67#125×C61—24no—
68#125×C61D414yes—
69C22⋊Q8×C61—28no—
70C22⋊Q8×C61—28no—
71C22⋊Q8×C61—24no—
72C22⋊Q8×C61—24no—
73C22⋊Q8×C61—22no—
74C22⋊Q8×C61—22no—
75C22⋊Q8×C61—24no—
76C4×Q8×C61—22no—
77C4×Q8×C61—22no—
78C4×D4×C61—24no—
79C4×D4×C61—24no—
80C4×D4×C61—22no—
81C4×D4×C61—22no—
82#117×C61C2314yes—
83C2×C4⋊C4×C61—22no—
84C2×C4⋊C4×C61—24no—
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C22⋊C4×C61—24no—
87C2×C22⋊C4×C61D412yes—
88C2×C42×C61C2311yescharacteristic
89C24×C61C2×D412yes—
90C2×Q8×C61—24no—
91C2×Q8×C61—22no—
92C2×Q8×C61—22no—
93C2×Q8×C61—84no—
94C2×D4×C61—22no—
95C2×D4×C61—22no—
96C2×D4×C61C2×Q812yes—
97C2×D4×C61#4112yes—
98C22×C4×C61—24no—
99C22×C4×C61—24no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—82no—
103C22×C4×C61C2×D412yes—
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2411yescharacteristic
107C22×Dic61—21no—
108C22×Dic61—21no—
109C22×Dic61—21no—
110C22×Dic61—21no—
111C2×Dic122—22no—
112C2×Dic122—22no—
113C2×Dic122—21no—
114C2×Dic122C2×Q812yescomplemented
115C4⋊C4×C61—24no—
116C4⋊C4×C61—28no—
117C4⋊C4×C61—24no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—24no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61C2×D414yes—
123C4⋊C4×C61C2×D414yes—
124C22⋊C4×C61—28no—
125C22⋊C4×C61—24no—
126C22⋊C4×C61—24no—
127C42×C61—24no—
128C42×C61C2×D412yes—
129C42×C61C2×D412yes—
130C22⋊Dic61—28no—
131C22⋊Dic61—82no—
132C4⋊Dic61—22no—
133#11758196—41no—
134#11758196—41no—
135#11758196—82no—
136C4×Dic61—21no—
137C4×Dic61—22no—
138C4×Dic61—42no—
139C4×Dic61C2×Q812yescomplemented
140#11758194—24no—
141#11758194—24no—
142C2×#92418—24no—
143C23×C61—22no—
144C23×C61—22no—
145C23×C61—42no—
146C23×C61—42no—
147C23×C61C22⋊Q814yes—
148C23×C61C22×D411yescharacteristic
149Q8×C61—44no—
150Q8×C61—82no—
151Q8×C61—84no—
152D4×C61—24no—
153D4×C61—24no—
154C2×C4×C61—24no—
155C2×C4×C61—24no—
156C2×C4×C61—22no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—44no—
160C2×C4×C61—42no—
161C2×C4×C61—82no—
162C2×C4×C61—84no—
163C2×C4×C61—84no—
164C2×C4×C61#12012yes—
165C2×C4×C61#12011yescharacteristic
166C2×C4×C61#12011yescharacteristic
167C2×C4×C61C22×D412yes—
168C2×C4×C61C22×Q811yescharacteristic
169C2×C4×C61C2×#4111yescharacteristic
170C2×C4×C612−1+412yes—
171C2×Dic61—22no—
172C2×Dic61—22no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—42no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182Dic122—24no—
183Dic122—42no—
184#92418—44no—
185#24318—611nomaximal, Sylow
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—24no—
189C22×C61—44no—
190C22×C61—42no—
191C22×C61—42no—
192C22×C61—82no—
193C22×C61C2×#12011yescharacteristic
194C22×C61C2×C22⋊Q811yescharacteristic
195C22×C61#53611yescharacteristic
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—22no—
199C4×C61—42no—
200C4×C61—44no—
201C4×C61—44no—
202C4×C61#54812yes—
203C4×C61D4×Q812yes—
204Dic61—42no—
205Dic61—42no—
206Dic61—81no—
207Dic61—81no—
208#3501D6111yescharacteristic
209#3078—612no—
210#2326—611no—
211#2060—612no—
212#2028—612no—
213#1995—612no—
214#1885—612no—
215#1856—612no—
216#1800—611no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—44no—
220C2×C61#240911yescharacteristic
221C2×C61#301911yescharacteristic
222C2×C61#302211yescharacteristic
223#554—22no—
224#554—24no—
225D4×Q8—22no—
226#544C2×D6112yes—
227#543—22no—
228#533C2×D6112yes—
229C2×#124—611no—
230C2×C22⋊Q8—22no—
231#518C2×D6112yes—
232C2×C4×Q8—611no—
233C2×C4×D4C2×D6111yescharacteristic
234#479—1224no—
235#477—1222no—
236#475—1222no—
237#428—614no—
238#423—612no—
239#419—614no—
240C2×#100—612no—
241#391—611no—
242#390—611no—
243#388—1222no—
244#385—1222no—
245#384—611no—
246#376—611no—
247#343—612no—
248#339—614no—
249#339—614no—
250C61#2431811yescharacteristic, complemented, Sylow
