Finite Group Databasebeta

← #29575969

Subgroups — #29575969

Order 15616.

500 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455291C211yesmaximal, characteristic
2#21444378C211yesmaximal, characteristic
3#21444313C211yesmaximal, characteristic
4#21444214C211yesmaximal, characteristic, complemented
5#21443722C211yesmaximal, characteristic, complemented
6#21443677C211yesmaximal, characteristic, complemented
7C61×#2356C211yesmaximal, characteristic, complemented
8C2×#14519808—21no—
9#16840312C2211yescharacteristic
10#16839984—22no—
11#16839972C412yescomplemented
12#16839962C412yescomplemented
13#16839815C2211yescharacteristic
14#16839815C2211yescharacteristic
15#16839786C2211yescharacteristic
16#16839761—21no—
17#16839495—22no—
18#16839495—21no—
19#16839495—21no—
20Q8⋊#92418—22no—
21C2×C4×Q8×C61C2211yescharacteristic
22C61×#391—21no—
23C61×#387—22no—
24C61×#383—21no—
25C61×#382C2211yescharacteristic
26C4×C4⋊C4×C61—21no—
27C4×C4⋊C4×C61C2211yescharacteristic
28C61×#376—21no—
29C2×#11758222—21no—
30#14519808—42no—
31#14519808—42no—
32C2×#11758202—22no—
33C2×#11758201—22no—
34C2×#11758194—21no—
35#14519777C2×C411yescharacteristic
36#14519774C2×C411yescharacteristic
37#14519765—21no—
38#14519751—21no—
39#14519751—21no—
40C4×D244—21no—
41C4×D244—21no—
42C4⋊#92418—21no—
43C4⋊#92418—21no—
44C4⋊#92418—21no—
45C4⋊#92418C2×C412yes—
46#14519700—22no—
47#14519697—22no—
48C22⋊#92418—21no—
49C22⋊#92418—21no—
50C4×#92418—21no—
51C4×#92418—21no—
52C22×Q8×C61D411yescharacteristic
53C4×Q8×C61—21no—
54C4×Q8×C61—21no—
55C4×Q8×C61C2×C412yes—
56C4×Q8×C61D412yes—
57C2×C4⋊C4×C61—21no—
58C2×C4⋊C4×C61—21no—
59C2×C4⋊C4×C61—21no—
60C2×C4⋊C4×C61D411yescharacteristic
61C2×C42×C61—21no—
62C2×C42×C61—22no—
63C2×C42×C61—21no—
64C2×C42×C61C2311yescharacteristic
65#95×C61—22no—
66#95×C61—22no—
67#95×C61—22no—
68#95×C61—21no—
69#95×C61D411yescharacteristic
70#95×C61D411yescharacteristic
71#95×C61D412yes—
72C2×Q8×C61—21no—
73C2×Q8×C61—21no—
74C2×Q8×C61—21no—
75C2×Q8×C61—21no—
76C2×Q8×C61—21no—
77C2×Q8×C61C22⋊C412yes—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61—22no—
82C22×C4×C61—21no—
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C2×C22⋊D61—41no—
87#11758222—41no—
88#11758222—41no—
89#11758222—41no—
90#11758222—41no—
91C2×D244—21no—
92C2×D244—21no—
93C2×D244—21no—
94C2×C4×D61—41no—
95C2×Dic122—21no—
96C4⋊C4×C61—22no—
97C4⋊C4×C61—22no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—42no—
100C4⋊C4×C61—41no—
101C4⋊C4×C61—41no—
102C4⋊C4×C61C22⋊C412yes—
103C42×C61—22no—
104C42×C61—22no—
105C42×C61—44no—
106C42×C61—41no—
107C42×C61—41no—
108C42×C61C22×C411yescharacteristic
109C42×C61C2×D411yescharacteristic
110C42×C61#4111yescharacteristic
111C42×C61#4111yescharacteristic
112#11758202—44no—
113#11758201—44no—
114#11758198—41no—
115#11758198—41no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118#11758194—41no—
119#11758194—41no—
120C2×#92418—22no—
121C2×#92418—41no—
122C2×#92418—41no—
123C23×C61#12011yescharacteristic
124C22×D61—41no—
125Q8×C61—22no—
126Q8×C61—22no—
127Q8×C61—41no—
128Q8×C61—41no—
129C2×C4×C61—22no—
130C2×C4×C61—22no—
131C2×C4×C61—22no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—42no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—42no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—42no—
143C2×C4×C61—41no—
144C2×C4×C61—42no—
145C2×C4×C61C2×C22⋊C411yescharacteristic
146C2×C4×C61C4×D411yescharacteristic
147C2×C4×C61C4×D411yescharacteristic
148C2×C4×C61#12111yescharacteristic
149C2×C4×C61#12111yescharacteristic
150C2×C4×C61#12311yescharacteristic
151C2×C4×C61#13511yescharacteristic
152C2×C4×C61#13511yescharacteristic
153C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
155C22⋊D61—81no—
156C22⋊D61—81no—
157C2×Dic61—41no—
158D244—41no—
159D244—41no—
160D244—81no—
161D244—81no—
162C4×D61—81no—
163C4×D61—81no—
164Dic122—41no—
165Dic122—41no—
166#92418—42no—
167#92418—81no—
168#92418—81no—
169#9182—611nomaximal, Sylow
170C22×C61—21no—
171C22×C61—21no—
172C22×C61#38611yescharacteristic
173C22×C61#44911yescharacteristic
174C22×C61#44911yescharacteristic
175C2×D61—81no—
176C2×D61—81no—
177C2×D61—81no—
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—42no—
183C4×C61—42no—
184C4×C61—42no—
185C4×C61—81no—
186C4×C61—82no—
187C4×C61#41611yescharacteristic
188C4×C61#41811yescharacteristic
189C4×C61#48211yescharacteristic
190C4×C61#48211yescharacteristic
