Finite Group Databasebeta

← #29584413

Subgroups — #29584413

Order 15616.

604 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 121 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#21446006C212yesmaximal
2#21444083C211yesmaximal, characteristic
3#21443697C212yesmaximal
4#21443662C211yesmaximal, characteristic
5C61×#2659C211yesmaximal, characteristic
6C22×C4⋊Dic61C2211yescharacteristic
7C22×#11758196—21no—
8C2×#14519710—22no—
9C2×C22⋊#92418C2212yes—
10C2×#14519684—21no—
11C2×#14519684C2211yescharacteristic
12#16839524—22no—
13C2×C22⋊Q8×C61—21no—
14C61×#398—22no—
15C61×#395—22no—
16C61×#380—21no—
17C61×#380C2212yes—
18C2×#95×C61C2211yescharacteristic
19C23×Dic61—21no—
20C23×Dic61—21no—
21C22×#92418—22no—
22C2×C4⋊Dic61—22no—
23C2×C4⋊Dic61—42no—
24C2×#11758196—42no—
25C2×#11758196—42no—
26C2×#11758196—42no—
27#14519710—44no—
28C22⋊#92418—22no—
29C22⋊#92418Q814yescomplemented
30#14519684—21no—
31#14519684—42no—
32#14519684Q812yescomplemented
33C22×Q8×C61—21no—
34C23×C4×C61D411yescharacteristic
35C23×C4×C61D411yescharacteristic
36C23×C4×C61C2311yescharacteristic
37C22⋊Q8×C61—42no—
38C22⋊Q8×C61—42no—
39C2×C4⋊C4×C61—22no—
40C2×C4⋊C4×C61—21no—
41C2×C4⋊C4×C61—21no—
42C2×C4⋊C4×C61—21no—
43C2×C4⋊C4×C61D412yes—
44C2×C4⋊C4×C61D412yes—
45C2×C22⋊C4×C61—22no—
46#95×C61—22no—
47#95×C61—22no—
48#95×C61—22no—
49C24×C61C2×D411yescharacteristic
50C2×Q8×C61—42no—
51C2×Q8×C61—81no—
52C2×Q8×C61—81no—
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—21no—
57C22×C4×C61—22no—
58C22×C4×C61—22no—
59C22×C4×C61—41no—
60C22×C4×C61—41no—
61C22×C4×C61—41no—
62C22×C4×C61—41no—
63C22×C4×C61—41no—
64C22×C4×C61—41no—
65C22×C4×C61D812yes—
66C22×C4×C61Q1612yes—
67C22×C4×C61C2×D412yes—
68C22×C4×C61C2×Q812yes—
69C22×C4×C61C2×Q811yescharacteristic
70C22×C4×C61#4111yescharacteristic
71C22×Dic61—22no—
72C22×Dic61—22no—
73C22×Dic61—22no—
74C22×Dic61—22no—
75C22×Dic61—41no—
76C22×Dic61—41no—
77C22×Dic61—41no—
78C22×Dic61—41no—
79C22×Dic61—41no—
80C22×Dic61—41no—
81C22×Dic61—41no—
82C22×Dic61—41no—
83C22×Dic61—41no—
84C22×Dic61—41no—
85C4⋊C4×C61—42no—
86C4⋊C4×C61—44no—
87C4⋊C4×C61—42no—
88C4⋊C4×C61—42no—
89C4⋊C4×C61—81no—
90C4⋊C4×C61—81no—
91C22⋊C4×C61—42no—
92C22⋊C4×C61—42no—
93C4⋊Dic61—24no—
94C4⋊Dic61—42no—
95#11758196—48no—
96C2×#92418—24no—
97C2×#92418—42no—
98C2×#92418—42no—
99C23×C61—21no—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61#12011yescharacteristic
104C23×C61C22⋊Q811yescharacteristic
105C23×C61C4⋊Q811yescharacteristic
106C23×C61C2×D811yescharacteristic
107C23×C61C2×Q1611yescharacteristic
108C23×C61AGL(1,ℤ/8)11yescharacteristic
109C23×C61#13711yescharacteristic
110Q8×C61—84no—
111C2×C4×C61—22no—
112C2×C4×C61—22no—
113C2×C4×C61—22no—
114C2×C4×C61—22no—
115C2×C4×C61—42no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—42no—
123C2×C4×C61—42no—
124C2×C4×C61—42no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61—42no—
129C2×C4×C61—42no—
130C2×C4×C61—81no—
131C2×C4×C61—81no—
132C2×C4×C61C22⋊Q812yes—
133C2×C4×C61C4⋊Q812yes—
134C2×Dic61—24no—
135C2×Dic61—42no—
136C2×Dic61—42no—
137C2×Dic61—42no—
138C2×Dic61—42no—
139C2×Dic61—42no—
140C2×Dic61—42no—
141C2×Dic61—42no—
142C2×Dic61—42no—
143C2×Dic61—42no—
144C2×Dic61—81no—
145C2×Dic61—81no—
146C2×Dic61—81no—
147C2×Dic61—81no—
148#92418—44no—
149#10567—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—21no—
153C22×C61—21no—
154C22×C61—21no—
155C22×C61—21no—
156C22×C61—21no—
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—21no—
162C22×C61#39511yescharacteristic
163C22×C61#44711yescharacteristic
164C22×C61#45111yescharacteristic
165C22×C61#45311yescharacteristic
166C22×C61#45611yescharacteristic
167C22×C61B(4,𝔽2)11yescharacteristic
168C22×C61#45811yescharacteristic
169C22×C61#47411yescharacteristic
170C22×C61#47711yescharacteristic
171C22×C61#50011yescharacteristic
172C22×C61#50111yescharacteristic
173C4×C61—22no—
174C4×C61—22no—
175C4×C61—42no—
176C4×C61—42no—
177C4×C61—82no—
178C4×C61—82no—
179Dic61—44no—
180Dic61—82no—
181#2659D6111yescharacteristic
