Finite Group Databasebeta

← #29595838

Subgroups — #29595838

Order 15616.

992 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#16839557C211yesmaximal, characteristic, complemented
2C2×#16839549C212yesmaximal, complemented
3#21446228C212yesmaximal, complemented
4#21446208C211yesmaximal, characteristic, complemented
5C2×C61×H(3,𝔽4)C211yesmaximal, characteristic
6#16840237C2211yescharacteristic
7#16839771—21no—
8C2×#14519694C2211yescharacteristic
9C2×#14519692C2212yes—
10#16839618—22no—
11#16839557—22no—
12#16839549—24no—
13#16839528—22no—
14#16839522—21no—
15C61×H(3,𝔽4)—22no—
16C61×H(3,𝔽4)—24no—
17C61×H(3,𝔽4)C414yescomplemented
18C2×#124×C61—21no—
19C2×#124×C61—22no—
20C2×#124×C61—21no—
21C2×#124×C61C2211yescharacteristic
22C2×#120×C61—22no—
23C2×#120×C61C2212yes—
24C2×#11758194—21no—
25D4×Dic61—42no—
26#14519736—21no—
27C2×C22⋊Dic61—22no—
28C2×C22⋊Dic61—22no—
29C2×C22⋊Dic61—44no—
30#14519701—42no—
31#14519694—22no—
32#14519692—22no—
33#14519692—22no—
34C25×C61D412yes—
35C22×Q8×C61—21no—
36C22×Q8×C61D411yescharacteristic
37C22×D4×C61—21no—
38C22×D4×C61—21no—
39C22×D4×C61—22no—
40C22×D4×C61C2311yescharacteristic
41#124×C61—28no—
42#124×C61—24no—
43#124×C61—24no—
44#124×C61—24no—
45#124×C61—22no—
46#124×C61—28no—
47#124×C61—24no—
48#124×C61C2×C412yes—
49#124×C61D412yes—
50#120×C61—28no—
51#120×C61—28no—
52#120×C61—22no—
53#120×C61—22no—
54#120×C61C2×C414yes—
55#120×C61D414yes—
56C2×C22⋊C4×C61—22no—
57C2×C22⋊C4×C61—22no—
58C2×C22⋊C4×C61—22no—
59C2×C22⋊C4×C61—22no—
60C2×C22⋊C4×C61D412yes—
61C2×C42×C61—21no—
62C2×C42×C61D411yescharacteristic
63C24×C61—22no—
64C24×C61—22no—
65C24×C61—42no—
66C24×C61—42no—
67C24×C61—84no—
68C24×C61C22⋊C414yescomplemented
69C24×C61C2×D412yes—
70C2×Q8×C61—24no—
71C2×Q8×C61—21no—
72C2×Q8×C61—41no—
73C2×Q8×C61—41no—
74C2×Q8×C61—82no—
75C2×Q8×C61C22⋊C412yescomplemented
76C2×D4×C61—24no—
77C2×D4×C61—24no—
78C2×D4×C61—24no—
79C2×D4×C61—24no—
80C2×D4×C61—41no—
81C2×D4×C61—41no—
82C2×D4×C61—84no—
83C2×D4×C61—82no—
84C2×D4×C61—81no—
85C2×D4×C61—81no—
86C2×D4×C61C22×C411yescharacteristic
87C2×D4×C61C2×D411yescharacteristic
88C2×D4×C61C2×D412yes—
89C22×C4×C61—21no—
90C22×C4×C61—22no—
91C22×C4×C61—21no—
92C22×C4×C61C2×D411yescharacteristic
93C22×Dic61—24no—
94C22×Dic61—42no—
95C22⋊C4×C61—28no—
96C22⋊C4×C61—28no—
97C22⋊C4×C61—24no—
98C22⋊C4×C61—24no—
99C22⋊C4×C61—22no—
100C22⋊C4×C61—24no—
101C22⋊C4×C61—24no—
102C22⋊C4×C61C22⋊C414yescomplemented
103C42×C61—24no—
104C42×C61—21no—
105C42×C61C22⋊C412yescomplemented
106C22⋊Dic61—42no—
107C22⋊Dic61—41no—
108C22⋊Dic61—42no—
109C22⋊Dic61—42no—
110C22⋊Dic61—88no—
111C4⋊Dic61—22no—
112C4×Dic61—41no—
113#11758194—41no—
114#11758194—41no—
115C2×#92418—41no—
116C23×C61—22no—
117C23×C61—22no—
118C23×C61—24no—
119C23×C61—44no—
