Finite Group Databasebeta

← #29575924

Subgroups — #29575924

Order 15616.

490 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 113 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#21455620C211yesmaximal, characteristic
2#21448736C211yesmaximal, characteristic
3#21444318C211yesmaximal, characteristic
4#21444215C211yesmaximal, characteristic
5#21443721C211yesmaximal, characteristic
6#21443677C211yesmaximal, characteristic
7C61×#1518C211yesmaximal, characteristic
8C2×C4×Dic122C2211yescharacteristic
9C2×#14519742—21no—
10#16840066—22no—
11#16840054C412yescomplemented
12#16840044C412yescomplemented
13#16839815C2211yescharacteristic
14#16839805—21no—
15#16839786C2211yescharacteristic
16#16839770—21no—
17#16839659—22no—
18#16839658—21no—
19#16839626—21no—
20C4×C4⋊Dic61C2211yescharacteristic
21#16839613—21no—
22#16839499—21no—
23#16839499—21no—
24#16839494—22no—
25C61×#533C2211yescharacteristic
26C61×#431C2211yescharacteristic
27C61×#382C2211yescharacteristic
28C61×#328—22no—
29C22×Dic122D411yescharacteristic
30C2×#11758194—21no—
31C4×Dic122—21no—
32C4×Dic122—21no—
33C4×Dic122C2×C412yes—
34C4×Dic122D412yes—
35#14519742—42no—
36#14519742—42no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61D411yescharacteristic
39C2×#11758196—21no—
40C2×#11758196—22no—
41C2×#11758196—22no—
42C2×C4×Dic61—21no—
43C2×C4×Dic61—22no—
44C2×C4×Dic61—21no—
45#14519710—22no—
46#14519708—22no—
47C4⋊#92418—21no—
48C4⋊#92418—21no—
49C4⋊#92418C2×C412yes—
50#14519700—22no—
51#14519700—22no—
52C22⋊#92418—21no—
53C4×#92418—21no—
54#14519684—22no—
55#14519684D411yescharacteristic
56#14519684D411yescharacteristic
57C4⋊Q8×C61C2×C411yescharacteristic
58C4⋊Q8×C61C2×C411yescharacteristic
59#125×C61—21no—
60C22⋊Q8×C61—21no—
61C22⋊Q8×C61—21no—
62C4×Q8×C61—21no—
63C4×Q8×C61—21no—
64#117×C61—21no—
65C2×C4⋊C4×C61—21no—
66C2×C42×C61C2311yescharacteristic
67C4⋊C8×C61—21no—
68C22⋊C8×C61—21no—
69C4×C8×C61—21no—
70#95×C61—21no—
71#95×C61D412yes—
72C2×Q8×C61—21no—
73C2×Q8×C61—21no—
74C22×C4×C61—22no—
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—21no—
82C2×Dic122—21no—
83C2×Dic122—21no—
84C2×Dic122—21no—
85C2×Dic122C22⋊C412yes—
86C2×Dic122SD1612yes—
87C2×Dic122Q1612yes—
88C2×C8×C61—41no—
89C4⋊C4×C61—21no—
90C4⋊C4×C61—21no—
91C4⋊C4×C61—21no—
92C4⋊C4×C61—21no—
93C4⋊C4×C61—41no—
94C4⋊C4×C61—41no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—41no—
97C22⋊C4×C61—41no—
98C42×C61—41no—
99C42×C61C22×C411yescharacteristic
100C42×C61C2×D411yescharacteristic
101C42×C61#4111yescharacteristic
102C42×C61#4111yescharacteristic
103C4⋊Dic61—21no—
104C4⋊Dic61—42no—
105C4⋊Dic61C22⋊C412yes—
106#11758196—22no—
107#11758196—22no—
108#11758196—44no—
109#11758196—44no—
110C4×Dic61—22no—
111C4×Dic61—22no—
112C4×Dic61—44no—
113C4×Dic61—41no—
114C4×Dic61—41no—
115#11758194—41no—
116#11758194—41no—
117C2×#92418—22no—
118C2×#92418—41no—
119C23×C61#12011yescharacteristic
120Q8×C61—81no—
121Q8×C61—81no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—42no—
125C2×C4×C61—42no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—42no—
129C2×C4×C61C2×C22⋊C411yescharacteristic
130C2×C4×C61C4×D411yescharacteristic
131C2×C4×C61C4×D411yescharacteristic
132C2×C4×C61#12111yescharacteristic
133C2×C4×C61#12111yescharacteristic
134C2×C4×C61#12311yescharacteristic
135C2×C4×C61C2×SD1611yescharacteristic
136C2×C4×C61C2×Q1611yescharacteristic
137C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C61#13711yescharacteristic
139C2×Dic61—22no—
140C2×Dic61—22no—
141C2×Dic61—22no—
142C2×Dic61—22no—
143C2×Dic61—22no—
144C2×Dic61—42no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150Dic122—22no—
151Dic122—22no—
152Dic122—22no—
153Dic122#10314yes—
154C8×C61—81no—
155#92418—42no—
156#92418—81no—
157#9176—611nomaximal, Sylow
158C22×C61—21no—
159C22×C61—21no—
160C22×C61#38611yescharacteristic
161C22×C61#45011yescharacteristic
162C22×C61#45111yescharacteristic
163C4×C61—21no—
164C4×C61—21no—
165C4×C61—21no—
166C4×C61—21no—
167C4×C61—82no—
168C4×C61—81no—
169C4×C61—81no—
170C4×C61C2×#10311yescharacteristic
171C4×C61#41711yescharacteristic
172C4×C61#48311yescharacteristic
173C4×C61#48411yescharacteristic
