Finite Group Databasebeta

← #29580731

Subgroups — #29580731

Order 15616.

536 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 87 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#21456625C211yesmaximal, characteristic
2C2×#16840098C211yesmaximal, characteristic, complemented
3C2×#16840073C211yesmaximal, characteristic
4#21445897C211yesmaximal, characteristic, complemented
5#21444184C211yesmaximal, characteristic, complemented
6#21443707C211yesmaximal, characteristic, complemented
7C61×#1471C211yesmaximal, characteristic
8C2×#14519756C2211yescharacteristic
9C22×C4⋊Dic61C2211yescharacteristic
10#16840098—22no—
11#16840098—22no—
12#16840097—21no—
13#16840073—22no—
14#16840073—21no—
15#16840073—22no—
16#16840006—22no—
17#16839913—21no—
18#16839900—21no—
19C2×#14519711—21no—
20C2×#14519710—21no—
21C2×#14519702—21no—
22C2×#14519698—21no—
23#16839726C2211yescharacteristic
24C2×C22⋊#92418C2211yescharacteristic
25#16839530—22no—
26#16839487—22no—
27C2×#121×C61C2211yescharacteristic
28C61×#407C2211yescharacteristic
29C2×C22⋊C8×C61C2211yescharacteristic
30C61×#323—22no—
31C22×Dic122D411yescharacteristic
32C23×Dic61—21no—
33C2×#11758200—22no—
34C2×#11758194D412yes—
35C22×#92418—21no—
36#14519756—21no—
37#14519756—21no—
38#14519756—21no—
39#14519756—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61D412yes—
46C2×C4⋊Dic61D411yescharacteristic
47C2×#11758196—21no—
48C2×C22⋊Dic61—21no—
49#14519711—24no—
50#14519710—22no—
51#14519710—24no—
52#14519702—24no—
53#14519700—22no—
54#14519698—24no—
55#14519698—22no—
56C22⋊#92418—21no—
57C22⋊#92418—21no—
58C22⋊#92418—21no—
59C22⋊#92418D412yes—
60C22×D4×C61—21no—
61C22×D4×C61D411yescharacteristic
62C23×C4×C61C2311yescharacteristic
63C2×M16×C61—21no—
64C22×C8×C61—21no—
65#121×C61—21no—
66#121×C61—21no—
67#121×C61—21no—
68#121×C61—21no—
69C2×C4⋊C4×C61D411yescharacteristic
70C2×C22⋊C4×C61—21no—
71C22⋊C8×C61—21no—
72C22⋊C8×C61—21no—
73C22⋊C8×C61—21no—
74C22⋊C8×C61D412yes—
75C24×C61—21no—
76C24×C61#4111yescharacteristic
77C2×D4×C61—21no—
78C2×D4×C61—21no—
79C2×D4×C61—42no—
80C2×D4×C61—41no—
81C2×D4×C61—41no—
82C2×D4×C61—82no—
83C22×C4×C61—21no—
84C22×C4×C61—42no—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×Dic61—21no—
92C22×Dic61—22no—
93C22×Dic61—21no—
94C22×Dic61—41no—
95C22×Dic61—41no—
96C22×Dic61—82no—
97C2×Dic122—21no—
98C2×Dic122—21no—
99C2×Dic122—42no—
100M16×C61—42no—
101C2×C8×C61—22no—
102C2×C8×C61—42no—
103C2×C8×C61—41no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C22⋊C4×C61—41no—
107C22⋊C4×C61—41no—
108C22⋊Dic61—41no—
109C22⋊Dic61—41no—
110#11758200—82no—
111C4⋊Dic61—22no—
112C4⋊Dic61—21no—
113C4⋊Dic61—22no—
114C4⋊Dic61—21no—
115C4⋊Dic61—41no—
116C4⋊Dic61—41no—
117#11758196—41no—
118#11758196—41no—
119#11758194—42no—
120C2×#92418—22no—
121C2×#92418—22no—
122C2×#92418—42no—
123C23×C61—22no—
124C23×C61—22no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61—42no—
128C23×C61—81no—
129C23×C61—81no—
130C23×C61#12111yescharacteristic
131C23×C61#12111yescharacteristic
132C23×C61#12111yescharacteristic
133D4×C61—42no—
134D4×C61—42no—
135D4×C61—82no—
136D4×C61—82no—
137C2×C4×C61—22no—
138C2×C4×C61—22no—
139C2×C4×C61—22no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—81no—
145C2×C4×C61#12011yescharacteristic
146C2×C4×C61#12011yescharacteristic
147C2×C4×C61#12011yescharacteristic
148C2×C4×C61#12711yescharacteristic
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—42no—
152C2×Dic61—42no—
153C2×Dic61—82no—
154C2×Dic61—82no—
155C2×Dic61—81no—
156C2×Dic61—82no—
157Dic122—42no—
158Dic122—42no—
159C8×C61—42no—
160C8×C61—81no—
161#92418—42no—
162#92418—42no—
163#9994—611nomaximal, Sylow
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—44no—
172C22×C61—81no—
173C22×C61—81no—
174C22×C61—81no—
175C22×C61—162no—
176C22×C61#39211yescharacteristic
177C22×C61#47212yes—
178C22×C61#47312yes—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—81no—
183C4×C61#45812yes—
184Dic61—81no—
185Dic61—82no—
186Dic61—81no—
187#3193—611no—
188C2×#484—611no—
189C2×#480—611no—
190#1821—611no—
191#1472—611no—
192#1471D6111yescharacteristic
193#1298—611no—
194C2×C61—22no—
195C2×C61—22no—
196C2×C61—41no—
