Finite Group Databasebeta

← #29596259

Subgroups — #29596259

Order 15616.

784 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 107 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#21445740C211yesmaximal, characteristic
2#21445734C211yesmaximal, characteristic
3#21443937C214yesmaximal, complemented
4C61×#2308C211yesmaximal, characteristic
5C2×#14519694—21no—
6C2×#14519694C2212yes—
7C2×#14519692—21no—
8C2×#14519692C2212yes—
9#16839726—21no—
10#16839618—21no—
11C23×D4×C61C2211yescharacteristic
12C22×C22⋊C4×C61C411yescharacteristic, complemented
13C22×C22⋊C4×C61C411yescharacteristic, complemented
14C61×#386—21no—
15C61×#386—21no—
16C61×#386C412yescomplemented
17C61×#386C2212yes—
18C61×#381—21no—
19C61×#379—21no—
20C2×#11758194—22no—
21C2×C22⋊Dic61—22no—
22C2×C22⋊Dic61—42no—
23#14519694—24no—
24#14519694—42no—
25#14519692—24no—
26#14519692—42no—
27C22⋊#92418—42no—
28C25×C61C2×C411yescharacteristic
29C25×C61D411yescharacteristic
30C22×D4×C61—24no—
31C22×D4×C61—42no—
32C22×D4×C61—42no—
33C22×D4×C61C2×C411yescharacteristic
34C22×D4×C61D411yescharacteristic
35C22×D4×C61D411yescharacteristic
36C22×D4×C61C2311yescharacteristic
37C23×C4×C61—21no—
38C23×C4×C61C2×C412yes—
39C23×C4×C61D411yescharacteristic
40C2×C22⋊C4×C61—21no—
41C2×C22⋊C4×C61—21no—
42C2×C22⋊C4×C61—21no—
43C2×C22⋊C4×C61—22no—
44C2×C22⋊C4×C61—22no—
45C2×C22⋊C4×C61—42no—
46C2×C22⋊C4×C61—42no—
47C2×C22⋊C4×C61C2×C412yes—
48#95×C61—22no—
49#95×C61—22no—
50C24×C61—22no—
51C24×C61—22no—
52C24×C61—22no—
53C24×C61—22no—
54C24×C61—22no—
55C24×C61—41no—
56C24×C61—82no—
57C24×C61—82no—
58C24×C61C22⋊C411yescharacteristic
59C24×C61C22⋊C411yescharacteristic
60C24×C61C22⋊C411yescharacteristic
61C24×C61C22×C411yescharacteristic
62C24×C61C2×D411yescharacteristic
63C2×D4×C61—22no—
64C2×D4×C61—22no—
65C2×D4×C61—22no—
66C2×D4×C61—22no—
67C2×D4×C61—22no—
68C2×D4×C61—41no—
69C2×D4×C61—41no—
70C2×D4×C61—41no—
71C2×D4×C61—41no—
72C2×D4×C61—41no—
73C2×D4×C61—41no—
74C2×D4×C61—416no—
75C2×D4×C61#4114yes—
76C22×C4×C61—22no—
77C22×C4×C61—22no—
78C22×C4×C61—22no—
79C22×C4×C61—22no—
80C22×C4×C61—41no—
81C22×C4×C61—42no—
82C22×C4×C61—42no—
83C22×C4×C61—82no—
84C22×C4×C61—82no—
85C22×C4×C61C22⋊C411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×Dic61—42no—
88C22⋊C4×C61—42no—
89C22⋊C4×C61—42no—
90C22⋊C4×C61—42no—
91C22⋊C4×C61—42no—
92C22⋊C4×C61—82no—
93C22⋊C4×C61—82no—
94C22⋊Dic61—42no—
95C22⋊Dic61—42no—
96C22⋊Dic61—84no—
97#11758194—42no—
98#11758194—42no—
99C2×#92418—42no—
100C23×C61—28no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—21no—
104C23×C61—21no—
105C23×C61—21no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61—21no—
112C23×C61—42no—
113C23×C61—44no—
114C23×C61—42no—
115C23×C61—42no—
116C23×C61—42no—
117C23×C61—42no—
118C23×C61—44no—
119C23×C61—82no—
120C23×C61—82no—
121C23×C61—82no—
122C23×C61—84no—
123C23×C61—84no—
124C23×C61—84no—
125C23×C61—84no—
126C23×C61#9912yes—
127C23×C61#9912yes—
128C23×C61C2×C22⋊C411yescharacteristic
129C23×C61#11712yes—
130C23×C61#12312yes—
131D4×C61—44no—
132D4×C61—44no—
133D4×C61—44no—
134D4×C61—44no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—44no—
141C2×C4×C61—42no—
142C2×C4×C61—42no—
143C2×C4×C61—42no—
144C2×C4×C61—42no—
145C2×C4×C61—42no—
146C2×C4×C61—42no—
147C2×C4×C61—84no—
148C2×C4×C61—84no—
149C2×C4×C61#12312yes—
150C2×Dic61—82no—
151C2×Dic61—84no—
152#92418—82no—
153#12854—611nomaximal, Sylow
154C22×C61—22no—
155C22×C61—22no—
156C22×C61—22no—
157C22×C61—22no—
158C22×C61—22no—
159C22×C61—22no—
160C22×C61—22no—
161C22×C61—42no—
162C22×C61—42no—
163C22×C61—42no—
164C22×C61—42no—
165C22×C61—42no—
166C22×C61—42no—
167C22×C61—42no—
168C22×C61—42no—
169C22×C61—42no—
170C22×C61—42no—
171C22×C61—42no—
172C22×C61—41no—
173C22×C61—42no—
174C22×C61—81no—
175C22×C61—81no—
176C22×C61—88no—