191Dic61—81no—
192#3037—611no—
193#2356D6111yescharacteristic, complemented
194#1517—611no—
195#1502—611no—
196#1479—611no—
197#1303—611no—
198#1286—611no—
199C2×C61—21no—
200C2×C61—21no—
201C2×C61#187511yescharacteristic
202C2×C61#190511yescharacteristic
203C2×C61#190611yescharacteristic
204D61—161no—
205C2×#137—1221no—
206#532—611no—
207C2×C4×Q8C2×D6111yescharacteristic
208#464—1222no—
209#462—612no—
210#460—612no—
211#431—611no—
212#431—611no—
213#423—611no—
214#417—1221no—
215#391—21no—
216#387—22no—
217#383—21no—
218#382C2×D6111yescharacteristic
219C4×C4⋊C4—21no—
220C4×C4⋊C4C2×D6111yescharacteristic
221#376—21no—
222#327—1221no—
223#327—1221no—
224#327—1222no—
225Q8⋊C8—1222no—
226C61#918211yescharacteristic, complemented, Sylow
227C2×#41—1221no—
228C22×Q8C22⋊D6111yescharacteristic
229#137—2442no—
230#137—2442no—
231C2×Q16—1222no—
232C2×SD16—1222no—
233C2×M16—1221no—
234C4⋊Q8—611no—
235#127—611no—
236#124—1221no—
237#121—1221no—
238#121—1221no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8C4×D6112yes—
242C4×Q8D24412yes—
243C4×D4—1221no—
244C4×D4—1221no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C22⋊D6111yescharacteristic
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—21no—
252C2×C42C22×D6111yescharacteristic
253C4⋊C8—612no—
254C4⋊C8—1221no—
255C4⋊C8—1221no—
256C4⋊C8—1221no—
257#103—1222no—
258#102—1222no—
259C22⋊C8—1221no—
260C22⋊C8—1221no—
261C4×C8—1221no—
262C4×C8—1221no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—21no—
267#95C22⋊D6111yescharacteristic
268#95C22⋊D6111yescharacteristic
269#95C22⋊D6112yes—
270#41—2441no—
271#41—2441no—
272#41—2441no—
273#41—2441no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—1221no—
280C2×Q8#1175819812yes—
281C2×D4—1221no—
282C2×D4—1221no—
283C2×D4—1221no—
284C2×D4—2441no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—2441no—
291C22×C4C2×C22⋊D6111yescharacteristic
292C22×C4C2×C22⋊D6111yescharacteristic
293C22×C4C2×C22⋊D6111yescharacteristic
294Q16—2444no—
295SD16—2444no—
296M16—2441no—
297M16—2441no—
298C2×C8—1222no—
299C2×C8—2441no—
300C2×C8—2441no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—1221no—
308C4⋊C4—1221no—
309C4⋊C4#1175819812yes—
310C22⋊C4—2441no—
311C22⋊C4—2441no—
312C42—22no—
313C42—22no—
314C42—44no—
315C42—41no—
316C42—41no—
317C42C2×C4×D6111yescharacteristic
318C42C2×D24411yescharacteristic
319C42#1175822211yescharacteristic
320C42#1175822211yescharacteristic
321C23—2441no—
322C23#1451974611yescharacteristic
323Q8—22no—
324Q8—22no—
325Q8—41no—
326Q8—41no—
327Q8—2441no—
328Q8—2441no—
329D4—4881no—
330D4—4881no—
331D4—4881no—
332D4—4881no—
333D4—2441no—
334D4—2441no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—4881no—
352C2×C4—4881no—
353C2×C4—2441no—
354C2×C4C2×#1175819811yescharacteristic
355C2×C4C4×C22⋊D6111yescharacteristic
356C2×C4C4×C22⋊D6111yescharacteristic
357C2×C4#1451975111yescharacteristic
358C2×C4#1451975111yescharacteristic
359C2×C4#1451976011yescharacteristic
360C2×C4#1451979711yescharacteristic
361C2×C4#1451979711yescharacteristic
362C2×C4#1451980211yescharacteristic
363C2×C4#1451980211yescharacteristic
364C8—4881no—
365C8—4881no—
366C8—2442no—
367C22—21no—
368C22—21no—
369C22—4881no—
370C22—4881no—
371C22—4881no—
372C22#1683964111yescharacteristic
373C22#1683990711yescharacteristic
374C22#1683990711yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—42no—
380C4—42no—
381C4—42no—
382C4—81no—
383C4—82no—
384C4—4881no—
385C4#1683975611yescharacteristic
386C4#1683976611yescharacteristic
387C4#1684008611yescharacteristic
388C4#1684008611yescharacteristic
389C2—21no—
390C2—21no—
391C2—9761no—
392C2#2144614111yescharacteristic
393C2#2144631311yescharacteristic
394C2#2144632011yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2356C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#2356D61
hypocenterC61#9182
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2356

Complements

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

NH ≅ G/Ntype
4. #21444214C2semidirect
5. #21443722C2semidirect
6. #21443677C2semidirect
7. C61×#2356C2semidirect
11. #16839972C4semidirect
12. #16839962C4semidirect
193. #2356D61semidirect
226. C61#9182semidirect

Maximal subgroups

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

subgroupindex
#214552912
#214443782
#214443132
#214442142
#214437222
#214436772
C61×#23562
#918261