Finite Group Databasebeta

← #29652798

Subgroups — #29652798

Order 15616.

1106 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1106 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, complemented
2#21455918C212yesmaximal, complemented
3#21444843C214yesmaximal, complemented
4#21444741C214yesmaximal, complemented
5#21444330C212yesmaximal, complemented
6#21444160C211yesmaximal, characteristic, complemented
7C61×#2397C211yesmaximal, characteristic
8#16840471—24no—
9#16840469—22no—
10D4×Dic122—22no—
11#16840393—21no—
12#16840388C2214yes—
13#16840319C2212yes—
14C2×#14519756—22no—
15#16840237—21no—
16#16840097C2214yes—
17#16840042—22no—
18#16839967—22no—
19#16839913C2214yes—
20#16839862—24no—
21#16839819C2212yes—
22#16839498C2218yes—
23#16839486C2214yes—
24C2×C4×D4×C61—22no—
25C2×C4×D4×C61C2211yescharacteristic
26C2×C4×D4×C61C2211yescharacteristic
27C61×#399—21no—
28C61×#399C2212yes—
29C61×#392—21no—
30C61×#392C2212yes—
31C61×#376C2211yescharacteristic
32C22×Dic122—22no—
33C2×#11758200—24no—
34#14519777D412yescomplemented
35#14519777D412yescomplemented
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×Dic122D414yescomplemented
48D4×Dic61—24no—
49D4×Dic61—24no—
50#14519736—22no—
51C2×C4⋊Dic61—22no—
52C2×#11758196—24no—
53C2×C22⋊Dic61—24no—
54#14519710—24no—
55C4⋊#92418—22no—
56#14519700—24no—
57#14519698—24no—
58C22⋊#92418C2314yes—
59#14519688—22no—
60C22×D4×C61—22no—
61C22×D4×C61D411yescharacteristic
62C22×D4×C61C2311yescharacteristic
63C23×C4×C61—21no—
64C23×C4×C61D412yes—
65C4×D4×C61—22no—
66C4×D4×C61—22no—
67C4×D4×C61—22no—
68C4×D4×C61—22no—
69C4×D4×C61—48no—
70C2×C4⋊C4×C61—22no—
71C2×C4⋊C4×C61D411yescharacteristic
72C2×C4⋊C4×C61C2311yescharacteristic
73C2×C22⋊C4×C61—22no—
74C2×C22⋊C4×C61—21no—
75C2×C22⋊C4×C61—22no—
76C2×C22⋊C4×C61C2312yes—
77C2×C42×C61—21no—
78C2×C42×C61C2311yescharacteristic
79#95×C61—21no—
80#95×C61C2312yes—
81C24×C61—21no—
82C24×C61C2×D412yes—
83C2×D4×C61—24no—
84C2×D4×C61—22no—
85C2×D4×C61—22no—
86C2×D4×C61—44no—
87C2×D4×C61—42no—
88C2×D4×C61—44no—
89C2×D4×C61—44no—
90C22×C4×C61—22no—
91C22×C4×C61—22no—
92C22×C4×C61—22no—
93C22×C4×C61—21no—
94C22×C4×C61—44no—
95C22×C4×C61—44no—
96C22×C4×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D412yes—
101C22×C4×C61C2411yescharacteristic
102C22×Dic61—24no—
103C22×Dic61—24no—
104C2×Dic122—22no—
105C2×Dic122—24no—
106C2×Dic122—84no—
107C2×Dic122C2×D414yes—
108C4⋊C4×C61—22no—
109C4⋊C4×C61—22no—
110C4⋊C4×C61—44no—
111C22⋊C4×C61—24no—
112C22⋊C4×C61—44no—
113C22⋊C4×C61—44no—
114C22⋊C4×C61—42no—
115C42×C61—24no—
116C42×C61C2×D412yes—
117C42×C61C2×D412yes—
118C22⋊Dic61—28no—
119C22⋊Dic61—24no—
120#11758200—48no—
121C4⋊Dic61—24no—
122C4⋊Dic61—22no—
123C4⋊Dic61—22no—
124C4⋊Dic61—22no—
125C4⋊Dic61C2×D414yes—
126#11758196—28no—
127#11758196—24no—
128#11758196—24no—
129C4×Dic61—24no—
130C2×#92418—24no—
131C23×C61—22no—
132C23×C61—22no—
133C23×C61—22no—
134C23×C61—42no—
135C23×C61—42no—
136C23×C61—42no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61—42no—
140C23×C61#13714yes—
141C23×C61C22×D411yescharacteristic
142D4×C61—28no—
143D4×C61—24no—
144D4×C61—42no—
145D4×C61—416no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—24no—
149C2×C4×C61—22no—
150C2×C4×C61—41no—
151C2×C4×C61—42no—
152C2×C4×C61—42no—
153C2×C4×C61—44no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—41no—
157C2×C4×C61—82no—
158C2×C4×C61—82no—
159C2×C4×C61—82no—
160C2×C4×C61#12012yes—
161C2×C4×C61#12011yescharacteristic
162C2×C4×C61#12011yescharacteristic
163C2×C4×C61C22×D412yes—
164C2×Dic61—24no—
165C2×Dic61—24no—
166C2×Dic61—84no—
167C2×Dic61—84no—
168Dic122—44no—
169Dic122—44no—
170Dic122—84no—
171#92418—44no—
172#18180—611nomaximal, Sylow
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—24no—
176C22×C61—44no—
177C22×C61—44no—
178C22×C61—42no—
179C22×C61—42no—
180C22×C61—42no—
181C22×C61—82no—
182C22×C61—82no—
183C22×C61—81no—
184C22×C61—81no—
185C22×C61C2×#12011yescharacteristic
186C22×C61C2×#13712yes—
187C4×C61—22no—
188C4×C61—21no—
189C4×C61—22no—
190C4×C61—44no—
191C4×C61—42no—
192C4×C61—81no—
193C4×C61#45312yes—
194Dic61—44no—
195Dic61—44no—
196#3501—611no—
197#3110—612no—
198#2397D6111yescharacteristic
