Finite Group Databasebeta

← #29665339

Subgroups — #29665339

Order 15616.

632 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#21456748C211yesmaximal, characteristic
2#21455850C211yesmaximal, characteristic, complemented
3#21454900C211yesmaximal, characteristic, complemented
4#21445279C212yesmaximal, complemented
5#21445237C212yesmaximal, complemented
6#21445177C212yesmaximal, complemented
7#21445165C212yesmaximal, complemented
8#21444600C211yesmaximal, characteristic, complemented
9#21444525C211yesmaximal, characteristic, complemented
10#21444481C211yesmaximal, characteristic, complemented
11C61×#3112C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13#16840138C2211yescharacteristic
14#16840134C2211yescharacteristic
15#16840114C2211yescharacteristic
16#16840110C2211yescharacteristic
17#16840080—21no—
18#16840047C2212yes—
19#16839977—22no—
20#16839962C2212yes—
21#16839905—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#16839511C2212yes—
34#16839507C2212yes—
35#16839506C2212yes—
36#16839505C2212yes—
37C61×#532C2211yescharacteristic
38C2×C4×Q8×C61C2211yescharacteristic
39C61×#480—21no—
40C61×#477C2212yes—
41C61×#460C2212yes—
42C61×#448—21no—
43C61×#440—22no—
44C61×#432C2211yescharacteristic
45C2×#11758222—21no—
46C2×#11758200—22no—
47C2×#11758201—22no—
48C2×#11758199—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
75C2×#41×C61—21no—
76C22×Q8×C61D411yescharacteristic
77C2×SD16×C61—22no—
78C4⋊Q8×C61C2311yescharacteristic
79#127×C61C2311yescharacteristic
80#124×C61—21no—
81#121×C61—21no—
82#121×C61—21no—
83C4×Q8×C61—21no—
84C4×Q8×C61—21no—
85C4×Q8×C61D412yes—
86C4×Q8×C61C2312yes—
87C4×D4×C61—21no—
88C4×D4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61D411yescharacteristic
91C2×C42×C61—21no—
92C2×C42×C61C2311yescharacteristic
93#107×C61—22no—
94C4⋊C8×C61C2312yes—
95#103×C61—22no—
96#102×C61—22no—
97C22⋊C8×C61—21no—
98#97×C61—21no—
99#41×C61—82no—
100C2×Q8×C61—24no—
101C2×Q8×C61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61C2×D412yes—
104C2×D4×C61—21no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—41no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—41no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C2×C22⋊D61—41no—
115#11758222—82no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×C4×D61—41no—
120C2×Dic122—21no—
121SD16×C61—44no—
122C2×C8×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61C2×D412yes—
129C22⋊C4×C61—41no—
130C22⋊C4×C61—41no—
131C42×C61—22no—
132C42×C61—22no—
133C42×C61C2×D411yescharacteristic
134C42×C61C2×D411yescharacteristic
135C42×C61#4111yescharacteristic
136C42×C61C2411yescharacteristic
137#11758201—44no—
138#11758200—82no—
139#11758199—82no—
140#11758198—41no—
141#11758198—41no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C2×#92418—22no—
145C2×#92418—22no—
146C2×#92418C2×D412yes—
147C2×#92418C2×D412yes—
148C23×C61—41no—
149C23×C61#12111yescharacteristic
150C22×D61—41no—
151Q8×C61—24no—
152Q8×C61—22no—
153Q8×C61—81no—
154Q8×C61#13514yes—
155D4×C61—42no—
156D4×C61—42no—
157D4×C61—81no—
158D4×C61—82no—
159C2×C4×C61—24no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—22no—
163C2×C4×C61—22no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—82no—
167C2×C4×C61—81no—
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61C22×D411yescharacteristic
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C2×#4111yescharacteristic
174C22⋊D61—82no—
175C2×Dic61—41no—
176D244—42no—
177D244—42no—
178D244—81no—
179C4×D61—82no—
180Dic122—42no—
181C8×C61—42no—
182#92418—22no—
183#92418—22no—
184#92418—42no—
185#92418—42no—
186#19401—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61C2×#12111yescharacteristic
194C2×D61—81no—
195C2×D61—81no—
196C2×D61—81no—
197C4×C61—21no—
198C4×C61—24no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—42no—
203C4×C61—81no—
204C4×C61C2×#13512yes—
205C4×C61#57712yes—
206Dic61—81no—
207#3207—611no—
208#3112D6111yescharacteristic, complemented
209#3103—611no—
210#2982—611no—
211#1684—612no—
212#1678—612no—
213#1668—612no—
214#1666—612no—
215#1572—611no—
216#1553—611no—
217#1543—611no—
218C2×C61—41no—
219C2×C61—161no—
220C2×C61#304311yescharacteristic
221C2×C61#305011yescharacteristic
222C2×C61#306811yescharacteristic
223D61—161no—
224#532—611no—
225#532C2×D6111yescharacteristic
226C2×C4×Q8C2×D6111yescharacteristic
