Finite Group Databasebeta

← #29656299

Subgroups — #29656299

Order 15616.

1040 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456279C211yesmaximal, characteristic, complemented
2#21455634C212yesmaximal, complemented
3#21444964C212yesmaximal, complemented
4#21444959C212yesmaximal, complemented
5#21444946C214yesmaximal, complemented
6#21444554C211yesmaximal, characteristic, complemented
7#21444258C212yesmaximal, complemented
8C61×#3483C211yesmaximal, characteristic
9C2×#14519807—22no—
10C2×#14519777C2211yescharacteristic
11#16840157—22no—
12#16840141C2212yes—
13#16840134C2212yes—
14#16840057C2214yes—
15#16840052C2214yes—
16#16840047—22no—
17#16840042—22no—
18#16839805—22no—
19#16839786C2212yes—
20#16839763—22no—
21C4×#11758194C2211yescharacteristic
22#16839511C2212yes—
23#16839506C2212yes—
24Q8⋊#92418C2214yes—
25D4⋊#92418C2214yes—
26Q82×C61C2211yescharacteristic
27C61×#555—22no—
28C61×#555—22no—
29C61×#555C2211yescharacteristic
30C61×#550—22no—
31C61×#550C2212yes—
32C61×#532—22no—
33C61×#532—22no—
34C61×#532C2211yescharacteristic
35C4×#41×C61—22no—
36C4×#41×C61C2212yes—
37C22×Dic122—21no—
38#14519807—48no—
39C2×#11758202—24no—
40C2×#11758200—24no—
41C2×#11758194—21no—
42C2×#11758194—22no—
43#14519777—22no—
44#14519777—21no—
45#14519777C2312yes—
46C2×C4⋊Dic61—22no—
47#14519710—24no—
48#14519708—24no—
49C4⋊#92418—22no—
50C4⋊#92418C2314yes—
51#14519701—24no—
52#14519698—24no—
53#14519688—21no—
54C4×#92418C2312yes—
55C2×#41×C61—22no—
56C2×#41×C61—22no—
57C2×#41×C61D412yes—
58C2×#41×C61D412yes—
59C4⋊Q8×C61—22no—
60C4⋊Q8×C61—22no—
61C4⋊Q8×C61C2311yescharacteristic
62#127×C61—22no—
63#127×C61—22no—
64#127×C61C2311yescharacteristic
65#124×C61—22no—
66#124×C61—24no—
67#124×C61—22no—
68#124×C61D412yes—
69#121×C61—24no—
70#121×C61—24no—
71#121×C61—22no—
72#121×C61—24no—
73#121×C61—22no—
74#121×C61—22no—
75C4×Q8×C61—22no—
76C4×Q8×C61D412yes—
77C4×Q8×C61C2312yes—
78C4×D4×C61—24no—
79C4×D4×C61—24no—
80C4×D4×C61—22no—
81C4×D4×C61—22no—
82C4×D4×C61—22no—
83C4×D4×C61—22no—
84C4×D4×C61D412yes—
85C4×D4×C61C2312yes—
86#117×C61—24no—
87#117×C61—22no—
88#117×C61—22no—
89#117×C61D412yes—
90C2×C42×C61—22no—
91C2×C42×C61—22no—
92C2×C42×C61C2311yescharacteristic
93#41×C61—24no—
94#41×C61—42no—
95#41×C61—44no—
96#41×C61—44no—
97#41×C61—44no—
98#41×C61—84no—
99C2×Q8×C61—22no—
100C2×Q8×C61C2×D412yes—
101C2×Q8×C61C2×D412yes—
102C2×D4×C61—24no—
103C2×D4×C61—22no—
104C2×D4×C61—22no—
105C2×D4×C61—24no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61C2×D412yes—
109C2×D4×C61C2×D412yes—
110C22×C4×C61—24no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D412yes—
117C22×Dic61—41no—
118C2×Dic122—22no—
119C2×Dic122—44no—
120C2×Dic122—41no—
121C4⋊C4×C61—24no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61C2×D412yes—
126C4⋊C4×C61C2×D412yes—
127C22⋊C4×C61—24no—
128C22⋊C4×C61—24no—
129C22⋊C4×C61—22no—
130C22⋊C4×C61—24no—