120C23×C61—44no—
121C23×C61—44no—
122C23×C61—44no—
123C23×C61—48no—
124C23×C61—42no—
125C23×C61—42no—
126C23×C61—82no—
127C23×C61—84no—
128C23×C61—84no—
129C23×C61—88no—
130C23×C61—84no—
131C23×C61C2×C22⋊C412yes—
132C23×C61#12011yescharacteristic
133C23×C61#12012yes—
134Q8×C61—42no—
135Q8×C61—81no—
136Q8×C61—84no—
137D4×C61—41no—
138D4×C61—42no—
139D4×C61—41no—
140D4×C61—84no—
141D4×C61—84no—
142D4×C61—84no—
143D4×C61—84no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—22no—
147C2×C4×C61—24no—
148C2×C4×C61—22no—
149C2×C4×C61—44no—
150C2×C4×C61—42no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61C2×C22⋊C411yescharacteristic
154C2×C4×C61#12011yescharacteristic
155C2×Dic61—42no—
156C2×Dic61—84no—
157C2×Dic61—82no—
158C2×Dic61—84no—
159#92418—81no—
160#12763—611nomaximal, Sylow
161C22×C61—22no—
162C22×C61—22no—
163C22×C61—22no—
164C22×C61—42no—
165C22×C61—42no—
166C22×C61—44no—
167C22×C61—44no—
168C22×C61—44no—
169C22×C61—48no—
170C22×C61—44no—
171C22×C61—82no—
172C22×C61—84no—
173C22×C61—84no—
174C22×C61—88no—
175C22×C61—84no—
176C22×C61#37911yescharacteristic
177C22×C61B(4,𝔽2)11yescharacteristic
178C22×C61#45811yescharacteristic
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—44no—
182C4×C61—42no—
183C4×C61—42no—
184Dic61—82no—
185Dic61—162no—
186C2×H(3,𝔽4)D6111yescharacteristic
187C2×#354—611no—
188C2×AΣL(1,𝔽16)—612no—
189#1889—612no—
190#1885—611no—
191C2×C61—21no—
192C2×C61—21no—
193C2×C61—42no—
194C2×C61—42no—
195C2×C61—84no—
196C2×C61#179011yescharacteristic
197C2×C61#219912yes—
198H(3,𝔽4)—22no—
199H(3,𝔽4)—24no—
200H(3,𝔽4)Dic6114yescomplemented
201C2×#124—21no—
202C2×#124—22no—
203C2×#124—21no—
204C2×#124C2×D6111yescharacteristic
205C2×#120—22no—
206C2×#120C2×D6112yes—
207#518—611no—
208#419—1221no—
209C2×#100—611no—
210C2×#99—612no—
211#379—1222no—
212#354—1222no—
213AΣL(1,𝔽16)—1224no—
214#342—1222no—
215#339—1221no—
216C61#1276311yescharacteristic, complemented, Sylow
217C25C22⋊D6112yes—
218C22×Q8—21no—
219C22×Q8C22⋊D6111yescharacteristic
220C22×D4—21no—
221C22×D4—21no—
222C22×D4—22no—
223C22×D4C22×D6111yescharacteristic
224C2×M16—1221no—
225#124—28no—
226#124—24no—
227#124—24no—
228#124—24no—
229#124—22no—
230#124—28no—
231#124—24no—
232#124C2×Dic6112yes—
233#124C22⋊D6112yes—
234#120—28no—
235#120—28no—
236#120—22no—
237#120—22no—
238#120C2×Dic6114yes—
239#120C22⋊D6114yes—
240C4×D4—2442no—
241#117—1221no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—1222no—
247C2×C22⋊C4—1222no—
248C2×C22⋊C4—2444no—
249C2×C22⋊C4C22⋊D6112yes—
250C2×C42—21no—
251C2×C42C22⋊D6111yescharacteristic
252#103—2442no—
253#100—1222no—
254#99—1222no—
255#99—1222no—
256C24—22no—
257C24—22no—
258C24—42no—
259C24—42no—
260C24—84no—
261C24C22⋊Dic6114yescomplemented
262C24C2×C22⋊D6112yes—
263C2×Q8—24no—
