Finite Group Databasebeta

← #29576742

Subgroups — #29576742

Order 15616.

632 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448545C211yesmaximal, characteristic, complemented
2#21446652C211yesmaximal, characteristic, complemented
3#21444487C211yesmaximal, characteristic, complemented
4#21444379C211yesmaximal, characteristic, complemented
5#21443678C211yesmaximal, characteristic, complemented
6#21443647C211yesmaximal, characteristic, complemented
7C61×#3205C211yesmaximal, characteristic
8C2×D4×Dic61—21no—
9#16839815C2211yescharacteristic
10#16839786C2211yescharacteristic
11C2×#14519704—21no—
12C2×#14519704—21no—
13#16839665—21no—
14#16839642—22no—
15#16839642—22no—
16#16839628C2211yescharacteristic
17#16839608C2211yescharacteristic
18#16839506—22no—
19#16839496—22no—
20C61×#532C2211yescharacteristic
21C2×#127×C61C2211yescharacteristic
22C61×#493C412yescomplemented
23C61×#486C412yescomplemented
24C61×#459—22no—
25C61×#447—22no—
26C61×#431C2211yescharacteristic
27C23×Dic61—22no—
28C2×#11758194—21no—
29D4×Dic61—42no—
30D4×Dic61—42no—
31C2×C4⋊Dic61—21no—
32C2×C22⋊Dic61—22no—
33C2×C22⋊Dic61—22no—
34C2×C22⋊Dic61—22no—
35C2×C4×Dic61D411yescharacteristic
36C2×C4×Dic61D411yescharacteristic
37C4⋊#92418—21no—
38C4⋊#92418—21no—
39C4⋊#92418—21no—
40#14519704—42no—
41#14519704—42no—
42C22⋊#92418—21no—
43C4×#92418—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684D412yes—
48C2×#41×C61—21no—
49C22×D4×C61—21no—
50C22×D4×C61—21no—
51C22×D4×C61D412yes—
52C2×D8×C61—24no—
53C4⋊Q8×C61C2×C411yescharacteristic, complemented
54#127×C61—21no—
55#127×C61—21no—
56#127×C61C2×C412yes—
57#127×C61C2×C411yescharacteristic, complemented
58#127×C61D412yes—
59#124×C61—21no—
60#121×C61—21no—
61#121×C61—21no—
62C4×D4×C61—21no—
63C4×D4×C61—21no—
64C2×C42×C61C2311yescharacteristic
65C4⋊C8×C61—21no—
66#102×C61—24no—
67C22⋊C8×C61—21no—
68C4×C8×C61C2×C412yescomplemented
69C24×C61—22no—
70#41×C61—41no—
71#41×C61—41no—
72#41×C61—41no—
73#41×C61—41no—
74C2×Q8×C61—21no—
75C2×D4×C61—21no—
76C2×D4×C61—22no—
77C2×D4×C61—21no—
78C2×D4×C61—22no—
79C2×D4×C61—21no—
80C2×D4×C61—22no—
81C2×D4×C61—22no—
82C2×D4×C61—22no—
83C2×D4×C61—41no—
84C2×D4×C61—41no—
85C2×D4×C61—41no—
86C2×D4×C61—41no—
87C2×D4×C61—41no—
88C2×D4×C61—82no—
89C2×D4×C61—82no—
90C2×D4×C61C22⋊C414yes—
91C22×C4×C61—41no—
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×Dic61—21no—
96C22×Dic61—22no—
97C22×Dic61—21no—
98C22×Dic61—44no—
99C22×Dic61—82no—
100C22×Dic61—82no—
101D8×C61—84no—
102C2×C8×C61—22no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—21no—
105C22⋊C4×C61—41no—
106C22⋊C4×C61—41no—
107C42×C61C22×C411yescharacteristic
108C42×C61C2×D411yescharacteristic
109C42×C61#4111yescharacteristic
110C42×C61#4111yescharacteristic
111C22⋊Dic61—44no—
112C22⋊Dic61—82no—
113C22⋊Dic61—82no—
114C4⋊Dic61—41no—
115C4⋊Dic61—41no—
116C4×Dic61—21no—
117C4×Dic61—21no—
118C4×Dic61—21no—
119C4×Dic61D812yescomplemented
120#11758194—41no—
121#11758194—41no—
122C2×#92418—41no—
123C2×#92418—41no—
124C23×C61—24no—
125C23×C61—42no—
126C23×C61—42no—
127C23×C61—42no—
128C23×C61—42no—
129C23×C61—42no—
130C23×C61—42no—
131C23×C61—41no—
132C23×C61#12011yescharacteristic