177C22×C61—88no—
178C22×C61#38111yescharacteristic
179C22×C61C2×#9911yescharacteristic
180C22×C61C2×#9911yescharacteristic
181C4×C61—22no—
182C4×C61—41no—
183C4×C61—82no—
184C4×C61—82no—
185Dic61—162no—
186#2308D6111yescharacteristic
187#1783—611no—
188#1781—611no—
189#1390—614no—
190C2×C61—21no—
191C2×C61—21no—
192C2×C61—42no—
193C2×C61—42no—
194C2×C61—42no—
195C2×C61—42no—
196C2×C61—81no—
197C2×C61#181911yescharacteristic
198C2×C61#212112yes—
199C23×D4C2×D6111yescharacteristic
200C22×C22⋊C4Dic6111yescharacteristic, complemented
201C22×C22⋊C4Dic6111yescharacteristic, complemented
202C2×#100—612no—
203C2×#100—1221no—
204C2×#99—612no—
205C2×#99—1221no—
206#407—1221no—
207#386—21no—
208#386—21no—
209#386Dic6112yescomplemented
210#386C2×D6112yes—
211#381—21no—
212#379—21no—
213#379—1221no—
214C61#1285411yescharacteristic, complemented, Sylow
215C25C2×Dic6111yescharacteristic
216C25C22⋊D6111yescharacteristic
217C22×D4—24no—
218C22×D4—42no—
219C22×D4—42no—
220C22×D4C2×Dic6111yescharacteristic
221C22×D4C22⋊D6111yescharacteristic
222C22×D4C22⋊D6111yescharacteristic
223C22×D4C22×D6111yescharacteristic
224C23×C4—21no—
225C23×C4C2×Dic6112yes—
226C23×C4C22⋊D6111yescharacteristic
227C2×M16—1222no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—1222no—
236C2×C22⋊C4—2442no—
237C2×C22⋊C4C2×Dic6112yes—
238#100—1224no—
239#100—2442no—
240#99—1224no—
241#99—2442no—
242C22⋊C8—2442no—
243#95—22no—
244#95—22no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—41no—
251C24—82no—
252C24—82no—
253C24C22⋊Dic6111yescharacteristic
254C24C22⋊Dic6111yescharacteristic
255C24C22⋊Dic6111yescharacteristic
256C24C22×Dic6111yescharacteristic
257C24C2×C22⋊D6111yescharacteristic
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—416no—
270C2×D4#1175822214yes—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—2442no—
281C22×C4C22⋊Dic6111yescharacteristic
282C22×C4C2×C22⋊D6111yescharacteristic
283M16—2442no—
284M16—2442no—
285C2×C8—2442no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—82no—
291C22⋊C4—82no—
292C22⋊C4—4884no—
293C22⋊C4—2442no—
294C22⋊C4—2442no—
295C23—28no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—42no—
308C23—44no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—44no—
314C23—82no—
315C23—82no—
316C23—82no—
317C23—84no—
318C23—84no—
319C23—84no—
320C23—84no—
321C23#1451969212yes—
322C23#1451969212yes—
323C23C2×C22⋊Dic6111yescharacteristic
324C23#1451973612yes—
325C23#1451976012yes—
326D4—44no—
327D4—44no—
328D4—44no—
329D4—44no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—4882no—
345C2×C4—4884no—
346C2×C4#1451976012yes—
347C8—4882no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—42no—
368C22—81no—
369C22—81no—
370C22—88no—
371C22—88no—
372C22#1683962511yescharacteristic
373C22C2×#1451969211yescharacteristic
374C22C2×#1451969211yescharacteristic
375C4—22no—
376C4—41no—
377C4—82no—
378C4—82no—
379C4—9762no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—42no—
384C2—42no—
385C2—42no—
386C2—81no—
387C2#2144589311yescharacteristic
388C2#2144746512yes—
subgroupG/H
centerC22C2×#14519692
commutatorC24×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#2308C2
socleC22×C61C2×#99
prime radicalC23×D4×C61C22
cosocleC22×D4×C61C23
hypercenter#2308D61
hypocenterC61#12854
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #2308

Complements

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

NH ≅ G/Ntype
3. #21443937C2semidirect
12. C22×C22⋊C4×C61C4semidirect
13. C22×C22⋊C4×C61C4semidirect
16. C61×#386C4semidirect
200. C22×C22⋊C4Dic61semidirect
201. C22×C22⋊C4Dic61semidirect
209. #386Dic61semidirect
214. C61#12854semidirect

Maximal subgroups

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

subgroupindex
#214457402
#214457342
#214439372
C61×#23082
#1285461