Finite Group Databasebeta

← #29733737

Subgroups — #29733737

Order 15616.

1052 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456279C211yesmaximal, characteristic, complemented
2#21455306C211yesmaximal, characteristic, complemented
3#21451770C211yesmaximal, characteristic, complemented
4#21446866C214yesmaximal, complemented
5#21446813C211yesmaximal, characteristic, complemented
6#21446652C211yesmaximal, characteristic, complemented
7#21446188C214yesmaximal, complemented
8#21445629C211yesmaximal, characteristic, complemented
9C61×#3512C211yesmaximal, characteristic
10C2×#14519807C2214yes—
11C2×#14519807C2212yes—
12C2×#14519777C2211yescharacteristic
13#16840157—22no—
14#16840141—21no—
15#16840134—21no—
16#16839972—22no—
17#16839967—22no—
18#16839786C2211yescharacteristic
19C2×#14519704C2212yes—
20C2×#14519704C2214yes—
21#16839770C2212yes—
22C4×#11758194C2211yescharacteristic
23#16839701—24no—
24#16839667C2214yes—
25#16839608C2211yescharacteristic
26#16839531C2214yes—
27#16839521C2212yes—
28C61×#553—22no—
29C61×#553—24no—
30C61×#552—22no—
31C61×#541—21no—
32C61×#540—21no—
33C61×#537C2212yes—
34C61×#532C2211yescharacteristic
35C61×#529—22no—
36C2×#125×C61C2211yescharacteristic
37C61×#520C2212yes—
38C4×#41×C61C2211yescharacteristic
39C22×Dic122C2312yes—
40#14519807—28no—
41#14519807—44no—
42#14519807—44no—
43C2×#11758202—22no—
44C2×#11758202—24no—
45C2×#11758200—22no—
46C2×#11758200—24no—
47C2×#11758194C2312yes—
48C2×#11758194C2312yes—
49#14519777—22no—
50#14519777—21no—
51#14519777—21no—
52C2×C4⋊Dic61—22no—
53C2×C4×Dic61C2312yes—
54C4⋊#92418—22no—
55#14519704—24no—
56#14519704—44no—
57#14519700—24no—
58#14519688—21no—
59C4×#92418—21no—
60#14519684—22no—
61#14519684—28no—
62C2×#41×C61C2312yes—
63C2×#41×C61C2311yescharacteristic
64C4⋊Q8×C61D411yescharacteristic
65#127×C61D411yescharacteristic
66#126×C61—24no—
67#126×C61—24no—
68#125×C61—24no—
69#125×C61—21no—
70#125×C61—22no—
71#125×C61—21no—
72#124×C61D412yes—
73#123×C61—24no—
74#123×C61—24no—
75C22⋊Q8×C61—22no—
76C22⋊Q8×C61—22no—
77#121×C61—24no—
78#121×C61—22no—
79#121×C61—22no—
80#121×C61—22no—
81#121×C61—24no—
82C4×Q8×C61—22no—
83C4×Q8×C61D412yes—
84C4×D4×C61—24no—
85C4×D4×C61—22no—
86C4×D4×C61—22no—
87C4×D4×C61—24no—
88C4×D4×C61—22no—
89C4×D4×C61D412yes—
90#117×C61—24no—
91#117×C61—22no—
92#117×C61—21no—
93#117×C61C2311yescharacteristic
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61D414yes—
96C2×C4⋊C4×C61C2312yes—
97C2×C42×C61—21no—
98C2×C42×C61C2311yescharacteristic
99#41×C61—22no—
100#41×C61—24no—
101#41×C61—24no—
102#41×C61—22no—
103C2×Q8×C61C2×D412yes—
104C2×Q8×C61C2×D411yescharacteristic
105C2×D4×C61—21no—
106C2×D4×C61—22no—
107C2×D4×C61—24no—
108C2×D4×C61—22no—
109C2×D4×C61C2×D412yes—
110C2×D4×C61C2×D411yescharacteristic
111C22×C4×C61—24no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61C2×D414yes—
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2411yescharacteristic
118C22×Dic61—22no—
119C22×Dic61—22no—
120C2×Dic122—22no—
121C2×Dic122—22no—
122C2×Dic122—24no—
123C2×Dic122—24no—
124C4⋊C4×C61—24no—
125C4⋊C4×C61—24no—
126C4⋊C4×C61—24no—
127C4⋊C4×C61—24no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61C2×D412yes—