133Q8×C61—41no—
134Q8×C61—41no—
135D4×C61—44no—
136D4×C61—42no—
137D4×C61—42no—
138D4×C61—41no—
139D4×C61—42no—
140D4×C61—41no—
141D4×C61—42no—
142D4×C61—42no—
143D4×C61—82no—
144D4×C61—82no—
145D4×C61—82no—
146D4×C61—81no—
147D4×C61—81no—
148D4×C61—81no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—81no—
153C2×C4×C61—81no—
154C2×C4×C61C2×C22⋊C411yescharacteristic
155C2×C4×C61C4×D411yescharacteristic
156C2×C4×C61C4×D411yescharacteristic
157C2×C4×C61#12111yescharacteristic
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61#12311yescharacteristic
160C2×C4×C61C2×D811yescharacteristic
161C2×C4×C61#13511yescharacteristic
162C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
164C2×Dic61—22no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—42no—
170C2×Dic61—42no—
171C2×Dic61—42no—
172C2×Dic61—41no—
173C2×Dic61—88no—
174C8×C61—42no—
175#92418—81no—
176#92418—81no—
177#9299—611nomaximal, Sylow
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—44no—
181C22×C61—44no—
182C22×C61—44no—
183C22×C61—82no—
184C22×C61—82no—
185C22×C61—82no—
186C22×C61—82no—
187C22×C61—81no—
188C22×C61—81no—
189C22×C61—81no—
190C22×C61#38611yescharacteristic
191C22×C61#44711yescharacteristic
192C22×C61#44911yescharacteristic
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—81no—
198C4×C61C4×D811yescharacteristic
199C4×C61#44211yescharacteristic
200C4×C61#45911yescharacteristic
201C4×C61#46311yescharacteristic
202Dic61—42no—
203Dic61—81no—
204Dic61—82no—
205#3205D6111yescharacteristic
206#2328—611no—
207#1963—611no—
208#1545—611no—
209#1517—611no—
210#1286—611no—
211#1274—611no—
212C2×C61—21no—
213C2×C61—21no—
214C2×C61—84no—
215C2×C61—161no—
216C2×C61#187911yescharacteristic
217C2×C61#190411yescharacteristic
218C2×C61#191111yescharacteristic
219#532C2×D6111yescharacteristic
220C2×#127C2×D6111yescharacteristic
221C2×C4×D4—1221no—
222#493Dic6112yescomplemented
223#486Dic6112yescomplemented
224#459—22no—
225#447—22no—
226#431—611no—
227#431C2×D6111yescharacteristic
228#423—611no—
229C2×#104—1221no—
230C2×#104—1221no—
231#390—1221no—
232#386—1222no—
233#386—1222no—
234#382—611no—
235#376—611no—
236#331—1222no—
237#327—1222no—
238C61#929911yescharacteristic, complemented, Sylow
239C2×#41—21no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C22⋊D6112yes—
243C23×C4—1222no—
244C2×D8—24no—
245C2×M16—1221no—
246C4⋊Q8C2×Dic6111yescharacteristic, complemented
247#127—21no—
248#127—21no—
249#127C2×Dic6112yes—
250#127C2×Dic6111yescharacteristic, complemented
251#127C22⋊D6112yes—
252#124—21no—
253#121—21no—
254#121—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—2442no—
258C4×D4—2442no—
259C2×C4⋊C4—1221no—
260C2×C22⋊C4—1222no—
261C2×C22⋊C4—1222no—
262C2×C22⋊C4—1222no—
263C2×C42—611no—
264C2×C42—611no—
265C2×C42C22×D6111yescharacteristic
266C4⋊C8—21no—
267C4⋊C8—1221no—
268C4⋊C8—1221no—
269C4⋊C8—1221no—
270#104—2442no—
271#104—2442no—
272#102—24no—
273C22⋊C8—21no—
274C22⋊C8—1221no—
275C4×C8—1221no—
276C4×C8C2×Dic6112yescomplemented
277#95—612no—
278#95—1221no—
279#95—1221no—
280#95—1221no—
281C24—22no—
282#41—41no—
283#41—41no—
284#41—41no—
285#41—41no—
286C2×Q8—21no—
287C2×D4—21no—
