Finite Group Databasebeta

← #29652805

Subgroups — #29652805

Order 15616.

970 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 970 conjugacy classes of subgroups — too many to draw. Only its 167 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#21459622C211yesmaximal, characteristic
2#21455949C212yesmaximal, complemented
3#21444809C214yesmaximal, complemented
4#21444775C214yesmaximal, complemented
5#21444330C212yesmaximal, complemented
6#21444160C211yesmaximal, characteristic
7C61×#2398C211yesmaximal, characteristic
8#16840471—24no—
9#16840469—22no—
10D4×Dic122—22no—
11#16840393—21no—
12#16840388C2214yes—
13#16840319C2212yes—
14C2×#14519756—22no—
15#16840237—21no—
16#16840092C2214yes—
17#16840047—22no—
18#16839972—22no—
19#16839918C2214yes—
20#16839867—24no—
21#16839819C2212yes—
22#16839498C2218yes—
23#16839486C2214yes—
24C2×C4×Q8×C61C2211yescharacteristic
25C2×C4×D4×C61C2211yescharacteristic
26C61×#400—21no—
27C61×#397C2212yes—
28C61×#394—21no—
29C61×#393C2212yes—
30C4×C22⋊C4×C61—22no—
31C61×#376C2211yescharacteristic
32C22×Dic122—22no—
33C2×#11758202—24no—
34#14519777D412yes—
35#14519777D412yes—
36#14519776—24no—
37#14519769—22no—
38#14519768—24no—
39#14519756—24no—
40#14519756—24no—
41#14519756—22no—
42#14519756—24no—
43#14519756C2314yes—
44#14519754—28no—
45#14519754—28no—
46C4×Dic122—22no—
47C4×Dic122D414yes—
48D4×Dic61—24no—
49D4×Dic61—24no—
50#14519736—22no—
51C2×C4⋊Dic61—22no—
52C2×#11758196—24no—
53C2×C22⋊Dic61—24no—
54#14519708—24no—
55C4⋊#92418—22no—
56#14519701—24no—
57#14519700—24no—
58C22⋊#92418C2314yes—
59#14519688—22no—
60C22×Q8×C61C2311yescharacteristic
61C22×D4×C61D411yescharacteristic
62C23×C4×C61D412yes—
63C4×Q8×C61—22no—
64C4×Q8×C61—22no—
65C4×D4×C61—22no—
66C4×D4×C61—22no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61D411yescharacteristic
69C2×C4⋊C4×C61C2311yescharacteristic
70C2×C22⋊C4×C61—22no—
71C2×C22⋊C4×C61—22no—
72C2×C22⋊C4×C61C2312yes—
73C2×C42×C61—21no—
74C2×C42×C61—21no—
75C2×C42×C61C2311yescharacteristic
76#95×C61—21no—
77#95×C61—22no—
78#95×C61—22no—
79#95×C61C2312yes—
80C24×C61C2×D412yes—
81C2×Q8×C61—22no—
82C2×Q8×C61—42no—
83C2×D4×C61—24no—
84C2×D4×C61—22no—
85C22×C4×C61—21no—
86C22×C4×C61—22no—
87C22×C4×C61—22no—
88C22×C4×C61—22no—
89C22×C4×C61—21no—
90C22×C4×C61—44no—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D412yes—
94C22×C4×C61C2411yescharacteristic
95C22×Dic61—24no—
96C22×Dic61—24no—
97C2×Dic122—22no—
98C2×Dic122—24no—
99C2×Dic122—84no—
100C2×Dic122C2×D414yes—
101C4⋊C4×C61—22no—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—42no—
104C22⋊C4×C61—24no—
105C22⋊C4×C61—44no—
106C22⋊C4×C61—44no—
107C42×C61—42no—
108C42×C61—41no—
109C42×C61—41no—
110C42×C61C2×D412yes—
111C42×C61C2×D412yes—
112C22⋊Dic61—28no—
113C22⋊Dic61—24no—
114#11758202—48no—
115C4⋊Dic61—24no—
116C4⋊Dic61—22no—
117C4⋊Dic61—22no—
118C4⋊Dic61—22no—
119C4⋊Dic61C2×D414yes—
120#11758196—28no—
121#11758196—24no—
122#11758196—24no—
123C4×Dic61—24no—
124C2×#92418—24no—
125C23×C61—22no—
126C23×C61—22no—
127C23×C61—42no—
128C23×C61—42no—
129C23×C61#13714yes—
130C23×C61C22×D411yescharacteristic
131Q8×C61—24no—
132Q8×C61—42no—
133D4×C61—28no—
134C2×C4×C61—22no—
135C2×C4×C61—22no—
136C2×C4×C61—22no—
137C2×C4×C61—22no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—44no—
142C2×C4×C61—42no—
143C2×C4×C61—42no—
144C2×C4×C61—42no—
145C2×C4×C61—41no—
146C2×C4×C61—42no—
147C2×C4×C61—42no—
148C2×C4×C61—82no—
149C2×C4×C61#12012yes—
150C2×C4×C61#12011yescharacteristic
151C2×C4×C61#12011yescharacteristic
152C2×C4×C61C22×D412yes—
153C2×Dic61—24no—
154C2×Dic61—24no—
155C2×Dic61—84no—
156C2×Dic61—84no—
157Dic122—44no—
158Dic122—44no—
159Dic122—84no—
160#92418—44no—
161#18181—611nomaximal, Sylow
162C22×C61—21no—
163C22×C61—21no—
164C22×C61—24no—
165C22×C61—44no—
166C22×C61—42no—
167C22×C61—42no—
168C22×C61—82no—
169C22×C61C2×#12011yescharacteristic
170C22×C61C2×#13712yes—
171C4×C61—22no—
172C4×C61—21no—
173C4×C61—22no—
174C4×C61—42no—
175C4×C61—42no—
176C4×C61—82no—
177C4×C61—81no—
178C4×C61#45512yes—
179Dic61—44no—