133C22⋊C4×C61—24no—
134C22⋊C4×C61—24no—
135C22⋊C4×C61—24no—
136C22⋊C4×C61—21no—
137C22⋊C4×C61—22no—
138C22⋊C4×C61—22no—
139C22⋊C4×C61—22no—
140C42×C61—24no—
141C42×C61—22no—
142C42×C61—21no—
143C42×C61C2×D411yescharacteristic
144C42×C61C2×D411yescharacteristic
145C42×C61C2×D412yes—
146#11758202—48no—
147#11758202—44no—
148#11758200—48no—
149#11758200—44no—
150C4⋊Dic61—42no—
151C4⋊Dic61—42no—
152C4×Dic61—22no—
153C4×Dic61—22no—
154#11758194—22no—
155#11758194—24no—
156#11758194#4114yes—
157C2×#92418—22no—
158C2×#92418—22no—
159C23×C61—22no—
160C23×C61—21no—
161C23×C61C22×D411yescharacteristic
162Q8×C61—21no—
163Q8×C61—21no—
164Q8×C61—24no—
165D4×C61—21no—
166D4×C61—21no—
167D4×C61—24no—
168D4×C61—42no—
169D4×C61—44no—
170D4×C61—82no—
171D4×C61—84no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—21no—
175C2×C4×C61—24no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—44no—
181C2×C4×C61—82no—
182C2×C4×C61—84no—
183C2×C4×C61#12012yes—
184C2×C4×C61#12012yes—
185C2×C4×C61C22×D411yescharacteristic
186C2×C4×C61C22×D411yescharacteristic
187C2×C4×C61C2×#4112yes—
188C2×C4×C61H(5,𝔽2)12yes—
189C2×Dic61—44no—
190C2×Dic61—42no—
191C2×Dic61—42no—
192C2×Dic61—44no—
193Dic122—42no—
194Dic122—42no—
195Dic122—44no—
196#92418—24no—
197#92418—42no—
198#25619—611nomaximal, Sylow
199C22×C61—22no—
200C22×C61—42no—
201C22×C61—41no—
202C22×C61—42no—
203C22×C61—42no—
204C22×C61—42no—
205C22×C61C2×#12011yescharacteristic
206C22×C61#55012yes—
207C4×C61—22no—
208C4×C61—22no—
209C4×C61—42no—
210C4×C61—41no—
211C4×C61—44no—
212C4×C61—42no—
213C4×C61#53211yescharacteristic
214C4×C61#53511yescharacteristic
215C4×C61#54612yes—
216Dic61—82no—
217Dic61—82no—
218#3512D6111yescharacteristic
219#3152—611no—
220#3039—611no—
221#2642—611no—
222#2007—614no—
223#1998—611no—
224#1963—611no—
225#1882—614no—
226#1759—611no—
227C2×C61—22no—
228C2×C61—82no—
229C2×C61—81no—
230C2×C61#243111yescharacteristic
231C2×C61#302212yes—
232C2×#137—614no—
233C2×#137—612no—
234#553—22no—
235#553—24no—
236#552—22no—
237#541—21no—
238#540—21no—
239#537C2×D6112yes—
240#532C2×D6111yescharacteristic
241C2×C4⋊Q8—611no—
242#529—22no—
243C2×#125C2×D6111yescharacteristic
244#520C2×D6112yes—
245C4×#41C2×D6111yescharacteristic
246#497—1222no—
247#494—1221no—
248#492—1221no—
249#462—1222no—
250#461—1222no—
251#423—611no—
252C2×#104—612no—
253C2×#104—614no—
254#419—612no—
255C4×M16—611no—
256#400—1224no—
257#391—614no—
258#376—611no—
259#343—614no—
260#339—612no—
261C61#2561911yescharacteristic, complemented, Sylow
262C2×#41C22×D6111yescharacteristic
263C2×#41C22×D6112yes—
264C22×Q8—612no—
265#137—1228no—
266#137—2444no—
267#137—2444no—
268C2×Q16—1222no—
269C2×Q16—1224no—
270C2×SD16—1222no—
271C2×SD16—1224no—
272C2×M16—612no—
273C2×M16—612no—
274C4⋊Q8—1222no—
275C4⋊Q8—1221no—
276C4⋊Q8—1221no—
277C4⋊Q8C22⋊D6111yescharacteristic
278#127C22⋊D6111yescharacteristic
279#126—24no—
280#126—24no—
281#125—24no—
282#125—21no—
283#125—21no—
284#125—22no—
285#124C22⋊D6112yes—
286#123—24no—
287#123—24no—
288C22⋊Q8—22no—
289C22⋊Q8—22no—
290#121—24no—
291#121—22no—
292#121—22no—
293#121—22no—
294#121—24no—
295C4×Q8—22no—
296C4×Q8C22⋊D6112yes—
