Finite Group Databasebeta

← #29584396

Subgroups — #29584396

Order 15616.

574 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 99 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#21446656C212yesmaximal, complemented
2#21446248C212yesmaximal, complemented
3#21445996C211yesmaximal, characteristic, complemented
4#21443687C211yesmaximal, characteristic, complemented
5C61×#3186C211yesmaximal, characteristic
6C23×#92418C2211yescharacteristic
7C2×#14519706—21no—
8C2×#14519698—22no—
9C2×#14519692—22no—
10C2×#14519692—22no—
11#16839622C2212yes—
12C4×C22⋊Dic61C2211yescharacteristic
13#16839524—22no—
14#16839516—21no—
15C2×#121×C61C2212yes—
16C61×#480—22no—
17C61×#466—21no—
18C61×#466—21no—
19C61×#447—22no—
20C2×C22⋊C8×C61C2211yescharacteristic
21C22×#92418—22no—
22C22×#92418—21no—
23C22×#92418—42no—
24C2×C4⋊Dic61—22no—
25C2×C22⋊Dic61—22no—
26C2×C22⋊Dic61D411yescharacteristic
27C2×C22⋊Dic61D411yescharacteristic
28C2×C4×Dic61—21no—
29#14519706—42no—
30#14519698—44no—
31#14519692—44no—
32#14519692—44no—
33#14519684—21no—
34#14519684—22no—
35C22×D4×C61—22no—
36C22×D4×C61D412yes—
37C23×C4×C61C2311yescharacteristic
38C2×D8×C61—21no—
39C2×D8×C61—21no—
40C22×C8×C61D411yescharacteristic
41C22×C8×C61D411yescharacteristic
42#121×C61—22no—
43#121×C61—22no—
44#121×C61—22no—
45#121×C61—22no—
46C2×C4⋊C4×C61D412yes—
47C2×C22⋊C4×C61—22no—
48#107×C61—21no—
49#107×C61—21no—
50#102×C61—22no—
51#102×C61—22no—
52C22⋊C8×C61—22no—
53C24×C61—22no—
54C24×C61C2×D411yescharacteristic
55C2×D4×C61—22no—
56C2×D4×C61—22no—
57C2×D4×C61—42no—
58C2×D4×C61—42no—
59C2×D4×C61—42no—
60C2×D4×C61—42no—
61C2×D4×C61—42no—
62C2×D4×C61—42no—
63C2×D4×C61—42no—
64C2×D4×C61—42no—
65C22×C4×C61—22no—
66C22×C4×C61—41no—
67C22×C4×C61—41no—
68C22×C4×C61C2×D412yes—
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61#4111yescharacteristic
71C22×C4×C61#4112yes—
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—42no—
75D8×C61—81no—
76D8×C61—81no—
77C2×C8×C61—21no—
78C2×C8×C61—21no—
79C2×C8×C61—41no—
80C2×C8×C61—41no—
81C4⋊C4×C61—22no—
82C4⋊C4×C61—22no—
83C22⋊C4×C61—42no—
84C22⋊C4×C61—42no—
85C22⋊Dic61—21no—
86C22⋊Dic61—22no—
87C22⋊Dic61—42no—
88C22⋊Dic61—42no—
89C22⋊Dic61D812yescomplemented
90C4⋊Dic61—42no—
91C4⋊Dic61—42no—
92C4×Dic61—41no—
93C4×Dic61—41no—
94C2×#92418—24no—
95C2×#92418—22no—
96C2×#92418—41no—
97C2×#92418—42no—
98C2×#92418—41no—
99C23×C61—41no—
100C23×C61—41no—
101C23×C61—42no—
102C23×C61—42no—
103C23×C61—42no—
104C23×C61—82no—
105C23×C61—82no—
106C23×C61#12112yes—
107C23×C61#12111yescharacteristic
108C23×C61C2×D811yescharacteristic
109C23×C61#13511yescharacteristic
110C23×C61AGL(1,ℤ/8)12yes—
111D4×C61—44no—
112D4×C61—42no—
113D4×C61—42no—
114D4×C61—82no—
115D4×C61—82no—
116D4×C61—82no—
117D4×C61—82no—
118C2×C4×C61—21no—
119C2×C4×C61—22no—
120C2×C4×C61—21no—
121C2×C4×C61—22no—
122C2×C4×C61—41no—
123C2×C4×C61—42no—
124C2×C4×C61—42no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61#12011yescharacteristic
128C2×C4×C61#12312yes—
129C2×C4×C61#12411yescharacteristic
130C2×Dic61—22no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—82no—
137C2×Dic61—82no—
138C2×Dic61—82no—
139C8×C61—41no—
140C8×C61—41no—
141#92418—42no—
142#10564—611nomaximal, Sylow
143C22×C61—21no—
144C22×C61—22no—
145C22×C61—21no—
146C22×C61—22no—
147C22×C61—41no—
148C22×C61—41no—
149C22×C61—42no—
150C22×C61—82no—
151C22×C61—82no—
152C22×C61—82no—
153C22×C61—162no—
154C22×C61—162no—
155C22×C61#39411yescharacteristic
156C22×C61#45912yes—
157C22×C61#46312yes—
158C22×C61#46611yescharacteristic
159C22×C61#46911yescharacteristic
160C4×C61—22no—
161C4×C61—21no—
162C4×C61—82no—
163C4×C61B(4,𝔽2)11yescharacteristic
164C4×C61#45811yescharacteristic
165Dic61—42no—
166Dic61—81no—
167Dic61—162no—
168#3186D6111yescharacteristic
169#1964—612no—
