Finite Group Databasebeta

← #29580527

Subgroups — #29580527

Order 15616.

568 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 109 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#21456594C211yesmaximal, characteristic
2C2×#16840098C211yesmaximal, characteristic
3#21450297C211yesmaximal, characteristic
4#21446804C211yesmaximal, characteristic
5#21445910C211yesmaximal, characteristic
6#21444392C211yesmaximal, characteristic
7C61×#1282C211yesmaximal, characteristic
8C2×#14519756C2211yescharacteristic
9C2×#14519756C2211yescharacteristic
10C22×C4⋊Dic61C2211yescharacteristic
11#16840098—22no—
12#16840098—21no—
13#16840098—21no—
14#16840098—22no—
15#16840003—22no—
16#16839920—21no—
17#16839920—21no—
18C2×#14519711—21no—
19C2×#14519702—21no—
20#16839694—22no—
21#16839668—21no—
22#16839637—22no—
23#16839632—21no—
24#16839622C2211yescharacteristic
25#16839528—22no—
26#16839525—22no—
27#16839500—22no—
28C2×C22⋊C8×C61C2211yescharacteristic
29C2×C22⋊C8×C61C2211yescharacteristic
30C4×C22⋊C4×C61C2211yescharacteristic
31C22×Dic122D411yescharacteristic
32C22×Dic122D411yescharacteristic
33C23×Dic61—21no—
34C2×Dic244—22no—
35#14519756—21no—
36#14519756—21no—
37#14519756—21no—
38#14519756—21no—
39#14519756—21no—
40#14519756—21no—
41#14519756—21no—
42#14519756—21no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61—21no—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61—21no—
48C2×C4⋊Dic61—21no—
49C2×C4⋊Dic61—24no—
50C2×C4⋊Dic61D412yes—
51C2×C4⋊Dic61D411yescharacteristic
52C2×C4⋊Dic61D411yescharacteristic
53C2×#11758196—21no—
54C2×#11758196—21no—
55C2×C22⋊Dic61—21no—
56C2×C22⋊Dic61—21no—
57C2×C22⋊Dic61—21no—
58#14519711—22no—
59#14519711—24no—
60#14519702—22no—
61#14519702—22no—
62#14519702—24no—
63#14519684—24no—
64#14519684—21no—
65C23×C4×C61C2311yescharacteristic
66C22×C8×C61—21no—
67C22×C8×C61D412yes—
68C2×C22⋊C4×C61D412yes—
69C2×C42×C61—21no—
70C22⋊C8×C61—21no—
71C22⋊C8×C61—21no—
72C22⋊C8×C61—21no—
73C22⋊C8×C61D412yes—
74#95×C61—21no—
75C24×C61C2×D411yescharacteristic
76C22×C4×C61—22no—
77C22×C4×C61—42no—
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61#4111yescharacteristic
84C22×Dic61—22no—
85C22×Dic61—21no—
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—41no—
90C22×Dic61—41no—
91C22×Dic61—41no—
92C22×Dic61—82no—
93C2×Dic122—21no—
94C2×Dic122—21no—
95C2×Dic122—21no—
96C2×Dic122—21no—
97C2×Dic122—42no—
98C2×Dic122—42no—
99C2×C8×C61—22no—
100C2×C8×C61—22no—
101C2×C8×C61—42no—
102C2×C8×C61—42no—
103C22⋊C4×C61—22no—
104C22⋊C4×C61Q1614yescomplemented
105C42×C61—42no—
106C22⋊Dic61—41no—
107C22⋊Dic61—41no—
108C22⋊Dic61—41no—
109C22⋊Dic61—41no—
110C22⋊Dic61—42no—
111C4⋊Dic61—22no—
112C4⋊Dic61—21no—
113C4⋊Dic61—21no—
114C4⋊Dic61—21no—
115C4⋊Dic61—22no—
116C4⋊Dic61—21no—
117C4⋊Dic61—41no—
118C4⋊Dic61—41no—
119C4⋊Dic61—42no—
120C4⋊Dic61—84no—
121#11758196—41no—