182#1852—612no—
183#1444—611no—
184#1294—612no—
185#1280—611no—
186C2×C61—21no—
187C2×C61—21no—
188C2×C61—21no—
189C2×C61—21no—
190C2×C61#159511yescharacteristic
191C2×C61#160511yescharacteristic
192C2×C61#161011yescharacteristic
193C2×C61#161411yescharacteristic
194C2×C61#205711yescharacteristic
195C2×C61#206011yescharacteristic
196C2×C61#206211yescharacteristic
197C2×C22⋊Q8—21no—
198C22×C4⋊C4—611no—
199C22×C4⋊C4—1221no—
200C2×#107—1222no—
201C2×C22⋊C8—612no—
202#398—22no—
203#395—22no—
204#380—21no—
205#380C2×D6112yes—
206C2×#95—611no—
207C2×#95—1221no—
208C2×#95C2×D6111yescharacteristic
209#340—1222no—
210C61#1056711yescharacteristic, complemented, Sylow
211C22×Q8—21no—
212C23×C4—1221no—
213C23×C4—1221no—
214C23×C4C22⋊D6111yescharacteristic
215C23×C4C22⋊D6111yescharacteristic
216C23×C4C22×D6111yescharacteristic
217C22×C8—1222no—
218C22⋊Q8—42no—
219C22⋊Q8—42no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—1222no—
225C2×C4⋊C4—2442no—
226C2×C4⋊C4—2442no—
227C2×C4⋊C4—2442no—
228C2×C4⋊C4—2442no—
229C2×C4⋊C4C22⋊D6112yes—
230C2×C4⋊C4C22⋊D6112yes—
231C2×C22⋊C4—22no—
232#107—2444no—
233C22⋊C8—614no—
234C22⋊C8—1222no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—612no—
239#95—1221no—
240#95—2442no—
241C24C2×C22⋊D6111yescharacteristic
242C2×Q8—42no—
243C2×Q8—81no—
244C2×Q8—81no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—1222no—
258C22×C4—1222no—
259C22×C4—1222no—
260C22×C4—1222no—
261C22×C4—2441no—
262C22×C4—2441no—
263C22×C4—2441no—
264C22×C4—2441no—
265C22×C4—2441no—
266C22×C4—2441no—
267C22×C4—2441no—
268C22×C4—2441no—
269C22×C4—2441no—
270C22×C4—2441no—
271C22×C4#1175819912yes—
272C22×C4#1175820212yes—
273C22×C4Q8×D6112yes—
274C22×C4Q8×D6111yescharacteristic
275C22×C4#1175822611yescharacteristic
276C22×C4C2×C22⋊D6112yes—
277C2×C8—1224no—
278C2×C8—2442no—
279C2×C8—2442no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—81no—
285C4⋊C4—81no—
286C4⋊C4—1224no—
287C4⋊C4—2442no—
288C4⋊C4—2448no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23#1451974611yescharacteristic
296C23#1451975311yescharacteristic
297C23#1451977511yescharacteristic
298C23C2×#1175819911yescharacteristic
299C23C2×#1175820211yescharacteristic
300C23#1451980311yescharacteristic
301C23#1451980811yescharacteristic
302Q8—84no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—1224no—
325C2×C4—4881no—
326C2×C4—4881no—
327C2×C4—4881no—
328C2×C4—4881no—
329C2×C4—2442no—
330C2×C4—2442no—
331C2×C4—2442no—
332C2×C4—2442no—
333C2×C4—2442no—
334C2×C4—2442no—
335C2×C4—2442no—
336C2×C4—2442no—
337C2×C4—2442no—
338C2×C4#1451975312yes—
339C2×C4#1451977512yes—
340C8—2444no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22#1683968111yescharacteristic
354C22#1683989611yescharacteristic
355C22#1683991811yescharacteristic
356C22#1683993011yescharacteristic
357C22#1683994511yescharacteristic
358C22#1683994911yescharacteristic
359C22#1683995211yescharacteristic
360C22#1684003911yescharacteristic
361C22#1684005411yescharacteristic
362C22#1684016511yescharacteristic
363C22#1684016811yescharacteristic
364C4—22no—
365C4—22no—
366C4—42no—
367C4—42no—
368C4—82no—
369C4—82no—
370C4—4882no—
371C4—2444no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2#2144473711yescharacteristic
377C2#2144479311yescharacteristic
378C2#2144482211yescharacteristic
379C2#2144484411yescharacteristic
380C2#2144714411yescharacteristic
381C2#2144716111yescharacteristic
382C2#2144717111yescharacteristic
subgroupG/H
centerC23#14519746
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#2659C2
socleC23×C61#120
prime radicalC24×C61C2×D4
hypercenter#2659D61
hypocenterC61#10567
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2659

Complements

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

NH ≅ G/Ntype
29. C22⋊#92418Q8semidirect
32. #14519684Q8semidirect
210. C61#10567semidirect

Maximal subgroups

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

subgroupindex
#214460062
#214440832
#214436972
#214436622
C61×#26592
#1056761