170#1893—612no—
171#1850—611no—
172#1289—611no—
173C2×C61—21no—
174C2×C61—21no—
175C2×C61—22no—
176C2×C61—22no—
177C2×C61—162no—
178C2×C61#204411yescharacteristic
179C2×C61#205311yescharacteristic
180C2×C61#205411yescharacteristic
181C23×C8—611no—
182C2×#121C2×D6112yes—
183#480—22no—
184#466—21no—
185#466—21no—
186#447—22no—
187C2×C4⋊C8—1221no—
188C2×#102—1222no—
189C2×#99—1222no—
190C2×#99—1222no—
191C2×C22⋊C8C2×D6111yescharacteristic
192#380—612no—
193C4×C22⋊C4—611no—
194#340—1222no—
195#336—1221no—
196C61#1056411yescharacteristic, complemented, Sylow
197C22×D4—22no—
198C22×D4C22⋊D6112yes—
199C23×C4C22×D6111yescharacteristic
200C2×D8—21no—
201C2×D8—21no—
202C22×C8—1221no—
203C22×C8—1222no—
204C22×C8—2442no—
205C22×C8C22⋊D6111yescharacteristic
206C22×C8C22⋊D6111yescharacteristic
207#121—22no—
208#121—22no—
209#121—22no—
210#121—22no—
211C2×C4⋊C4—1222no—
212C2×C4⋊C4C22⋊D6112yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—611no—
215C2×C22⋊C4—611no—
216C2×C22⋊C4—1222no—
217C2×C42—1221no—
218#107—21no—
219#107—21no—
220C4⋊C8—2442no—
221#102—22no—
222#102—22no—
223#102—2444no—
224#99—2444no—
225#99—2444no—
226C22⋊C8—22no—
227#95—1221no—
228#95—1222no—
229C24—22no—
230C24D4×D6111yescharacteristic
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C22×C4—22no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—1221no—
245C22×C4—1221no—
246C22×C4—2442no—
247C22×C4#1175822411yescharacteristic
248C22×C4#1175822412yes—
249C22×C4C2×C22⋊D6112yes—
250C22×C4C2×C22⋊D6111yescharacteristic
251D8—81no—
252D8—81no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—1224no—
258C2×C8—1222no—
259C2×C8—2441no—
260C2×C8—2442no—
261C2×C8—2441no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—2442no—
265C4⋊C4—2442no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—612no—
269C22⋊C4—1221no—
270C22⋊C4—1222no—
271C22⋊C4—2442no—
272C22⋊C4—2442no—
273C42—2441no—
274C42—2441no—
275C23—41no—
276C23—41no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—82no—
281C23—82no—
282C23#1451974811yescharacteristic
283C23#1451974912yes—
284C23D8×D6111yescharacteristic
285C23#1451979611yescharacteristic
286C23#1451980112yes—
287D4—44no—
288D4—42no—
289D4—42no—
290D4—82no—
291D4—82no—
292D4—82no—
293D4—82no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—1222no—
304C2×C4—4882no—
305C2×C4—4882no—
306C2×C4—4882no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4#1451974611yescharacteristic
313C2×C4#1451975812yes—
314C2×C4#1451976411yescharacteristic
315C8—41no—
316C8—41no—
317C8—2442no—
318C22—22no—
319C22—21no—
320C22—22no—
321C22—21no—
322C22—41no—
323C22—41no—
324C22—42no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—162no—
329C22—162no—
330C22#1683967611yescharacteristic
331C22#1683995512yes—
332C22#1683997512yes—
333C22#1683999511yescharacteristic
334C22#1684001211yescharacteristic
335C4—22no—
336C4—21no—
337C4—82no—
338C4—9762no—
339C4—4881no—
340C4—2442no—
341C4#1683994911yescharacteristic
342C4#1683995211yescharacteristic
343C2—22no—
344C2—21no—
345C2—21no—
346C2—22no—
347C2—162no—
348C2#2144706711yescharacteristic
349C2#2144712211yescharacteristic
350C2#2144712711yescharacteristic
subgroupG/H
centerC22#16839676
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
Fitting, prime radicalC61×#3186C2
socleC22×C61#394
hypercenter#3186D61
hypocenterC61#10564
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3186

Complements

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

NH ≅ G/Ntype
1. #21446656C2semidirect
2. #21446248C2semidirect
3. #21445996C2semidirect
4. #21443687C2semidirect
89. C22⋊Dic61D8semidirect
196. C61#10564semidirect

Maximal subgroups

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

subgroupindex
#214466562
#214462482
#214459962
#214436872
C61×#31862
#1056461