180Dic61—44no—
181#3501—611no—
182#3113—612no—
183#2398D6111yescharacteristic
184#1608—614no—
185#1602—614no—
186#1506—612no—
187#1466—611no—
188C2×C61—22no—
189C2×C61—44no—
190C2×C61#300411yescharacteristic
191C2×C61C2×#45511yescharacteristic
192C2×C61#302511yescharacteristic
193#554—1224no—
194#554—1222no—
195D4×Q8—1222no—
196#544—1221no—
197#543—614no—
198#533—612no—
199C2×C22⋊Q8—1222no—
200#518—1221no—
201C2×C4×Q8C2×D6111yescharacteristic
202C2×C4×D4C2×D6111yescharacteristic
203#483—614no—
204#475—1222no—
205#462—1222no—
206#451—614no—
207#441—1224no—
208#432—612no—
209#400—21no—
210#397C2×D6112yes—
211#394—21no—
212#393C2×D6112yes—
213C4×C22⋊C4—22no—
214#376C2×D6111yescharacteristic
215#328—618no—
216#323—614no—
217C61#1818111yescharacteristic, complemented, Sylow
218C22×Q8—1222no—
219C22×Q8C22×D6111yescharacteristic
220C22×D4C22⋊D6111yescharacteristic
221C23×C4C22⋊D6112yes—
222C2×Q16—1224no—
223C4⋊Q8—612no—
224C4⋊Q8—612no—
225C4⋊Q8—1224no—
226#125—1224no—
227#125—1222no—
228C22⋊Q8—614no—
229C22⋊Q8—1228no—
230C22⋊Q8—1228no—
231C22⋊Q8—1224no—
232C22⋊Q8—1224no—
233C22⋊Q8—1222no—
234C22⋊Q8—1224no—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×Q8—614no—
238C4×Q8—1222no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—1224no—
242C4×D4—1224no—
243#117—1222no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—1224no—
246C2×C4⋊C4—1222no—
247C2×C4⋊C4C22⋊D6111yescharacteristic
248C2×C4⋊C4C22×D6111yescharacteristic
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—1224no—
252C2×C22⋊C4C22×D6112yes—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42C22×D6111yescharacteristic
256#106—1224no—
257C4⋊C8—1222no—
258#103—1224no—
259#103—1224no—
260C22⋊C8—614no—
261#97—1222no—
262#95—21no—
263#95—22no—
264#95—22no—
265#95C22×D6112yes—
266C24C2×C22⋊D6112yes—
267C2×Q8—22no—
268C2×Q8—42no—
269C2×Q8—614no—
270C2×Q8—1224no—
271C2×Q8—1222no—
272C2×Q8—4884no—
273C2×D4—24no—
274C2×D4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—44no—
281C22×C4—1224no—
282C22×C4—1224no—
283C22×C4C2×C22⋊D6111yescharacteristic
284C22×C4C2×C22⋊D6111yescharacteristic
285C22×C4C2×C22⋊D6112yes—
286C22×C4C23×D6111yescharacteristic
287Q16—2448no—
288C2×C8—1224no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—42no—
292C4⋊C4—614no—
293C4⋊C4—1228no—
294C4⋊C4—1224no—
295C4⋊C4—1224no—
296C4⋊C4—1224no—
297C4⋊C4—1222no—
298C4⋊C4—1222no—
299C4⋊C4—1222no—
300C22⋊C4—24no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C22⋊C4—1228no—
304C22⋊C4—1224no—
305C42—42no—
306C42—41no—
307C42—41no—
308C42—1224no—
309C42D4×D6112yes—
310C42D4×D6112yes—
311C23—22no—
312C23—22no—
313C23—42no—
314C23—42no—
315C23#1451980714yes—
316C23C22×C22⋊D6111yescharacteristic
317Q8—24no—
318Q8—42no—
319Q8—4884no—
320Q8—2444no—
321Q8—2444no—
322D4—28no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—82no—
338C2×C4—1224no—
339C2×C4—1224no—
340C2×C4—4884no—
341C2×C4—4884no—
342C2×C4#1451974511yescharacteristic
343C2×C4#1451974512yes—
344C2×C4#1451974511yescharacteristic
345C2×C4C2×D4×D6112yes—
346C8—2444no—
347C22—21no—
348C22—21no—
349C22—24no—
350C22—44no—
351C22—42no—
352C22—42no—
353C22—82no—
354C22C2×#1451974511yescharacteristic
355C22C2×#1451980712yes—
356C4—21no—
357C4—22no—
358C4—22no—
359C4—42no—
360C4—42no—
361C4—82no—
362C4—81no—
363C4—2444no—
364C4—2444no—
365C4#1683993912yes—
366C2—22no—
367C2—44no—
368C2#2145505211yescharacteristic
369C2C2×#1683993911yescharacteristic
370C2#2145521011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2398C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenter#2398D61
hypocenterC61#18181
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2398

Complements

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

NH ≅ G/Ntype
2. #21455949C2semidirect
3. #21444809C2semidirect
4. #21444775C2semidirect
5. #21444330C2semidirect
217. C61#18181semidirect

Maximal subgroups

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

subgroupindex
#214596222
#214559492
#214448092
#214447752
#214443302
#214441602
C61×#23982
#1818161