227#493—611no—
228#492—611no—
229#487—611no—
230#486—611no—
231#481—1221no—
232#480—21no—
233#477C2×D6112yes—
234#475—612no—
235#463—1222no—
236#460—612no—
237#460C2×D6112yes—
238#449—1222no—
239#448—21no—
240#448—1221no—
241C8×Q8—612no—
242C8×Q8—612no—
243C8×Q8—1222no—
244#440—22no—
245C4×SD16—1222no—
246#432C2×D6111yescharacteristic
247#431—611no—
248#431—611no—
249#431—611no—
250#431—1222no—
251#335—612no—
252#334—612no—
253#333—612no—
254#332—612no—
255#331—612no—
256#331—612no—
257C61#1940111yescharacteristic, complemented, Sylow
258C2×#41—21no—
259C2×#41—1221no—
260C22×Q8C22⋊D6111yescharacteristic
261C2×SD16—22no—
262C2×SD16—1222no—
263C2×SD16—1222no—
264C2×D8—1222no—
265C4⋊Q8—611no—
266C4⋊Q8C22×D6111yescharacteristic
267#127—611no—
268#127C22×D6111yescharacteristic
269#124—21no—
270#124—1221no—
271#121—21no—
272#121—21no—
273#121—1221no—
274#121—1221no—
275C4×Q8—21no—
276C4×Q8—21no—
277C4×Q8C22⋊D6112yes—
278C4×Q8C22×D6112yes—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—1221no—
282C4×D4—1221no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4C22⋊D6111yescharacteristic
285C2×C42—21no—
286C2×C42C22×D6111yescharacteristic
287#107—22no—
288#106—1222no—
289C4⋊C8—612no—
290C4⋊C8—612no—
291C4⋊C8—612no—
292C4⋊C8—1222no—
293C4⋊C8—1222no—
294C4⋊C8C22×D6112yes—
295#103—22no—
296#103—1222no—
297#103—1222no—
298#102—22no—
299#102—1222no—
300#102—1222no—
301C22⋊C8—21no—
302C22⋊C8—612no—
303C22⋊C8—1221no—
304C22⋊C8—1221no—
305#97—21no—
306C4×C8—611no—
307C4×C8—611no—
308C4×C8—1222no—
309C4×C8—1222no—
310C4×C8—1221no—
311C4×C8—1221no—
312#41—82no—
313#41—4882no—
314C2×Q8—24no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—1221no—
318C2×Q8C2×C22⋊D6112yes—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—41no—
323C2×D4—1221no—
324C2×D4—1221no—
325C2×D4—1221no—
326C2×D4—2441no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—2441no—
331C22×C4D4×D6111yescharacteristic
332C22×C4#1175822411yescharacteristic
333C22×C4C2×C22⋊D6111yescharacteristic
334SD16—44no—
335SD16—4882no—
336SD16—2444no—
337D8—4882no—
338C2×C8—22no—
339C2×C8—612no—
340C2×C8—612no—
341C2×C8—1222no—
342C2×C8—1222no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—1221no—
349C4⋊C4—1221no—
350C4⋊C4C2×C22⋊D6112yes—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—2441no—
354C22⋊C4—2441no—
355C42—22no—
356C42—22no—
357C42D4×D6111yescharacteristic
358C42#1175822411yescharacteristic
359C42C2×C22⋊D6111yescharacteristic
360C42C23×D6111yescharacteristic
361C23—41no—
362C23—2441no—
363C23#1451974911yescharacteristic
364Q8—24no—
365Q8—22no—
366Q8—81no—
367Q8—2442no—
368Q8#1451979714yes—
369D4—42no—
370D4—42no—
371D4—81no—
372D4—82no—
373D4—4881no—
374D4—4882no—
375D4—2442no—
376D4—2442no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—82no—
385C2×C4—81no—
386C2×C4—4882no—
387C2×C4—2441no—
388C2×C4#1451974911yescharacteristic
389C2×C4#1451974911yescharacteristic
390C2×C4#1451974911yescharacteristic
391C2×C4C2×D4×D6111yescharacteristic
392C2×C4C22×C22⋊D6111yescharacteristic
393C2×C4C2×#1175822411yescharacteristic
394C8—42no—
395C8—1222no—
396C8—1222no—
397C8—2442no—
398C8—2442no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—4881no—
406C22—4881no—
407C22—4881no—
408C22C2×#1451974911yescharacteristic
409C4—24no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—42no—
415C4—81no—
416C4—4881no—
417C4C2×#1451979712yes—
418C4#1684057112yes—
419C2—41no—
420C2—161no—
421C2—9761no—
422C2#2145534211yescharacteristic
423C2#2145539711yescharacteristic
424C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3112C2
socleC22×C61C2×#121
prime radical#21456748C2
hypercenter#3112D61
hypocenterC61#19401
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3112

Complements

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

NH ≅ G/Ntype
2. #21455850C2semidirect
3. #21454900C2semidirect
4. #21445279C2semidirect
5. #21445237C2semidirect
6. #21445177C2semidirect
7. #21445165C2semidirect
8. #21444600C2semidirect
9. #21444525C2semidirect
10. #21444481C2semidirect
11. C61×#3112C2semidirect
208. #3112D61semidirect
257. C61#19401semidirect

Maximal subgroups

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

subgroupindex
#214567482
#214558502
#214549002
#214452792
#214452372
#214451772
#214451652
#214446002
#214445252
#214444812
C61×#31122
#1940161