297C4×D4—24no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—24no—
301C4×D4—22no—
302C4×D4C22⋊D6112yes—
303#117—24no—
304#117—22no—
305#117—21no—
306#117C22×D6111yescharacteristic
307C2×C4⋊C4—22no—
308C2×C4⋊C4—1222no—
309C2×C4⋊C4C22⋊D6114yes—
310C2×C4⋊C4C22×D6112yes—
311C2×C42—21no—
312C2×C42—612no—
313C2×C42C22×D6111yescharacteristic
314C4⋊C8—1222no—
315#104—1224no—
316#104—2444no—
317#103—1224no—
318#97—1221no—
319C4×C8—1221no—
320#95—1222no—
321#95—1228no—
322#41—22no—
323#41—24no—
324#41—24no—
325#41—22no—
326C2×Q8—1222no—
327C2×Q8—1222no—
328C2×Q8—1224no—
329C2×Q8—1224no—
330C2×Q8C2×C22⋊D6112yes—
331C2×Q8C2×C22⋊D6111yescharacteristic
332C2×D4—21no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—22no—
336C2×D4C2×C22⋊D6112yes—
337C2×D4C2×C22⋊D6111yescharacteristic
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—1222no—
343C22×C4—1222no—
344C22×C4C2×C22⋊D6114yes—
345C22×C4C2×C22⋊D6112yes—
346C22×C4C23×D6111yescharacteristic
347Q16—2448no—
348Q16—2444no—
349SD16—2448no—
350SD16—2444no—
351M16—614no—
352M16—1222no—
353M16—1224no—
354C2×C8—1222no—
355C2×C8—1222no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—2442no—
365C4⋊C4—2442no—
366C4⋊C4C2×C22⋊D6112yes—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—21no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C42—24no—
375C42—21no—
376C42—22no—
377C42—1222no—
378C42—1222no—
379C42C2×C22⋊D6111yescharacteristic
380C42C2×C22⋊D6111yescharacteristic
381C42C2×C22⋊D6112yes—
382C23—22no—
383C23—21no—
384C23C22×C22⋊D6111yescharacteristic
385Q8—21no—
386Q8—21no—
387Q8—24no—
388Q8—2442no—
389Q8—2442no—
390Q8—2444no—
391D4—21no—
392D4—21no—
393D4—24no—
394D4—42no—
395D4—44no—
396D4—82no—
397D4—84no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—24no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—82no—
408C2×C4—84no—
409C2×C4—2444no—
410C2×C4—2442no—
411C2×C4—2442no—
412C2×C4—2444no—
413C2×C4#1451974612yes—
414C2×C4#1451974612yes—
415C2×C4C22×C22⋊D6111yescharacteristic
416C2×C4C22×C22⋊D6111yescharacteristic
417C2×C4#41×D6112yes—
418C2×C4#1451982312yes—
419C8—1224no—
420C8—2442no—
421C22—22no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—41no—
427C22C2×#1451974611yescharacteristic
428C22#1684043112yes—
429C4—22no—
430C4—22no—
431C4—44no—
432C4—42no—
433C4—42no—
434C4—41no—
435C4—4882no—
436C4—4882no—
437C4#1684030811yescharacteristic
438C4#1684032611yescharacteristic
439C4#1684040012yes—
440C2—22no—
441C2—82no—
442C2—81no—
443C2#2144943411yescharacteristic
444C2#2145519812yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3512C2
socleC22×C61C2×#120
hypercenter#3512D61
hypocenterC61#25619
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3512

Complements

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

NH ≅ G/Ntype
1. #21456279C2semidirect
2. #21455306C2semidirect
3. #21451770C2semidirect
4. #21446866C2semidirect
5. #21446813C2semidirect
6. #21446652C2semidirect
7. #21446188C2semidirect
8. #21445629C2semidirect
261. C61#25619semidirect

Maximal subgroups

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

subgroupindex
#214562792
#214553062
#214517702
#214468662
#214468132
#214466522
#214461882
#214456292
C61×#35122
#2561961