199#1614—614no—
200#1596—614no—
201#1506—612no—
202#1466—611no—
203C2×C61—22no—
204C2×C61—44no—
205C2×C61—82no—
206C2×C61#300411yescharacteristic
207C2×C61C2×#45311yescharacteristic
208C2×C61#302511yescharacteristic
209#554—1224no—
210#554—1222no—
211D4×Q8—1222no—
212#544—1221no—
213#543—614no—
214#533—612no—
215C2×C22⋊Q8—1222no—
216#518—1221no—
217C2×C4×D4—22no—
218C2×C4×D4C2×D6111yescharacteristic
219C2×C4×D4C2×D6111yescharacteristic
220#484—614no—
221#474—1222no—
222#461—1222no—
223#450—614no—
224#440—1224no—
225#432—612no—
226#399—21no—
227#399C2×D6112yes—
228#392—21no—
229#392C2×D6112yes—
230#376C2×D6111yescharacteristic
231#328—618no—
232#323—614no—
233C61#1818011yescharacteristic, complemented, Sylow
234C22×Q8—1222no—
235C22×D4—22no—
236C22×D4C22⋊D6111yescharacteristic
237C22×D4C22×D6111yescharacteristic
238C23×C4—21no—
239C23×C4C22⋊D6112yes—
240C2×SD16—1224no—
241C4⋊Q8—612no—
242C4⋊Q8—612no—
243C4⋊Q8—1224no—
244#125—1224no—
245#125—1222no—
246C22⋊Q8—614no—
247C22⋊Q8—1228no—
248C22⋊Q8—1228no—
249C22⋊Q8—1224no—
250C22⋊Q8—1224no—
251C22⋊Q8—1222no—
252C22⋊Q8—1224no—
253C4×Q8—614no—
254C4×Q8—1222no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—48no—
260C4×D4—1224no—
261C4×D4—1224no—
262#117—1222no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—1224no—
265C2×C4⋊C4—1222no—
266C2×C4⋊C4C22⋊D6111yescharacteristic
267C2×C4⋊C4C22×D6111yescharacteristic
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—1224no—
272C2×C22⋊C4C22×D6112yes—
273C2×C42—21no—
274C2×C42C22×D6111yescharacteristic
275#107—1224no—
276C4⋊C8—1222no—
277#103—1224no—
278#102—1224no—
279C22⋊C8—614no—
280#97—1222no—
281#95—21no—
282#95C22×D6112yes—
283C24—21no—
284C24C2×C22⋊D6112yes—
285C2×Q8—614no—
286C2×Q8—1224no—
287C2×Q8—1222no—
288C2×Q8—4884no—
289C2×D4—24no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—44no—
293C2×D4—42no—
294C2×D4—44no—
295C2×D4—44no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—1224no—
305C22×C4—1224no—
306C22×C4C2×C22⋊D6111yescharacteristic
307C22×C4C2×C22⋊D6111yescharacteristic
308C22×C4C2×C22⋊D6112yes—
309C22×C4C23×D6111yescharacteristic
310SD16—2448no—
311C2×C8—1224no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—44no—
315C4⋊C4—614no—
316C4⋊C4—1228no—
317C4⋊C4—1224no—
318C4⋊C4—1224no—
319C4⋊C4—1224no—
320C4⋊C4—1222no—
321C4⋊C4—1222no—
322C4⋊C4—1222no—
323C22⋊C4—24no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—42no—
327C22⋊C4—1228no—
328C22⋊C4—1224no—
329C42—24no—
330C42—1224no—
331C42D4×D6112yes—
332C42D4×D6112yes—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—41no—
340C23—41no—
341C23—42no—
342C23#1451980714yes—
343C23C22×C22⋊D6111yescharacteristic
344Q8—4884no—
345Q8—2444no—
346Q8—2444no—
347D4—28no—
348D4—24no—
349D4—42no—
350D4—416no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—1224no—
366C2×C4—1224no—
367C2×C4—4884no—
368C2×C4—4884no—
369C2×C4#1451974512yes—
370C2×C4#1451974511yescharacteristic
371C2×C4#1451974511yescharacteristic
372C2×C4C2×D4×D6112yes—
373C8—2444no—
374C22—21no—
375C22—21no—
376C22—24no—
377C22—44no—
378C22—44no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—82no—
383C22—82no—
384C22—81no—
385C22—81no—
386C22C2×#1451974511yescharacteristic
387C22C2×#1451980712yes—
388C4—21no—
389C4—22no—
390C4—22no—
391C4—44no—
392C4—42no—
393C4—81no—
394C4—2444no—
395C4—2444no—
396C4#1683992912yes—
397C2—22no—
398C2—44no—
399C2—82no—
400C2#2145505211yescharacteristic
401C2C2×#1683992911yescharacteristic
402C2#2145521011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#2397C2
socleC22×C61C2×#120
hypercenter#2397D61
hypocenterC61#18180
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2397

Complements

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

NH ≅ G/Ntype
1. #21459622C2semidirect
2. #21455918C2semidirect
3. #21444843C2semidirect
4. #21444741C2semidirect
5. #21444330C2semidirect
6. #21444160C2semidirect
34. #14519777D4semidirect
35. #14519777D4semidirect
47. C4×Dic122D4semidirect
233. C61#18180semidirect

Maximal subgroups

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

subgroupindex
#214596222
#214559182
#214448432
#214447412
#214443302
#214441602
C61×#23972
#1818061