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—82no—
301C2×D4—82no—
302C2×D4C22⋊Dic6114yes—
303C22×C4—41no—
304C22×C4—1221no—
305C22×C4—1222no—
306C22×C4—1221no—
307C22×C4—4882no—
308C22×C4—4882no—
309C22×C4—2444no—
310C22×C4D4×D6111yescharacteristic
311C22×C4D4×D6111yescharacteristic
312C22×C4C2×C22⋊D6111yescharacteristic
313D8—84no—
314M16—2441no—
315M16—2441no—
316C2×C8—22no—
317C2×C8—2441no—
318C2×C8—2441no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—2441no—
322C4⋊C4—2441no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—4882no—
326C22⋊C4—4882no—
327C22⋊C4—2444no—
328C42—612no—
329C42—1221no—
330C42—1221no—
331C42—1221no—
332C42#1175822411yescharacteristic
333C42#1175822411yescharacteristic
334C42C22×Dic6111yescharacteristic
335C42C2×C22⋊D6111yescharacteristic
336C23—24no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—41no—
344C23#1451974511yescharacteristic
345Q8—41no—
346Q8—41no—
347D4—44no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—42no—
352D4—41no—
353D4—42no—
354D4—41no—
355D4—82no—
356D4—82no—
357D4—82no—
358D4—81no—
359D4—81no—
360D4—81no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—1222no—
367C2×C4—4888no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4—2442no—
371C2×C4—2442no—
372C2×C4—2442no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4C2×C22⋊Dic6111yescharacteristic
377C2×C4D4×Dic6111yescharacteristic
378C2×C4D4×Dic6111yescharacteristic
379C2×C4#1451974911yescharacteristic
380C2×C4#1451974911yescharacteristic
381C2×C4#1451975811yescharacteristic
382C2×C4D8×D6111yescharacteristic
383C2×C4#1451979611yescharacteristic
384C2×C4#1451980111yescharacteristic
385C2×C4#1451980111yescharacteristic
386C8—42no—
387C8—4881no—
388C8—4881no—
389C22—21no—
390C22—21no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22—82no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22#1683964211yescharacteristic
402C22#1683989411yescharacteristic
403C22#1683990411yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—81no—
409C4—4882no—
410C4—4881no—
411C4—2442no—
412C4D8×Dic6111yescharacteristic
413C4#1683987411yescharacteristic
414C4#1683995511yescharacteristic
415C4#1683997511yescharacteristic
416C2—21no—
417C2—21no—
418C2—84no—
419C2—161no—
420C2#2144616911yescharacteristic
421C2#2144630811yescharacteristic
422C2#2144635211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
Fitting, prime radicalC61×#3205C2
socleC22×C61#386
cosocleC2×C42×C61C23
hypercenter#3205D61
hypocenterC61#9299
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #21448545C2semidirect
2. #21446652C2semidirect
3. #21444487C2semidirect
4. #21444379C2semidirect
5. #21443678C2semidirect
6. #21443647C2semidirect
22. C61×#493C4semidirect
23. C61×#486C4semidirect
53. C4⋊Q8×C61C2×C4semidirect
57. #127×C61C2×C4semidirect
68. C4×C8×C61C2×C4semidirect
119. C4×Dic61D8semidirect
222. #493Dic61semidirect
223. #486Dic61semidirect
238. C61#9299semidirect
246. C4⋊Q8C2×Dic61semidirect
250. #127C2×Dic61semidirect
276. C4×C8C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214485452
#214466522
#214444872
#214443792
#214436782
#214436472
C61×#32052
#929961