174Dic61—24no—
175Dic61—42no—
176Dic61—42no—
177Dic61—81no—
178#3078—611no—
179#2351—611no—
180#1518D6111yescharacteristic
181#1503—611no—
182#1479—611no—
183#1303—611no—
184#1286—611no—
185C2×C61—21no—
186C2×C61—21no—
187C2×C61#187211yescharacteristic
188C2×C61#190511yescharacteristic
189C2×C61#190611yescharacteristic
190#533C2×D6111yescharacteristic
191C2×C4×Q8—611no—
192C2×C4×Q8—1221no—
193#479—1222no—
194#477—612no—
195#475—612no—
196#431—611no—
197#431C2×D6111yescharacteristic
198#428—1221no—
199#423—611no—
200#419—1221no—
201#389—1221no—
202#389—1222no—
203#382—1221no—
204#382C2×D6111yescharacteristic
205C4×C4⋊C4—611no—
206C4×C4⋊C4—1221no—
207#328—22no—
208#328—1221no—
209#328—1221no—
210Q8⋊C8—1222no—
211C61#917611yescharacteristic, complemented, Sylow
212C22×Q8—611no—
213C2×M16—1221no—
214C4⋊Q8C4×D6111yescharacteristic
215C4⋊Q8C4×D6111yescharacteristic
216#125—21no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—612no—
222C4×Q8—612no—
223C4×Q8—1221no—
224C4×Q8—1221no—
225C4×Q8—2442no—
226C4×Q8—2442no—
227#117—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—611no—
230C2×C4⋊C4—1221no—
231C2×C4⋊C4—1222no—
232C2×C4⋊C4—1222no—
233C2×C4⋊C4—1221no—
234C2×C42—1221no—
235C2×C42—1222no—
236C2×C42—1221no—
237C2×C42C22×D6111yescharacteristic
238#107—1222no—
239#106—1222no—
240C4⋊C8—21no—
241C4⋊C8—612no—
242C4⋊C8—1221no—
243C4⋊C8—1221no—
244#103—1222no—
245#103—1222no—
246C22⋊C8—21no—
247C22⋊C8—1221no—
248C4×C8—21no—
249C4×C8—1221no—
250#95—21no—
251#95—611no—
252#95—611no—
253#95—1222no—
254#95C22⋊D6112yes—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—612no—
258C2×Q8—612no—
259C2×Q8—612no—
260C2×Q8—1221no—
261C2×Q8—1221no—
262C2×Q8—1221no—
263C22×C4—22no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—1221no—
267C22×C4—1221no—
268C22×C4D4×D6111yescharacteristic
269C22×C4D4×D6111yescharacteristic
270C22×C4C2×C22⋊D6111yescharacteristic
271M16—2441no—
272M16—2441no—
273C2×C8—41no—
274C2×C8—1222no—
275C2×C8—2441no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—612no—
285C4⋊C4—1222no—
286C4⋊C4—1222no—
287C4⋊C4—1221no—
288C4⋊C4—2444no—
289C4⋊C4—2444no—
290C4⋊C4—2442no—
291C22⋊C4—41no—
292C42—41no—
293C42—1222no—
294C42—1222no—
295C42—2444no—
296C42—2441no—
297C42—2441no—
298C42C2×C4×D6111yescharacteristic
299C42#1175822211yescharacteristic
300C42D4×D6111yescharacteristic
301C42#1175822411yescharacteristic
302C23#1451974511yescharacteristic
303Q8—81no—
304Q8—81no—
305Q8—614no—
306Q8—1222no—
307Q8—1222no—
308Q8—1222no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—1222no—
317C2×C4—1222no—
318C2×C4—1222no—
319C2×C4—1222no—
320C2×C4—1222no—
321C2×C4—2441no—
322C2×C4—2441no—
323C2×C4—2441no—
324C2×C4—2442no—
325C2×C4—2441no—
326C2×C4—2441no—
327C2×C4C22⋊C4×D6111yescharacteristic
328C2×C4#1451973711yescharacteristic
329C2×C4C4×C22⋊D6111yescharacteristic
330C2×C4#1451974811yescharacteristic
331C2×C4#1451975011yescharacteristic
332C2×C4#1451975911yescharacteristic
333C2×C4SD16×D6111yescharacteristic
334C2×C4Q16×D6111yescharacteristic
335C2×C4#1451980411yescharacteristic
336C2×C4#1451980911yescharacteristic
337C8—81no—
338C8—4881no—
339C8—2442no—
340C22—21no—
341C22—21no—
342C22#1683963811yescharacteristic
343C22#1683991111yescharacteristic
344C22#1683991611yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—82no—
350C4—81no—
351C4—81no—
352C4—1224no—
353C4—4881no—
354C4—2442no—
355C4—2442no—
356C4#103×D6111yescharacteristic
357C4#1683976211yescharacteristic
358C4#1684009111yescharacteristic
359C4#1684009611yescharacteristic
360C2—21no—
361C2—21no—
362C2#2144612311yescharacteristic
363C2#2144631011yescharacteristic
364C2#2144631711yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1518C2
socleC22×C61#386
prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenter#1518D61
hypocenterC61#9176
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
11. #16840054C4semidirect
12. #16840044C4semidirect
211. C61#9176semidirect

Maximal subgroups

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

subgroupindex
#214556202
#214487362
#214443182
#214442152
#214437212
#214436772
C61×#15182
#917661