131C22⋊C4×C61—22no—
132C22⋊C4×C61—22no—
133C42×C61—22no—
134C42×C61—22no—
135C42×C61—22no—
136C42×C61—22no—
137C42×C61—22no—
138C42×C61C2×D412yes—
139C42×C61C2×D411yescharacteristic
140C42×C61C2411yescharacteristic
141#11758202—48no—
142#11758200—48no—
143C4⋊Dic61—24no—
144C4⋊Dic61—42no—
145#11758194—24no—
146#11758194—44no—
147C2×#92418—22no—
148C2×#92418—24no—
149C23×C61—22no—
150C23×C61—22no—
151C23×C61#12011yescharacteristic
152Q8×C61—22no—
153Q8×C61—42no—
154Q8×C61—82no—
155Q8×C61#13714yescomplemented
156D4×C61—22no—
157D4×C61—22no—
158D4×C61—44no—
159D4×C61—42no—
160D4×C61—48no—
161D4×C61—42no—
162D4×C61—42no—
163D4×C61—82no—
164D4×C61—82no—
165D4×C61—84no—
166C2×C4×C61—22no—
167C2×C4×C61—22no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—44no—
174C2×C4×C61—48no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61—44no—
178C2×C4×C61—44no—
179C2×C4×C61#12012yes—
180C2×C4×C61#12011yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C22×D412yes—
183C2×Dic61—42no—
184C2×Dic61—82no—
185Dic122—44no—
186Dic122—82no—
187#92418—24no—
188#92418—44no—
189#18549—611nomaximal, Sylow
190C22×C61—22no—
191C22×C61—21no—
192C22×C61—42no—
193C22×C61—44no—
194C22×C61—42no—
195C22×C61—42no—
196C22×C61—42no—
197C22×C61C2×#12011yescharacteristic
198C4×C61—22no—
199C4×C61—24no—
200C4×C61—21no—
201C4×C61—42no—
202C4×C61—42no—
203C4×C61—42no—
204C4×C61#45312yes—
205C4×C61C2×#13712yes—
206C4×C61#57512yes—
207Dic61—82no—
208#3483D6111yescharacteristic
209#3152—611no—
210#3079—612no—
211#1634—612no—
212#1633—612no—
213#1631—614no—
214#1560—611no—
215#1489—612no—
216C2×C61—41no—
217C2×C61—82no—
218C2×C61—82no—
219C2×C61#300712yes—
220C2×C61C2×#45311yescharacteristic
221C2×#137—1222no—
222Q82C2×D6111yescharacteristic
223#555—22no—
224#555—22no—
225#555C2×D6111yescharacteristic
226#550—22no—
227#550C2×D6112yes—
228#532—22no—
229#532—22no—
230#532C2×D6111yescharacteristic
231C2×C4⋊Q8—611no—
232C4×#41—22no—
233C4×#41C2×D6112yes—
234#497—1222no—
235#494—612no—
236#492—612no—
237#477—614no—
238#476—614no—
239#475—1222no—
240#474—1222no—
241#428—1222no—
242#423—612no—
243#417—1222no—
244C4×M16—611no—
245#333—612no—
246#331—612no—
247Q8⋊C8—614no—
248D4⋊C8—614no—
249C61#1854911yescharacteristic, complemented, Sylow
250C2×#41—22no—
251C2×#41—22no—
252C2×#41C22⋊D6112yes—
253C2×#41C22⋊D6112yes—
254C22×Q8—1221no—
255#137—2448no—
256C2×Q16—1224no—
257C2×SD16—1224no—
258C2×M16—1221no—
259C2×M16—1222no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—612no—
263C4⋊Q8—1222no—
264C4⋊Q8—1221no—
265C4⋊Q8C22×D6111yescharacteristic
266#127—22no—
267#127—22no—
268#127C22×D6111yescharacteristic
269#124—22no—
270#124—24no—
271#124—22no—
272#124C22⋊D6112yes—
273#121—24no—
274#121—24no—
275#121—22no—
276#121—24no—
277#121—22no—
278#121—22no—
279C4×Q8—22no—
280C4×Q8C22⋊D6112yes—
