Finite Group Databasebeta

← #29665328

Subgroups — #29665328

Order 15616.

610 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456771C211yesmaximal, characteristic
2#21455850C211yesmaximal, characteristic
3#21454900C211yesmaximal, characteristic, complemented
4#21445265C212yesmaximal, complemented
5#21445251C212yesmaximal, complemented
6#21445187C212yesmaximal, complemented
7#21445169C212yesmaximal, complemented
8#21444600C211yesmaximal, characteristic, complemented
9#21444525C211yesmaximal, characteristic
10#21444484C211yesmaximal, characteristic, complemented
11C61×#3113C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13#16840141C2211yescharacteristic
14#16840136C2211yescharacteristic
15#16840114C2211yescharacteristic
16#16840110C2211yescharacteristic
17#16840080—21no—
18#16840047C2212yes—
19#16839984—22no—
20#16839962C2212yes—
21#16839922—22no—
22#16839901—21no—
23Q8×#92418—22no—
24Q8×#92418C2212yes—
25Q8×#92418C2212yes—
26C4×#11758201—22no—
27#16839815—22no—
28#16839815C2211yescharacteristic
29#16839815C2211yescharacteristic
30#16839815C2211yescharacteristic
31#16839514C2212yes—
32#16839512C2212yes—
33#16839508C2212yes—
34#16839508C2212yes—
35#16839507C2212yes—
36#16839505C2212yes—
37C61×#533C2211yescharacteristic
38C2×C4×Q8×C61C2211yescharacteristic
39C61×#483—21no—
40C61×#475C2212yes—
41C61×#462C2212yes—
42C61×#451—21no—
43C61×#441—22no—
44C61×#432C2211yescharacteristic
45C2×#11758222—21no—
46C2×#11758202—22no—
47C2×#11758201—22no—
48C2×#11758201—22no—
49#14519777C2311yescharacteristic
50#14519774C2311yescharacteristic
51#14519765—21no—
52#14519751—21no—
53#14519751—21no—
54C4×D244—21no—
55C4×D244—21no—
56#14519708—22no—
57C4⋊#92418—22no—
58C4⋊#92418—22no—
59C4⋊#92418D412yes—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519701—22no—
63#14519700—22no—
64#14519697—22no—
65#14519697—22no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C22⋊#92418D412yes—
69C4×#92418—21no—
70C4×#92418—21no—
71C4×#92418—22no—
72C4×#92418—22no—
73C4×#92418C2311yescharacteristic
74C4×#92418C2311yescharacteristic
75C22×Q8×C61D411yescharacteristic
76C2×Q16×C61—22no—
77C4⋊Q8×C61C2311yescharacteristic
78C4⋊Q8×C61C2311yescharacteristic
79#125×C61—21no—
80C22⋊Q8×C61—21no—
81C22⋊Q8×C61—21no—
82C4×Q8×C61—21no—
83C4×Q8×C61—21no—
84C4×Q8×C61—21no—
85C4×Q8×C61—21no—
86C4×Q8×C61D412yes—
87C4×Q8×C61C2312yes—
88#117×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61D411yescharacteristic
91C2×C42×C61—21no—
92C2×C42×C61C2311yescharacteristic
93#106×C61—22no—
94C4⋊C8×C61C2312yes—
95#103×C61—22no—
96#103×C61—22no—
97C22⋊C8×C61—21no—
98#97×C61—21no—
99C2×Q8×C61—24no—
100C2×Q8×C61—21no—
101C2×Q8×C61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61C2×D412yes—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61#4111yescharacteristic
109C2×C22⋊D61—41no—
110#11758222—82no—
111C2×D244—21no—
112C2×D244—21no—
113C2×D244—21no—
114C2×C4×D61—41no—
115C2×Dic122—21no—
116Q16×C61—44no—
117C2×C8×C61—22no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61—21no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—41no—
126C4⋊C4×C61—41no—
127C4⋊C4×C61C2×D412yes—
128C22⋊C4×C61—41no—
129C42×C61—22no—
130C42×C61—22no—
131C42×C61—41no—
132C42×C61C2×D411yescharacteristic
133C42×C61C2×D411yescharacteristic
134C42×C61#4111yescharacteristic
135C42×C61C2411yescharacteristic
136#11758202—82no—
137#11758201—44no—
138#11758201—82no—
139#11758198—41no—
140#11758198—41no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C2×#92418—22no—
144C2×#92418—22no—
145C2×#92418C2×D412yes—
146C2×#92418C2×D412yes—
147C23×C61#12111yescharacteristic
148C22×D61—41no—
149Q8×C61—24no—
150Q8×C61—22no—
151Q8×C61—42no—
152Q8×C61—42no—
153Q8×C61#13514yes—
154C2×C4×C61—24no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—22no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—81no—
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61C22×D411yescharacteristic
167C2×C4×C61C22×D411yescharacteristic
168C2×C4×C61C2×#4111yescharacteristic
169C22⋊D61—82no—
170C2×Dic61—41no—
171D244—42no—
172D244—42no—
173D244—81no—
174C4×D61—82no—
175Dic122—42no—
176C8×C61—42no—
177#92418—22no—
178#92418—22no—
179#92418—42no—
180#92418—42no—
181#19400—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61C2×#12111yescharacteristic
186C2×D61—81no—
187C2×D61—81no—
188C2×D61—81no—
189C4×C61—21no—
190C4×C61—24no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—42no—
195C4×C61—81no—
196C4×C61—81no—
197C4×C61C2×#13512yes—
