Finite Group Databasebeta

← #29577672

Subgroups — #29577672

Order 15616.

568 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450786C211yesmaximal, characteristic
2#21444442C212yesmaximal, complemented
3#21444160C211yesmaximal, characteristic, complemented
4#21443674C212yesmaximal, complemented
5C61×#2973C211yesmaximal, characteristic, complemented
6C2×#14519773C411yescharacteristic
7C2×#14519765C2211yescharacteristic
8C2×#14519764C411yescharacteristic
9#16839819C2212yes—
10#16839678—21no—
11#16839678—21no—
12#16839667C411yescharacteristic
13#16839663C411yescharacteristic
14#16839641—21no—
15#16839641—21no—
16C4×C4⋊Dic61C2211yescharacteristic
17#16839501—24no—
18#16839486—21no—
19#16839486—21no—
20C2×C4×D4×C61C2211yescharacteristic
21C61×#435—24no—
22C61×#432C2212yes—
23C61×#407—21no—
24C2×C22⋊C8×C61—21no—
25C22×Dic122C2×C411yescharacteristic
26C22×D244C2×C411yescharacteristic
27C22×C22⋊D61—21no—
28C22×C22⋊D61—21no—
29#14519773—42no—
30#14519765—22no—
31#14519765—22no—
32#14519764—42no—
33C2×C4⋊Dic61—21no—
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×#11758198—21no—
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×C4×Dic61C2×C411yescharacteristic
41C2×C4×Dic61C2×C411yescharacteristic
42C4⋊#92418—22no—
43C22⋊#92418—22no—
44#14519688—22no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48C22×D4×C61C2×C411yescharacteristic
49C23×C4×C61D411yescharacteristic
50C23×C4×C61D411yescharacteristic
51C2×M16×C61—22no—
52C22×C8×C61—22no—
53C4×D4×C61—22no—
54C4×D4×C61—22no—
55C2×C4⋊C4×C61C2×C411yescharacteristic
56C2×C22⋊C4×C61D411yescharacteristic
57C2×C22⋊C4×C61D411yescharacteristic
58C2×C42×C61C2311yescharacteristic
59C4⋊C8×C61—22no—
60C22⋊C8×C61—22no—
61C22⋊C8×C61—22no—
62C22⋊C8×C61—22no—
63#97×C61D414yescomplemented
64C24×C61C22⋊C411yescharacteristic
65C24×C61C22⋊C411yescharacteristic
66C23×D61—21no—
67C2×D4×C61—22no—
68C2×D4×C61—22no—
69C2×D4×C61—41no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—81no—
75C22×C4×C61—81no—
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22×C411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C2×C22⋊D61—42no—
82C2×C22⋊D61—42no—
83C2×C22⋊D61—42no—
84C2×C22⋊D61—42no—
85C2×C22⋊D61—81no—
86C2×C22⋊D61—81no—
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C2×D244—22no—
91C2×D244—41no—
92C2×D244—41no—
93C2×Dic122—22no—
94C2×Dic122—41no—
95C2×Dic122—41no—
96M16×C61—82no—
97C2×C8×C61—24no—
98C2×C8×C61—82no—
99C4⋊C4×C61—22no—
100C22⋊C4×C61—22no—
101C22⋊C4×C61—22no—
102C42×C61C2×D412yes—
103C42×C61#4112yes—
104C22⋊Dic61—42no—
105C22⋊Dic61—42no—
106#11758198—41no—
107#11758198—42no—
108#11758198—41no—
109#11758198—81no—
110#11758198—81no—
111C4⋊Dic61—42no—
112C4×Dic61—41no—
113C4×Dic61—41no—
114C2×#92418—42no—
115C23×C61—21no—
116C23×C61—21no—
117C23×C61—21no—
118C23×C61—21no—
119C23×C61—21no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61#10012yes—
125C23×C61C2×C22⋊C411yescharacteristic
126C22×D61—41no—
127C22×D61—41no—
128C22×D61—41no—
129C22×D61—41no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—41no—
133D4×C61—44no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—42no—
141C2×C4×C61—42no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—81no—
145C2×C4×C61—81no—
146C2×C4×C61C4×D412yes—
147C2×C4×C61#12011yescharacteristic
148C2×C4×C61#12112yes—
149C2×C4×C61#12311yescharacteristic
150C22⋊D61—84no—
151C22⋊D61—84no—
152C2×Dic61—22no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161D244—82no—
162Dic122—82no—
163C8×C61—44no—
164#92418—82no—
165#9447—611nomaximal, Sylow
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—22no—
169C22×C61—42no—
170C22×C61—42no—
171C22×C61—41no—
172C22×C61—41no—
173C22×C61—41no—
174C22×C61—41no—
175C22×C61—41no—
176C22×C61—81no—
177C22×C61—81no—
178C22×C61#38611yescharacteristic
179C22×C61C2×#10011yescharacteristic
180C22×C61#41311yescharacteristic
181C2×D61—81no—
182C2×D61—81no—
183C2×D61—81no—
184C2×D61—81no—
185C2×D61—81no—
186C2×D61—81no—
187C2×D61—81no—
188C4×C61—22no—
189C4×C61—22no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—41no—
193C4×C61—41no—
194Dic61—42no—
195Dic61—81no—
196Dic61—81no—
197#2973D6111yescharacteristic, complemented