281C4×Q8C22×D6112yes—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4C22⋊D6112yes—
289C4×D4C22×D6112yes—
290#117—24no—
291#117—22no—
292#117—22no—
293#117C22⋊D6112yes—
294C2×C4⋊C4—1222no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42C22×D6111yescharacteristic
298#107—1224no—
299#106—1224no—
300C4⋊C8—614no—
301C4⋊C8—1222no—
302#103—1224no—
303#102—1224no—
304#97—1221no—
305C4×C8—612no—
306#41—24no—
307#41—42no—
308#41—44no—
309#41—44no—
310#41—44no—
311#41—84no—
312C2×Q8—22no—
313C2×Q8—1222no—
314C2×Q8—2444no—
315C2×Q8—2441no—
316C2×Q8C2×C22⋊D6112yes—
317C2×Q8C2×C22⋊D6112yes—
318C2×D4—24no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—24no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4C2×C22⋊D6112yes—
325C2×D4C2×C22⋊D6112yes—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—2441no—
332C22×C4C2×C22⋊D6112yes—
333C22×C4C2×C22⋊D6111yescharacteristic
334Q16—2448no—
335SD16—2448no—
336M16—1224no—
337M16—2444no—
338C2×C8—1222no—
339C2×C8—1224no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—1224no—
345C4⋊C4—2442no—
346C4⋊C4C2×C22⋊D6112yes—
347C4⋊C4C2×C22⋊D6112yes—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—24no—
353C22⋊C4—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42C2×C22⋊D6111yescharacteristic
360C42C2×C22⋊D6112yes—
361C42C23×D6111yescharacteristic
362C23—22no—
363C23—22no—
364C23#1451974611yescharacteristic
365Q8—22no—
366Q8—42no—
367Q8—82no—
368Q8—4882no—
369Q8—2444no—
370Q8#1451980714yescomplemented
371D4—22no—
372D4—22no—
373D4—44no—
374D4—42no—
375D4—48no—
376D4—42no—
377D4—42no—
378D4—82no—
379D4—82no—
380D4—84no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—4882no—
395C2×C4—2442no—
396C2×C4#1451974612yes—
397C2×C4#1451974611yescharacteristic
398C2×C4C22×C22⋊D6111yescharacteristic
399C2×C4C22×C22⋊D6112yes—
400C8—1224no—
401C8—2444no—
402C22—21no—
403C22—22no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—44no—
408C22—42no—
409C22C2×#1451974611yescharacteristic
410C4—21no—
411C4—22no—
412C4—24no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—4882no—
417C4#1683993012yes—
418C4C2×#1451980712yes—
419C4#1684056012yes—
420C2—41no—
421C2—82no—
422C2—82no—
423C2#2145507912yes—
424C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3483C2
socleC22×C61C2×#120
hypercenter#3483D61
hypocenterC61#18549
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3483

Complements

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

NH ≅ G/Ntype
1. #21456279C2semidirect
2. #21455634C2semidirect
3. #21444964C2semidirect
4. #21444959C2semidirect
5. #21444946C2semidirect
6. #21444554C2semidirect
7. #21444258C2semidirect
155. Q8×C61#137semidirect
249. C61#18549semidirect
370. Q8#14519807semidirect

Maximal subgroups

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

subgroupindex
#214562792
#214556342
#214449642
#214449592
#214449462
#214445542
#214442582
C61×#34832
#1854961