Finite Group Databasebeta

← #29577218

Subgroups — #29577218

Order 15616.

542 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 107 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#21448704C211yesmaximal, characteristic, complemented
2#21446656C211yesmaximal, characteristic
3#21445918C211yesmaximal, characteristic, complemented
4#21444335C211yesmaximal, characteristic
5#21443724C211yesmaximal, characteristic, complemented
6#21443704C211yesmaximal, characteristic, complemented
7C61×#3094C211yesmaximal, characteristic
8C2×#14519692—22no—
9#16839726C2211yescharacteristic
10#16839661—21no—
11#16839655—22no—
12#16839628—21no—
13#16839622C2211yescharacteristic
14C4⋊C4×Dic61—21no—
15C4×C22⋊Dic61C2211yescharacteristic
16C4×C22⋊Dic61C2211yescharacteristic
17#16839608—21no—
18#16839521—22no—
19#16839496—21no—
20#16839496—21no—
21C2×#121×C61C2211yescharacteristic
22C2×#117×C61C2211yescharacteristic
23C61×#482—22no—
24C61×#481—21no—
25C61×#480—21no—
26C61×#428C412yescomplemented
27C61×#418C412yescomplemented
28C61×#407C2211yescharacteristic
29C2×#11758194—21no—
30C2×C4⋊Dic61—21no—
31C2×C4⋊Dic61—22no—
32C2×C22⋊Dic61—21no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61D412yes—
35C2×C4×Dic61—22no—
36C2×C4×Dic61—21no—
37C2×C4×Dic61—21no—
38#14519692—44no—
39C22⋊#92418—21no—
40C22⋊#92418—21no—
41#14519684—22no—
42#14519684—22no—
43#14519684—21no—
44#14519684—21no—
45#14519684D411yescharacteristic
46#14519684D411yescharacteristic
47C22×D4×C61—21no—
48C22×D4×C61C2×C411yescharacteristic
49C23×C4×C61C2311yescharacteristic
50C2×M16×C61C2×C412yes—
51#121×C61—21no—
52#121×C61—21no—
53#121×C61—21no—
54#121×C61—21no—
55#117×C61—21no—
56#117×C61—21no—
57#117×C61C2×C412yes—
58#117×C61D412yes—
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C4⋊C4×C61D411yescharacteristic
61C2×C4⋊C4×C61D411yescharacteristic
62C2×C22⋊C4×C61—21no—
63C2×C22⋊C4×C61—21no—
64C2×C42×C61—21no—
65#107×C61—22no—
66#106×C61—22no—
67#102×C61—22no—
68#102×C61—22no—
69C22⋊C8×C61—21no—
70C22⋊C8×C61—21no—
71C24×C61—21no—
72C24×C61C2×D411yescharacteristic
73C2×D4×C61—21no—
74C2×D4×C61—21no—
75C2×D4×C61—42no—
76C2×D4×C61—42no—
77C2×D4×C61—42no—
78C2×D4×C61—41no—
79C2×D4×C61—41no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—42no—
84C22×C4×C61C22×C411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61#4111yescharacteristic
89C22×C4×C61#4111yescharacteristic
90C22×Dic61—21no—
91C22×Dic61—22no—
92C22×Dic61—21no—
93C22×Dic61—41no—
94M16×C61—42no—
95C2×C8×C61—22no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—21no—
100C4⋊C4×C61C22⋊C412yes—
101C4⋊C4×C61C22⋊C412yes—
102C22⋊C4×C61—22no—
103C22⋊C4×C61—22no—
104C22⋊C4×C61—41no—
105C22⋊C4×C61—41no—
106C42×C61—22no—
107C42×C61—22no—
108C22⋊Dic61—22no—
109C22⋊Dic61—42no—
110C22⋊Dic61—41no—
111C22⋊Dic61—42no—
112C4⋊Dic61—42no—
113C4⋊Dic61—82no—
114C4×Dic61—44no—
115C4×Dic61—42no—
116C4×Dic61—42no—
117#11758194—42no—
118C2×#92418—41no—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—42no—
124C23×C61—41no—
125C23×C61—81no—
126C23×C61—81no—
127C23×C61C4×D411yescharacteristic
128C23×C61#12011yescharacteristic
129C23×C61#12111yescharacteristic
130D4×C61—42no—
131D4×C61—81no—
132D4×C61—82no—
133D4×C61—81no—
134D4×C61—81no—
135C2×C4×C61—22no—
136C2×C4×C61—22no—
137C2×C4×C61—22no—
138C2×C4×C61—22no—
139C2×C4×C61—22no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—42no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—42no—
147C2×C4×C61—41no—
148C2×C4×C61C2×C22⋊C411yescharacteristic
149C2×C4×C61C4×D411yescharacteristic
150C2×C4×C61#12111yescharacteristic
151C2×C4×C61#12311yescharacteristic
152C2×Dic61—22no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—42no—
157C2×Dic61—42no—
158C2×Dic61—42no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—42no—
162C2×Dic61—81no—
163C2×Dic61—81no—
164C8×C61—42no—
165#92418—81no—
166#9367—611nomaximal, Sylow
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—81no—
176C22×C61—81no—
177C22×C61—81no—
178C22×C61—81no—
179C22×C61—81no—
180C22×C61—162no—
181C22×C61#38611yescharacteristic
182C22×C61#45311yescharacteristic
183C22×C61#45411yescharacteristic
184C4×C61—21no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—42no—
188C4×C61—42no—
189C4×C61—81no—
190C4×C61#41012yes—
191Dic61—42no—
192Dic61—82no—
193Dic61—81no—