122#11758196—41no—
123#11758196—41no—
124#11758196—41no—
125Dic244—82no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61#12011yescharacteristic
129C23×C61#12111yescharacteristic
130C23×C61#12711yescharacteristic
131C23×C61C2×Q1612yes—
132C23×C61#13712yes—
133C2×C4×C61—22no—
134C2×C4×C61—22no—
135C2×C4×C61—22no—
136C2×C4×C61—24no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—42no—
141C2×C4×C61—42no—
142C2×C4×C61#12011yescharacteristic
143C2×C4×C61#12011yescharacteristic
144C2×C4×C61#12111yescharacteristic
145C2×C4×C61#12111yescharacteristic
146C2×Dic61—42no—
147C2×Dic61—44no—
148C2×Dic61—42no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—42no—
154C2×Dic61—82no—
155C2×Dic61—82no—
156C2×Dic61—81no—
157C2×Dic61—81no—
158C2×Dic61—81no—
159C2×Dic61—81no—
160C2×Dic61—81no—
161Dic122—42no—
162Dic122—42no—
163Dic122—42no—
164Dic122—81no—
165C8×C61—42no—
166C8×C61—42no—
167#9959—611nomaximal, Sylow
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—42no—
175C22×C61—42no—
176C22×C61#39211yescharacteristic
177C22×C61#45112yes—
178C22×C61#46212yes—
179C22×C61#49411yescharacteristic
180C22×C61#49711yescharacteristic
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—44no—
185C4×C61#45812yes—
186Dic61—81no—
187Dic61—82no—
188Dic61—81no—
189Dic61—81no—
190Dic61—81no—
191Dic61—161no—
192#3189—611no—
193C2×#484—611no—
194#2502—611no—
195#1996—611no—
196#1824—611no—
197#1520—611no—
198#1282D6111yescharacteristic
199C2×C61—21no—
200C2×C61—22no—
201C2×C61—22no—
202C2×C61—21no—
203C2×C61#200811yescharacteristic
204C2×C61#201611yescharacteristic
205C2×C61#202211yescharacteristic
206C2×C22⋊Q8—611no—
207C2×C22⋊Q8—611no—
208C22×C4⋊C4—611no—
209#484—1222no—
210#484—1221no—
211#484—1221no—
212#484—1222no—
213#467—1222no—
214#451—1221no—
215#451—1221no—
216C2×#107—1221no—
217C2×#103—1221no—
218C2×C22⋊C8C2×D6111yescharacteristic
219C2×C22⋊C8C2×D6111yescharacteristic
220#398—1222no—
221#391—1221no—
222#385—1222no—
223#384—1221no—
224#380—611no—
225C4×C22⋊C4C2×D6111yescharacteristic
226#342—1222no—
227#340—1222no—
228#328—1222no—
229C61#995911yescharacteristic, complemented, Sylow
230C22×Q8—611no—
231C22×Q8—611no—
232C23×C4—1221no—
233C23×C4C22×D6111yescharacteristic
234C2×Q16—1222no—
235C22×C8—21no—
236C22×C8D24412yes—
237C22⋊Q8—1221no—
238C22⋊Q8—1221no—
239C22⋊Q8—1221no—
240C22⋊Q8—1221no—
241C22⋊Q8—1221no—
242C22⋊Q8—1221no—
243C22⋊Q8—1221no—
244C22⋊Q8—1221no—
245C2×C4⋊C4—612no—
246C2×C4⋊C4—611no—
247C2×C4⋊C4—611no—
248C2×C4⋊C4—1221no—
249C2×C4⋊C4—1221no—
250C2×C4⋊C4—1221no—
251C2×C4⋊C4—1221no—
252C2×C4⋊C4—1224no—
253C2×C4⋊C4—1221no—
254C2×C4⋊C4—1221no—
255C2×C4⋊C4—1221no—
256C2×C4⋊C4—1221no—
257C2×C22⋊C4—1221no—
258C2×C22⋊C4—1221no—
259C2×C22⋊C4—1221no—
260C2×C22⋊C4D24412yes—
261C2×C42—21no—
262#107—1224no—
263#107—1222no—
264#103—1224no—
265#103—1222no—
266#103—1222no—