251C22×Q8—22no—
252C22×Q8—611no—
253C22×D4C22×D6111yescharacteristic
254C23×C4C22⋊D6112yes—
255C2×M16—614no—
256C4⋊Q8—24no—
257C4⋊Q8C22⋊D6112yes—
258C4⋊Q8C22⋊D6112yes—
259#125—24no—
260#125C22⋊D6114yes—
261#124—614no—
262#124—1222no—
263C22⋊Q8—24no—
264C22⋊Q8—24no—
265C22⋊Q8—28no—
266C22⋊Q8—28no—
267C22⋊Q8—22no—
268C22⋊Q8—24no—
269C22⋊Q8—22no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—1222no—
273C4×Q8—1222no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—22no—
277C4×D4—22no—
278#117C22×D6114yes—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—611no—
282C2×C4⋊C4—1222no—
283C2×C4⋊C4—1221no—
284C2×C4⋊C4C22×D6111yescharacteristic
285C2×C22⋊C4—24no—
286C2×C22⋊C4—1222no—
287C2×C22⋊C4—1222no—
288C2×C22⋊C4C22⋊D6112yes—
289C2×C42—611no—
290C2×C42—611no—
291C2×C42—1221no—
292C2×C42C22×D6111yescharacteristic
293#107—1224no—
294#106—1224no—
295C4⋊C8—1222no—
296C4⋊C8—1222no—
297#103—1224no—
298#103—1224no—
299#100—1224no—
300#95—1221no—
301#95—1221no—
302#95—1222no—
303C24C2×C22⋊D6112yes—
304C2×Q8—24no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—84no—
308C2×Q8—612no—
309C2×Q8—1221no—
310C2×Q8—1222no—
311C2×Q8—1222no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4Q8×D6112yes—
315C2×D4#1175822612yes—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—82no—
321C22×C4—1221no—
322C22×C4—1221no—
323C22×C4—1221no—
324C22×C4—1221no—
325C22×C4C2×C22⋊D6112yes—
326C22×C4C2×C22⋊D6111yescharacteristic
327C22×C4C2×C22⋊D6111yescharacteristic
328C22×C4C23×D6111yescharacteristic
329M16—1224no—
330M16—1224no—
331C2×C8—1224no—
332C4⋊C4—24no—
333C4⋊C4—28no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—1222no—
340C4⋊C4—4882no—
341C4⋊C4—2441no—
342C4⋊C4—2441no—
343C4⋊C4C2×C22⋊D6114yes—
344C4⋊C4C2×C22⋊D6114yes—
345C22⋊C4—28no—
346C22⋊C4—24no—
347C22⋊C4—24no—
348C22⋊C4—1228no—
349C22⋊C4—4882no—
350C42—24no—
351C42—612no—
352C42—1221no—
353C42—1222no—
354C42—2442no—
355C42C2×C22⋊D6112yes—
356C42C2×C22⋊D6112yes—
357C23—22no—
358C23—22no—
359C23—42no—
360C23—42no—
361C23#1451975314yes—
362C23C22×C22⋊D6111yescharacteristic
363Q8—44no—
364Q8—82no—
365Q8—84no—
366Q8—1224no—
367Q8—2442no—
368D4—24no—
369D4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—82no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—1222no—
381C2×C4—1222no—
382C2×C4—2441no—
383C2×C4—2442no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4#1451974612yes—
392C2×C4#1451974611yescharacteristic
393C2×C4#1451974611yescharacteristic
394C2×C4C22×C22⋊D6112yes—
395C2×C4C2×Q8×D6111yescharacteristic
396C2×C4C2×#1175822611yescharacteristic
397C2×C4#1451982512yes—
398C8—2444no—
399C22—21no—
400C22—21no—
401C22—24no—
402C22—44no—
403C22—42no—
404C22—42no—
405C22—82no—
406C22C2×#1451974611yescharacteristic
407C22C2×#1451975311yescharacteristic
408C22#1684033611yescharacteristic
409C4—21no—
410C4—21no—
411C4—22no—
412C4—42no—
413C4—44no—
414C4—44no—
415C4—4881no—
416C4—4881no—
417C4—2442no—
418C4—2442no—
419C4#1684042312yes—
420C4Q8×C22⋊D6112yes—
421C2—21no—
422C2—21no—
423C2—44no—
424C2#2144917811yescharacteristic
425C2#2145517611yescharacteristic
426C2#2145519811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radicalC22×D4×C61C23
hypercenter#3501D61
hypocenterC61#24318
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #21455620C2semidirect
3. #21447161C2semidirect
4. #21446977C2semidirect
5. #21446801C2semidirect
40. #14519764Q8semidirect
114. C2×Dic122C2×Q8semidirect
139. C4×Dic61C2×Q8semidirect
250. C61#24318semidirect

Maximal subgroups

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

subgroupindex
#214556202
#214485312
#214471612
#214469772
#214468012
#214462072
#214460292
#214458112
C61×#35012
#2431861