194Dic61—82no—
195#3094D6111yescharacteristic
196#2346—611no—
197#1964—611no—
198#1826—611no—
199#1507—611no—
200#1305—611no—
201#1297—611no—
202C2×C61—21no—
203C2×C61—21no—
204C2×C61—41no—
205C2×C61—41no—
206C2×C61—161no—
207C2×C61#188311yescharacteristic
208C2×C61#189911yescharacteristic
209C2×C61#190611yescharacteristic
210C2×#121C2×D6111yescharacteristic
211C2×#117C2×D6111yescharacteristic
212#482—22no—
213#481—21no—
214#480—21no—
215#428Dic6112yescomplemented
216#418Dic6112yescomplemented
217C2×#99—1222no—
218#407—611no—
219#407C2×D6111yescharacteristic
220#389—1221no—
221#388—1222no—
222#382—1221no—
223#380—611no—
224C4×C4⋊C4—1221no—
225C4×C22⋊C4—611no—
226C4×C22⋊C4—611no—
227#376—1221no—
228#339—1222no—
229#327—1221no—
230#327—1221no—
231C61#936711yescharacteristic, complemented, Sylow
232C22×D4—21no—
233C22×D4C2×Dic6111yescharacteristic
234C23×C4C22×D6111yescharacteristic
235C2×M16—1221no—
236C2×M16C2×Dic6112yes—
237#121—21no—
238#121—21no—
239#121—21no—
240#121—21no—
241#117—21no—
242#117—21no—
243#117C2×Dic6112yes—
244#117C22⋊D6112yes—
245C2×C4⋊C4—1222no—
246C2×C4⋊C4—1221no—
247C2×C4⋊C4C2×Dic6111yescharacteristic
248C2×C4⋊C4C22⋊D6111yescharacteristic
249C2×C4⋊C4C22⋊D6111yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—612no—
253C2×C22⋊C4—1221no—
254C2×C22⋊C4—1221no—
255C2×C42—21no—
256C2×C42—1222no—
257C2×C42—1221no—
258C2×C42—1221no—
259#107—22no—
260#106—22no—
261#102—22no—
262#102—22no—
263#99—2444no—
264C22⋊C8—21no—
265C22⋊C8—21no—
266C22⋊C8—1221no—
267C22⋊C8—1221no—
268#95—611no—
269#95—611no—
270#95—1222no—
271#95—1222no—
272#95—1221no—
273#95—1221no—
274C24—21no—
275C24D4×D6111yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—41no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—1221no—
288C22×C4—1221no—
289C22×C4—1222no—
290C22×C4—2441no—
291C22×C4D4×D6111yescharacteristic
292C22×C4#1175822411yescharacteristic
293C22×C4#1175822411yescharacteristic
294C22×C4C22×Dic6111yescharacteristic
295C22×C4C2×C22⋊D6111yescharacteristic
296C22×C4C2×C22⋊D6111yescharacteristic
297M16—42no—
298M16—2442no—
299C2×C8—22no—
300C2×C8—2441no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—4882no—
306C4⋊C4—2442no—
307C4⋊C4C22⋊Dic6112yes—
308C4⋊C4C22⋊Dic6112yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—1222no—
314C22⋊C4—2442no—
315C22⋊C4—2441no—
316C22⋊C4—2442no—
317C42—22no—
318C42—22no—
319C42—2442no—
320C42—2442no—
321C42—2444no—
322C23—21no—
323C23—21no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—41no—
328C23—81no—
329C23—81no—
330C23D4×Dic6111yescharacteristic
331C23#1451974511yescharacteristic
332C23#1451974911yescharacteristic
333D4—42no—
334D4—81no—
335D4—82no—
336D4—81no—
337D4—81no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—1222no—
352C2×C4—4881no—
353C2×C4—4881no—
354C2×C4—2441no—
355C2×C4—2442no—
356C2×C4—2442no—
357C2×C4—2441no—
358C2×C4—2442no—
359C2×C4—2442no—
360C2×C4—2441no—
361C2×C4—2441no—
362C2×C4—2441no—
363C2×C4C2×C22⋊Dic6111yescharacteristic
364C2×C4D4×Dic6111yescharacteristic
365C2×C4#1451974911yescharacteristic
366C2×C4#1451975811yescharacteristic
367C8—42no—
368C8—4881no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—162no—
383C22#1683964211yescharacteristic
384C22#1683992811yescharacteristic
385C22#1683993311yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—42no—
390C4—42no—
391C4—81no—
392C4—4881no—
393C4—4882no—
394C4—4882no—
395C4—2442no—
396C4#1683973612yes—
397C2—21no—
398C2—21no—
399C2—41no—
400C2—41no—
401C2—161no—
402C2#2144619711yescharacteristic
403C2#2144627811yescharacteristic
404C2#2144632211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#3094C2
socleC22×C61#386
prime radicalC2×#121×C61C22
cosocleC23×C4×C61C23
hypercenter#3094D61
hypocenterC61#9367
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3094

Complements

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

NH ≅ G/Ntype
1. #21448704C2semidirect
3. #21445918C2semidirect
5. #21443724C2semidirect
6. #21443704C2semidirect
26. C61×#428C4semidirect
27. C61×#418C4semidirect
215. #428Dic61semidirect
216. #418Dic61semidirect
231. C61#9367semidirect

Maximal subgroups

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

subgroupindex
#214487042
#214466562
#214459182
#214443352
#214437242
#214437042
C61×#30942
#936761