267C22⋊C8—21no—
268C22⋊C8—21no—
269C22⋊C8—21no—
270C22⋊C8D24412yes—
271#95—21no—
272#95—1224no—
273#95—1221no—
274C24C2×D24411yescharacteristic
275C2×Q8—1221no—
276C2×Q8—1221no—
277C2×Q8—1221no—
278C2×Q8—1221no—
279C2×Q8—2442no—
280C2×Q8—2442no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—1221no—
284C22×C4—1221no—
285C22×C4—1221no—
286C22×C4—1222no—
287C22×C4—1221no—
288C22×C4—4882no—
289C22×C4—2441no—
290C22×C4—2441no—
291C22×C4—2441no—
292C22×C4C2×D24411yescharacteristic
293C22×C4C2×D24411yescharacteristic
294C22×C4D4×D6111yescharacteristic
295C22×C4D4×D6111yescharacteristic
296C22×C4D4×D6111yescharacteristic
297C22×C4#1175822611yescharacteristic
298Q16—4882no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—42no—
302C2×C8—42no—
303C4⋊C4—1222no—
304C4⋊C4—1221no—
305C4⋊C4—1221no—
306C4⋊C4—1221no—
307C4⋊C4—1222no—
308C4⋊C4—1221no—
309C4⋊C4—4884no—
310C4⋊C4—2442no—
311C4⋊C4—2441no—
312C4⋊C4—2441no—
313C4⋊C4—2441no—
314C4⋊C4—2441no—
315C4⋊C4—2441no—
316C4⋊C4—2441no—
317C22⋊C4—22no—
318C22⋊C4—2442no—
319C22⋊C4—2441no—
320C22⋊C4—2441no—
321C22⋊C4—2441no—
322C22⋊C4—2441no—
323C22⋊C4Dic24414yescomplemented
324C42—42no—
325C23—41no—
326C23—41no—
327C23#1451974711yescharacteristic
328C23#1451975211yescharacteristic
329C23#1451977411yescharacteristic
330C23C2×Dic24412yes—
331C23#1451981012yes—
332Q8—4881no—
333Q8—2442no—
334Q8—2442no—
335Q8—2442no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—4881no—
346C2×C4—4881no—
347C2×C4—4881no—
348C2×C4—4882no—
349C2×C4—4882no—
350C2×C4—4881no—
351C2×C4—4881no—
352C2×C4—2442no—
353C2×C4—2444no—
354C2×C4—2442no—
355C2×C4—2441no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4—2441no—
359C2×C4—2442no—
360C2×C4#1451974711yescharacteristic
361C2×C4#1451974711yescharacteristic
362C2×C4#1451975211yescharacteristic
363C2×C4#1451975211yescharacteristic
364C8—42no—
365C8—42no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22#1683967211yescharacteristic
375C22#1683992012yes—
376C22#1683997312yes—
377C22#1684014211yescharacteristic
378C22#1684015611yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—44no—
383C4—9761no—
384C4—4881no—
385C4—4881no—
386C4—4882no—
387C4—4881no—
388C4—4881no—
389C4#1683995312yes—
390C2—22no—
391C2—21no—
392C2—22no—
393C2—21no—
394C2#2144687511yescharacteristic
395C2#2144692111yescharacteristic
396C2#2144695111yescharacteristic
subgroupG/H
centerC22#16839672
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1282C2
socleC22×C61#392
prime radicalC24×C61C2×D4
hypercenter#1282D61
hypocenterC61#9959
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
104. C22⋊C4×C61Q16semidirect
229. C61#9959semidirect
323. C22⋊C4Dic244semidirect

Maximal subgroups

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

subgroupindex
#214565942
C2×#168400982
#214502972
#214468042
#214459102
#214443922
C61×#12822
#995961