197C2×C61—161no—
198C2×C61#201811yescharacteristic
199C2×C61#202011yescharacteristic
200C2×C61#202211yescharacteristic
201C2×C22⋊Q8—611no—
202C2×#121C2×D6111yescharacteristic
203C22×C4⋊C4—611no—
204#484—1222no—
205#484—1221no—
206#484—1222no—
207#480—1222no—
208#480—1222no—
209#480—1221no—
210#468—1222no—
211#450—1221no—
212#448—1221no—
213C2×#107—1221no—
214C2×#107—1221no—
215C2×#103—1221no—
216C2×#102—1221no—
217#407—611no—
218#407C2×D6111yescharacteristic
219C2×C22⋊C8—611no—
220C2×C22⋊C8C2×D6111yescharacteristic
221#343—1222no—
222#324—1222no—
223#323—22no—
224C61#999411yescharacteristic, complemented, Sylow
225C22×Q8—611no—
226C22×D4—21no—
227C22×D4D24411yescharacteristic
228C23×C4—1221no—
229C23×C4C22×D6111yescharacteristic
230C2×SD16—1222no—
231C2×M16—21no—
232C2×M16—612no—
233C22×C8—21no—
234C22×C8—1221no—
235C22⋊Q8—1221no—
236C22⋊Q8—1221no—
237C22⋊Q8—1221no—
238C22⋊Q8—1221no—
239#121—21no—
240#121—21no—
241#121—21no—
242#121—21no—
243C2×C4⋊C4—612no—
244C2×C4⋊C4—611no—
245C2×C4⋊C4—1221no—
246C2×C4⋊C4—1221no—
247C2×C4⋊C4—1221no—
248C2×C4⋊C4—1221no—
249C2×C4⋊C4—1221no—
250C2×C4⋊C4—1221no—
251C2×C4⋊C4D24411yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—1221no—
254#107—1224no—
255#107—1222no—
256#107—1224no—
257#103—1222no—
258#103—1224no—
259#102—1224no—
260#102—1222no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—612no—
265C22⋊C8—1221no—
266C22⋊C8—1221no—
267C22⋊C8—1221no—
268C22⋊C8C22⋊D6112yes—
269C24—21no—
270C24#1175822211yescharacteristic
271C2×Q8—1221no—
272C2×Q8—1221no—
273C2×Q8—2442no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—42no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—82no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—1221no—
283C22×C4—1222no—
284C22×C4—1221no—
285C22×C4—4882no—
286C22×C4—2441no—
287C22×C4—2441no—
288C22×C4C2×D24411yescharacteristic
289C22×C4D4×D6111yescharacteristic
290C22×C4D4×D6111yescharacteristic
291C22×C4D4×D6111yescharacteristic
292C22×C4D4×D6111yescharacteristic
293C22×C4C2×C22⋊D6111yescharacteristic
294SD16—4882no—
295M16—42no—
296M16—2442no—
297C2×C8—22no—
298C2×C8—42no—
299C2×C8—41no—
300C2×C8—1222no—
301C2×C8—1222no—
302C2×C8—2442no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—1222no—
306C4⋊C4—1221no—
307C4⋊C4—1222no—
308C4⋊C4—1221no—
309C4⋊C4—2441no—
310C4⋊C4—2441no—
311C4⋊C4—2441no—
312C4⋊C4—2441no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—2441no—
316C22⋊C4—2441no—
317C23—22no—
318C23—22no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23—81no—
323C23—81no—
324C23#1451975011yescharacteristic
325C23#1451975011yescharacteristic
326C23#1451975111yescharacteristic
327Q8—2442no—
328Q8—2442no—
329D4—42no—
330D4—42no—
331D4—82no—
332D4—82no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—81no—
341C2×C4—4882no—
342C2×C4—4882no—
343C2×C4—4882no—
344C2×C4—4881no—
345C2×C4—2442no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4—2442no—
349C2×C4#1451974511yescharacteristic
350C2×C4#1451974711yescharacteristic
351C2×C4#1451974711yescharacteristic
352C2×C4#1451977311yescharacteristic
353C8—42no—
354C8—81no—
355C8—2442no—
356C8—2442no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—44no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—162no—
369C22#1683967111yescharacteristic
370C22#1684003112yes—
371C22#1684003712yes—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—81no—
376C4—4882no—
377C4—4881no—
378C4—4881no—
379C4#1683995312yes—
380C2—22no—
381C2—22no—
382C2—41no—
383C2—161no—
384C2#2144692711yescharacteristic
385C2#2144694011yescharacteristic
386C2#2144694911yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1471C2
socleC22×C61#392
prime radicalC2×#121×C61C22
hypercenter#1471D61
hypocenterC61#9994
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1471

Complements

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

NH ≅ G/Ntype
2. C2×#16840098C2semidirect
4. #21445897C2semidirect
5. #21444184C2semidirect
6. #21443707C2semidirect
224. C61#9994semidirect

Maximal subgroups

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

subgroupindex
#214566252
C2×#168400982
C2×#168400732
#214458972
#214441842
#214437072
C61×#14712
#999461