264C2×Q8—21no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—82no—
268C2×Q8C22⋊Dic6112yescomplemented
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—84no—
276C2×D4—82no—
277C2×D4—81no—
278C2×D4—81no—
279C2×D4C22×Dic6111yescharacteristic
280C2×D4C2×C22⋊D6111yescharacteristic
281C2×D4C2×C22⋊D6112yes—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—1224no—
286C22×C4—2442no—
287C22×C4C2×C22⋊D6111yescharacteristic
288M16—2441no—
289M16—2441no—
290C2×C8—2441no—
291C4⋊C4—1222no—
292C22⋊C4—28no—
293C22⋊C4—28no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—22no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—4888no—
300C22⋊C4—2442no—
301C22⋊C4—2442no—
302C22⋊C4—2442no—
303C22⋊C4—2441no—
304C22⋊C4C22⋊Dic6114yescomplemented
305C42—24no—
306C42—21no—
307C42—2441no—
308C42C22⋊Dic6112yescomplemented
309C23—22no—
310C23—22no—
311C23—24no—
312C23—44no—
313C23—44no—
314C23—44no—
315C23—44no—
316C23—48no—
317C23—42no—
318C23—42no—
319C23—82no—
320C23—84no—
321C23—84no—
322C23—84no—
323C23—88no—
324C23C2×C22⋊Dic6112yes—
325C23#1451974612yes—
326C23#1451974611yescharacteristic
327Q8—42no—
328Q8—81no—
329Q8—84no—
330D4—41no—
331D4—42no—
332D4—41no—
333D4—84no—
334D4—84no—
335D4—84no—
336D4—84no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—4882no—
347C2×C4—4884no—
348C2×C4—4884no—
349C2×C4—2442no—
350C2×C4C2×C22⋊Dic6111yescharacteristic
351C2×C4#1451974611yescharacteristic
352C8—4881no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—42no—
357C22—42no—
358C22—44no—
359C22—44no—
360C22—44no—
361C22—48no—
362C22—44no—
363C22—82no—
364C22—84no—
365C22—84no—
366C22—88no—
367C22—84no—
368C22#1683961811yescharacteristic
369C22#1683994911yescharacteristic
370C22#1683995211yescharacteristic
371C4—21no—
372C4—21no—
373C4—44no—
374C4—42no—
375C4—42no—
376C4—9762no—
377C4—4882no—
378C2—21no—
379C2—21no—
380C2—42no—
381C2—42no—
382C2—84no—
383C2#2144576711yescharacteristic
384C2#2144782012yes—
subgroupG/H
centerC22#16839949
commutatorC2×D4×C61C22×C4
FrattiniC22×D4C22×D61
Fitting, prime radicalC2×C61×H(3,𝔽4)C2
socleC22×C61B(4,𝔽2)
cosocleC22×D4×C61C23
hypercenterC2×H(3,𝔽4)D61
hypocenterC61#12763
LCSG ▷ C2×D4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. C2×#16839557C2semidirect
2. C2×#16839549C2semidirect
3. #21446228C2semidirect
4. #21446208C2semidirect
17. C61×H(3,𝔽4)C4semidirect
68. C24×C61C22⋊C4semidirect
75. C2×Q8×C61C22⋊C4semidirect
102. C22⋊C4×C61C22⋊C4semidirect
105. C42×C61C22⋊C4semidirect
200. H(3,𝔽4)Dic61semidirect
216. C61#12763semidirect
261. C24C22⋊Dic61semidirect
268. C2×Q8C22⋊Dic61semidirect
304. C22⋊C4C22⋊Dic61semidirect
308. C42C22⋊Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168395572
C2×#168395492
#214462282
#214462082
C2×C61×H(3,𝔽4)2
#1276361