Finite Group Databasebeta

← #29665332

Subgroups — #29665332

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#21455937C211yesmaximal, characteristic, complemented
2#21455846C211yesmaximal, characteristic, complemented
3#21454899C211yesmaximal, characteristic, complemented
4#21445274C212yesmaximal, complemented
5#21445232C212yesmaximal, complemented
6#21445176C212yesmaximal, complemented
7#21445164C212yesmaximal, complemented
8#21444599C211yesmaximal, characteristic, complemented
9#21444524C211yesmaximal, characteristic, complemented
10#21444481C211yesmaximal, characteristic, complemented
11C61×#3207C211yesmaximal, characteristic
12C2×C4×Dic122C2211yescharacteristic
13#16840078—21no—
14#16840073—21no—
15#16840054C2212yes—
16#16840044C2212yes—
17#16839959C2212yes—
18#16839959C2212yes—
19#16839900—21no—
20#16839900—21no—
21C8×Dic122C2212yes—
22C8×Dic122C2212yes—
23#16839886—22no—
24#16839862—22no—
25C4×#11758200—22no—
26#16839819C2211yescharacteristic
27#16839815C2211yescharacteristic
28#16839815C2211yescharacteristic
29#16839812—22no—
30#16839515C2212yes—
31#16839513C2212yes—
32#16839511C2212yes—
33#16839508C2212yes—
34#16839506C2212yes—
35#16839506C2212yes—
36C61×#532C2211yescharacteristic
37C61×#532C2211yescharacteristic
38C61×#493C2211yescharacteristic
39C61×#492C2211yescharacteristic
40C61×#487C2211yescharacteristic
41C61×#486C2211yescharacteristic
42C61×#463—22no—
43C61×#449—22no—
44C61×#431C2211yescharacteristic
45C22×Dic122D411yescharacteristic
46C2×#11758200—22no—
47C2×#11758200—22no—
48C4×Dic122—21no—
49C4×Dic122—21no—
50C4×Dic122D412yes—
51C4×Dic122C2312yes—
52C2×C4⋊Dic61D411yescharacteristic
53C2×#11758196—21no—
54C2×C4×Dic61—21no—
55#14519710—22no—
56#14519708—22no—
57C4⋊#92418C2312yes—
58C4⋊#92418C2312yes—
59C4⋊#92418C2312yes—
60#14519706—22no—
61#14519706—22no—
62#14519700—22no—
63#14519700—22no—
64#14519698—22no—
65#14519698—22no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69#14519688—21no—
70C4×#92418—21no—
71C4×#92418—21no—
72C8×Dic61—22no—
73C8×Dic61—22no—
74C2×#41×C61—21no—
75C2×#41×C61—21no—
76C2×SD16×C61—22no—
77C2×D8×C61—22no—
78C4⋊Q8×C61C2311yescharacteristic
79C4⋊Q8×C61C2311yescharacteristic
80#127×C61C2311yescharacteristic
81#127×C61C2311yescharacteristic
82#124×C61—21no—
83#124×C61—21no—
84#121×C61—21no—
85#121×C61—21no—
86#121×C61—21no—
87#121×C61—21no—
88C4×D4×C61—21no—
89C4×D4×C61—21no—
90C4×D4×C61—21no—
91C4×D4×C61—21no—
92C2×C42×C61C2311yescharacteristic
93C4⋊C8×C61D412yes—
94#103×C61—22no—
95#102×C61—22no—
96C22⋊C8×C61D412yes—
97C4×C8×C61C2311yescharacteristic
98C4×C8×C61C2311yescharacteristic
99#41×C61—82no—
100#41×C61—82no—
101C2×Q8×C61—21no—
102C2×Q8×C61—21no—
103C2×D4×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—41no—
110C2×D4×C61—41no—
111C22×C4×C61—41no—
112C22×C4×C61—41no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C22×Dic61—21no—
117C22×Dic61—21no—
118C2×Dic122—21no—
119C2×Dic122—24no—
120C2×Dic122C2×D412yes—
121SD16×C61—82no—
122D8×C61—82no—
123C2×C8×C61C2×D412yes—
124C2×C8×C61C2×D412yes—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C22⋊C4×C61—41no—
130C22⋊C4×C61—41no—
131C22⋊C4×C61—41no—
132C22⋊C4×C61—41no—
133C42×C61C2×D411yescharacteristic
134C42×C61C2×D411yescharacteristic
135C42×C61#4111yescharacteristic
136C42×C61C2411yescharacteristic
137#11758200—44no—
138#11758200—44no—
139C4⋊Dic61—21no—
140C4⋊Dic61C2×D412yes—
141#11758196—22no—
142#11758196—22no—
143C4×Dic61—22no—
144C4×Dic61—22no—
145C2×#92418—22no—
146C2×#92418—22no—
147C2×#92418—22no—
148C23×C61—41no—
149C23×C61—41no—
150C23×C61#12111yescharacteristic
151Q8×C61—42no—
152Q8×C61—81no—
153D4×C61—42no—
154D4×C61—42no—
155D4×C61—42no—
156D4×C61—42no—
157D4×C61—81no—
158D4×C61—81no—
159D4×C61—82no—
160D4×C61—82no—
161C2×C4×C61—22no—
162C2×C4×C61—22no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—82no—
167C2×C4×C61—82no—
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61C22×D411yescharacteristic
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C2×#4111yescharacteristic
174C2×Dic61—24no—
175C2×Dic61—22no—
176C2×Dic61—22no—
177C2×Dic61—81no—
178Dic122—24no—
179Dic122—22no—
180Dic122#13514yes—
181C8×C61—22no—
182C8×C61—22no—
183#92418—42no—
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×C61—81no—
194C22×C61—81no—
195C22×C61—81no—
196C22×C61C2×#12111yescharacteristic