198C4×C61#57712yes—
199Dic61—81no—
200#3210—611no—
201#3113D6111yescharacteristic, complemented
202#3103—611no—
203#2982—611no—
204#1682—612no—
205#1680—612no—
206#1670—612no—
207#1666—612no—
208#1572—611no—
209#1553—611no—
210#1544—611no—
211C2×C61—41no—
212C2×C61#304311yescharacteristic
213C2×C61#305011yescharacteristic
214C2×C61#306911yescharacteristic
215D61—161no—
216#533C2×D6111yescharacteristic
217#532—611no—
218C2×C4×Q8C2×D6111yescharacteristic
219#494—611no—
220#492—611no—
221#487—611no—
222#486—611no—
223#483—21no—
224#481—1221no—
225#475—612no—
226#475C2×D6112yes—
227#464—1222no—
228#462C2×D6112yes—
229#460—612no—
230#452—1222no—
231#451—21no—
232#448—1221no—
233C8×Q8—612no—
234C8×Q8—612no—
235C8×Q8—1222no—
236#441—22no—
237C4×SD16—1222no—
238#432C2×D6111yescharacteristic
239#431—611no—
240#431—611no—
241#431—611no—
242#431—1222no—
243#335—612no—
244#334—612no—
245#332—612no—
246#332—612no—
247#332—612no—
248#331—612no—
249C61#1940011yescharacteristic, complemented, Sylow
250C2×#41—1221no—
251C22×Q8C22⋊D6111yescharacteristic
252C2×Q16—22no—
253C2×Q16—1222no—
254C2×SD16—1222no—
255C2×SD16—1222no—
256C4⋊Q8—611no—
257C4⋊Q8C22×D6111yescharacteristic
258C4⋊Q8C22×D6111yescharacteristic
259#127—611no—
260#125—21no—
261#124—1221no—
262C22⋊Q8—21no—
263C22⋊Q8—21no—
264#121—1221no—
265#121—1221no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8C22⋊D6112yes—
271C4×Q8C22×D6112yes—
272C4×D4—1221no—
273C4×D4—1221no—
274#117—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4C22⋊D6111yescharacteristic
277C2×C42—21no—
278C2×C42C22×D6111yescharacteristic
279#106—22no—
280#106—1222no—
281C4⋊C8—612no—
282C4⋊C8—612no—
283C4⋊C8—612no—
284C4⋊C8—1222no—
285C4⋊C8—1222no—
286C4⋊C8C22×D6112yes—
287#103—22no—
288#103—22no—
289#103—1222no—
290#103—1222no—
291#102—1222no—
292#102—1222no—
293C22⋊C8—21no—
294C22⋊C8—612no—
295C22⋊C8—1221no—
296C22⋊C8—1221no—
297#97—21no—
298C4×C8—611no—
299C4×C8—611no—
300C4×C8—1222no—
301C4×C8—1222no—
302C4×C8—1221no—
303C4×C8—1221no—
304#41—4882no—
305C2×Q8—24no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—1221no—
310C2×Q8C2×C22⋊D6112yes—
311C2×D4—1221no—
312C2×D4—1221no—
313C2×D4—1221no—
314C2×D4—2441no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—2441no—
318C22×C4D4×D6111yescharacteristic
319C22×C4#1175822411yescharacteristic
320C22×C4C2×C22⋊D6111yescharacteristic
321Q16—44no—
322Q16—4882no—
323SD16—4882no—
324SD16—2444no—
325C2×C8—22no—
326C2×C8—612no—
327C2×C8—612no—
328C2×C8—1222no—
329C2×C8—1222no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—1221no—
340C4⋊C4—1221no—
341C4⋊C4C2×C22⋊D6112yes—
342C22⋊C4—41no—
343C22⋊C4—2441no—
344C22⋊C4—2441no—
345C42—22no—
346C42—22no—
347C42—41no—
348C42D4×D6111yescharacteristic
349C42#1175822411yescharacteristic
350C42C2×C22⋊D6111yescharacteristic
351C42C23×D6111yescharacteristic
352C23—2441no—
353C23#1451974911yescharacteristic
354Q8—24no—
355Q8—22no—
356Q8—42no—
357Q8—42no—
358Q8—2442no—
359Q8#1451979714yes—
360D4—4881no—
361D4—4882no—
362D4—2442no—
363D4—2442no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—81no—
373C2×C4—4882no—
374C2×C4—2441no—
375C2×C4#1451974911yescharacteristic
376C2×C4#1451974911yescharacteristic
377C2×C4#1451974911yescharacteristic
378C2×C4C2×D4×D6111yescharacteristic
379C2×C4C22×C22⋊D6111yescharacteristic
380C2×C4C2×#1175822411yescharacteristic
381C8—42no—
382C8—1222no—
383C8—1222no—
384C8—2442no—
385C8—2442no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—4881no—
390C22—4881no—
391C22—4881no—
392C22C2×#1451974911yescharacteristic
393C4—24no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—81no—
400C4—81no—
401C4—4881no—
402C4C2×#1451979712yes—
403C4#1684057112yes—
404C2—41no—
405C2—9761no—
406C2#2145534211yescharacteristic
407C2#2145539711yescharacteristic
408C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3113C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#3113D61
hypocenterC61#19400
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
3. #21454900C2semidirect
4. #21445265C2semidirect
5. #21445251C2semidirect
6. #21445187C2semidirect
7. #21445169C2semidirect
8. #21444600C2semidirect
10. #21444484C2semidirect
11. C61×#3113C2semidirect
201. #3113D61semidirect
249. C61#19400semidirect

Maximal subgroups

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

subgroupindex
#214567712
#214558502
#214549002
#214452652
#214452512
#214451872
#214451692
#214446002
#214445252
#214444842
C61×#31132
#1940061