198#2544—611no—
199#1533—612no—
200#1466—611no—
201#1285—612no—
202C2×C61—21no—
203C2×C61—21no—
204C2×C61—42no—
205C2×C61—81no—
206C2×C61#188411yescharacteristic
207C2×C61#190011yescharacteristic
208C2×C61#191111yescharacteristic
209D61—161no—
210C2×#127—611no—
211C2×#124—611no—
212C2×#124—611no—
213C2×C4×D4C2×D6111yescharacteristic
214#435—24no—
215#432—612no—
216#432C2×D6112yes—
217#407—21no—
218C2×C22⋊C8—21no—
219#394—1221no—
220#394—1221no—
221#391—611no—
222#390—611no—
223#386—1221no—
224#386—1221no—
225C4×C4⋊C4—611no—
226#329—1224no—
227#323—1221no—
228#323—1221no—
229C61#944711yescharacteristic, complemented, Sylow
230C22×Q8—611no—
231C22×D4—611no—
232C22×D4—1221no—
233C22×D4—1221no—
234C22×D4C4×D6111yescharacteristic
235C23×C4C22⋊D6111yescharacteristic
236C23×C4C22⋊D6111yescharacteristic
237C2×M16—22no—
238C22×C8—22no—
239#127—2442no—
240#124—1222no—
241#124—1222no—
242#124—2442no—
243C4×D4—22no—
244C4×D4—22no—
245C2×C4⋊C4—1221no—
246C2×C4⋊C4C4×D6111yescharacteristic
247C2×C22⋊C4—1221no—
248C2×C22⋊C4—1221no—
249C2×C22⋊C4—1221no—
250C2×C22⋊C4—1221no—
251C2×C22⋊C4—1221no—
252C2×C22⋊C4—1221no—
253C2×C22⋊C4D24411yescharacteristic
254C2×C22⋊C4D24411yescharacteristic
255C2×C42—611no—
256C2×C42—611no—
257C2×C42C22×D6111yescharacteristic
258C4⋊C8—22no—
259C4⋊C8—1222no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—1222no—
264#97—1222no—
265#97C22⋊D6114yescomplemented
266#95—1221no—
267#95—1221no—
268#95—1221no—
269C24—1221no—
270C24#1175819811yescharacteristic
271C24#1175819811yescharacteristic
272C2×Q8—1222no—
273C2×Q8—2441no—
274C2×Q8—2441no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—41no—
278C2×D4—1222no—
279C2×D4—4881no—
280C2×D4—4881no—
281C2×D4—2441no—
282C2×D4—2441no—
283C2×D4—2442no—
284C2×D4—2442no—
285C2×D4—2442no—
286C2×D4—2442no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—1221no—
294C22×C4—1221no—
295C22×C4—1221no—
296C22×C4#1175819811yescharacteristic
297C22×C4#1175819811yescharacteristic
298C22×C4C2×C4×D6111yescharacteristic
299C22×C4C2×D24411yescharacteristic
300C22×C4C2×C22⋊D6111yescharacteristic
301M16—82no—
302C2×C8—24no—
303C2×C8—82no—
304C2×C8—2442no—
305C4⋊C4—22no—
306C4⋊C4—2442no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—4881no—
310C22⋊C4—4881no—
311C22⋊C4—2442no—
312C22⋊C4—2442no—
313C22⋊C4—2441no—
314C22⋊C4—2442no—
315C22⋊C4—2441no—
316C42—2441no—
317C42—2441no—
318C42#1175822212yes—
319C42C2×C22⋊D6112yes—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—2441no—
330C23—2441no—
331C23—2441no—
332C23—2441no—
333C23—2441no—
334C23—2441no—
335C23—2441no—
336C23#1451969312yes—
337C23C2×#1175819811yescharacteristic
338Q8—4882no—
339D4—44no—
340D4—4882no—
341D4—4884no—
342D4—4884no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—1222no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4—2441no—
359C2×C4—2441no—
360C2×C4—2441no—
361C2×C4—2441no—
362C2×C4—2441no—
363C2×C4—2441no—
364C2×C4C4×C22⋊D6112yes—
365C2×C4#1451974611yescharacteristic
366C2×C4#1451975112yes—
367C2×C4#1451976011yescharacteristic
368C8—44no—
369C8—4882no—
370C22—21no—
371C22—21no—
372C22—22no—
373C22—42no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—81no—
382C22—4881no—
383C22—4881no—
384C22—4881no—
385C22—4881no—
386C22—4881no—
387C22—4881no—
388C22—4881no—
389C22#1683964111yescharacteristic
390C22C2×#1451969311yescharacteristic
391C22#1683974411yescharacteristic
392C4—22no—
393C4—22no—
394C4—21no—
395C4—21no—
396C4—41no—
397C4—41no—
398C4—4881no—
399C4—4881no—
400C4—2442no—
401C2—21no—
402C2—21no—
403C2—42no—
404C2—81no—
405C2—9761no—
406C2#2144620211yescharacteristic
407C2#2144628211yescharacteristic
408C2#2144635111yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2973C2
socleC22×C61#386
prime radicalC2×#14519773C4
cosocleC2×C42×C61C23
hypercenter#2973D61
hypocenterC61#9447
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #21444442C2semidirect
3. #21444160C2semidirect
4. #21443674C2semidirect
5. C61×#2973C2semidirect
63. #97×C61D4semidirect
197. #2973D61semidirect
229. C61#9447semidirect
265. #97C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214507862
#214444422
#214441602
#214436742
C61×#29732
#944761