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—81no—
202C4×C61—81no—
203C4×C61C2×#13512yes—
204C4×C61#57712yes—
205Dic61—24no—
206Dic61—42no—
207#3207D6111yescharacteristic
208#3112—611no—
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—161no—
221C2×C61#304311yescharacteristic
222C2×C61#305011yescharacteristic
223C2×C61#306811yescharacteristic
224#532C2×D6111yescharacteristic
225#532C2×D6111yescharacteristic
226C2×C4×Q8—611no—
227#493C2×D6111yescharacteristic
228#492C2×D6111yescharacteristic
229#487C2×D6111yescharacteristic
230#486C2×D6111yescharacteristic
231#481—1221no—
232#480—1221no—
233#477—612no—
234#475—612no—
235#463—22no—
236#460—612no—
237#460—612no—
238#449—22no—
239#448—1221no—
240#448—1221no—
241C8×Q8—612no—
242C8×Q8—612no—
243C8×Q8—1222no—
244#440—1222no—
245C4×SD16—1222no—
246#432—611no—
247#431—611no—
248#431—611no—
249#431—1222no—
250#431C2×D6111yescharacteristic
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—21no—
260C22×Q8—611no—
261C2×SD16—22no—
262C2×SD16—1222no—
263C2×SD16—1222no—
264C2×D8—22no—
265C4⋊Q8C22×D6111yescharacteristic
266C4⋊Q8C22×D6111yescharacteristic
267#127C22×D6111yescharacteristic
268#127C22×D6111yescharacteristic
269#124—21no—
270#124—21no—
271#121—21no—
272#121—21no—
273#121—21no—
274#121—21no—
275C4×Q8—612no—
276C4×Q8—612no—
277C4×Q8—1221no—
278C4×Q8—1221no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4—21no—
283C2×C4⋊C4—611no—
284C2×C4⋊C4—1221no—
285C2×C42—1221no—
286C2×C42C22×D6111yescharacteristic
287#107—1222no—
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—1221no—
302C22⋊C8—1221no—
303C22⋊C8—1221no—
304C22⋊C8C22⋊D6112yes—
305#97—1221no—
306C4×C8—1222no—
307C4×C8—1222no—
308C4×C8—1221no—
309C4×C8—1221no—
310C4×C8C22×D6111yescharacteristic
311C4×C8C22×D6111yescharacteristic
312#41—82no—
313#41—82no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—612no—
317C2×Q8—1224no—
318C2×Q8—1221no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—41no—
326C2×D4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—1221no—
330C22×C4—1221no—
331C22×C4D4×D6111yescharacteristic
332C22×C4#1175822411yescharacteristic
333C22×C4C2×C22⋊D6111yescharacteristic
334SD16—82no—
335SD16—2444no—
336SD16—2444no—
337D8—82no—
338C2×C8—1222no—
339C2×C8—1222no—
340C2×C8—1222no—
341C2×C8C2×C22⋊D6112yes—
342C2×C8C2×C22⋊D6112yes—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—612no—
348C4⋊C4—1222no—
349C4⋊C4—1221no—
350C4⋊C4—1222no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C42—1222no—
356C42—1222no—
357C42D4×D6111yescharacteristic
358C42#1175822411yescharacteristic
359C42C2×C22⋊D6111yescharacteristic
360C42C23×D6111yescharacteristic
361C23—41no—
362C23—41no—
363C23#1451974811yescharacteristic
364Q8—42no—
365Q8—81no—
366Q8—614no—
367Q8—1224no—
368Q8—1222no—
369D4—42no—
370D4—42no—
371D4—42no—
372D4—42no—
373D4—81no—
374D4—81no—
375D4—82no—
376D4—82no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—1224no—
385C2×C4—1222no—
386C2×C4—1222no—
387C2×C4—4881no—
388C2×C4#1451974811yescharacteristic
389C2×C4#1451974811yescharacteristic
390C2×C4#1451974811yescharacteristic
391C2×C4C2×D4×D6111yescharacteristic
392C2×C4C22×C22⋊D6111yescharacteristic
393C2×C4C2×#1175822411yescharacteristic
394C8—22no—
395C8—22no—
396C8—2442no—
397C8—2442no—
398C8—2442no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22C2×#1451974811yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—81no—
414C4—81no—
415C4—1224no—
416C4—2442no—
417C4#135×D6112yes—
418C4#1684057312yes—
419C2—41no—
420C2—161no—
421C2—161no—
422C2#2145533611yescharacteristic
423C2#2145539611yescharacteristic
424C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3207C2
socleC22×C61C2×#121
hypercenter#3207D61
hypocenterC61#19401
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3207

Complements

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

NH ≅ G/Ntype
1. #21455937C2semidirect
2. #21455846C2semidirect
3. #21454899C2semidirect
4. #21445274C2semidirect
5. #21445232C2semidirect
6. #21445176C2semidirect
7. #21445164C2semidirect
8. #21444599C2semidirect
9. #21444524C2semidirect
10. #21444481C2semidirect
257. C61#19401semidirect

Maximal subgroups

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

subgroupindex
#214559372
#214558462
#214548992
#214452742
#214452322
#214451762
#214451642
#214445992
#214445242
#214444812
C61×#32072
#1940161