Finite Group Databasebeta

← #29573255

Subgroups — #29573255

Order 15616.

430 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454531C211yesmaximal, characteristic
2#21445990C211yesmaximal, characteristic, complemented
3C2×#16839512C211yesmaximal, characteristic, complemented
4#21444496C211yesmaximal, characteristic
5#21443694C211yesmaximal, characteristic, complemented
6#21443693C211yesmaximal, characteristic, complemented
7C61×#1547C211yesmaximal, characteristic
8C2×#14519784—21no—
9#16839840—21no—
10#16839840C412yes—
11D4×#92418—21no—
12D4×#92418C412yes—
13C2×#14519712—21no—
14C2×#14519712—21no—
15#16839786C2211yescharacteristic
16#16839786C2211yescharacteristic
17C2×C4⋊#92418C2211yescharacteristic
18#16839729—21no—
19C2×C4×#92418C2211yescharacteristic
20#16839512—21no—
21#16839512—21no—
22#16839508—21no—
23#16839506—21no—
24C61×#532C2211yescharacteristic
25C61×#423C2211yescharacteristic
26C2×C4⋊C8×C61C2211yescharacteristic
27C61×#332—21no—
28C61×#331—21no—
29#14519784—42no—
30#14519784—42no—
31C2×#11758194—21no—
32C2×#11758194—22no—
33C2×#11758194—21no—
34C22×#92418—22no—
35C22×#92418—21no—
36C22×#92418D411yescharacteristic
37C22×#92418D411yescharacteristic
38#14519712—42no—
39#14519712—42no—
40C4⋊#92418—21no—
41C4⋊#92418—21no—
42C4⋊#92418—21no—
43C4⋊#92418—21no—
44C4⋊#92418—21no—
45C4⋊#92418C2×C412yes—
46C22⋊#92418—22no—
47C22⋊#92418—22no—
48C4×#92418C2×C411yescharacteristic
49C4×#92418C2×C411yescharacteristic
50C4×#92418D411yescharacteristic
51C4×#92418D411yescharacteristic
52C2×#41×C61—21no—
53C2×M16×C61—21no—
54C22×C8×C61—21no—
55C4⋊Q8×C61D411yescharacteristic
56#127×C61D411yescharacteristic
57#124×C61—21no—
58#121×C61—21no—
59#121×C61Q812yes—
60C4×D4×C61—21no—
61C4×D4×C61C2×C412yes—
62C2×C42×C61C2311yescharacteristic
63C4⋊C8×C61—21no—
64C4⋊C8×C61—21no—
65C4⋊C8×C61—21no—
66C4⋊C8×C61—21no—
67#41×C61—41no—
68#41×C61—41no—
69#41×C61—41no—
70#41×C61—41no—
71C2×Q8×C61—21no—
72C2×D4×C61—22no—
73C2×D4×C61—21no—
74C2×D4×C61—21no—
75C2×D4×C61C4⋊C412yes—
76C22×C4×C61—22no—
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×Q811yescharacteristic
79C22×C4×C61#4111yescharacteristic
80M16×C61—41no—
81M16×C61—41no—
82C2×C8×C61—41no—
83C2×C8×C61—41no—
84C2×C8×C61—81no—
85C4⋊C4×C61—21no—
86C4⋊C4×C61C4⋊C412yes—
87C22⋊C4×C61—22no—
88C22⋊C4×C61—22no—
89C42×C61C22×C411yescharacteristic
90C42×C61C2×D411yescharacteristic
91C42×C61C2×D411yescharacteristic
92C42×C61#4111yescharacteristic
93#11758194—41no—
94#11758194—41no—
95#11758194—44no—
96#11758194—41no—
97#11758194—41no—
98C2×#92418—21no—
99C2×#92418—21no—
100C2×#92418—22no—
101C2×#92418—44no—
102C2×#92418—41no—
103C2×#92418—41no—
104C2×#92418—81no—
105C2×#92418C22⋊C411yescharacteristic
106C2×#92418C22⋊C411yescharacteristic
107C2×#92418C22⋊C411yescharacteristic
108C2×#92418C22⋊C411yescharacteristic
109C2×#92418D812yes—
110C2×#92418SD1612yes—
111C23×C61—22no—
112C23×C61C22⋊Q811yescharacteristic
113Q8×C61—41no—
114Q8×C61—41no—
115D4×C61—42no—
116D4×C61—42no—
117D4×C61—42no—
118D4×C61—41no—
119D4×C61—41no—
120D4×C61—41no—
121D4×C61—41no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—22no—
125C2×C4×C61—41no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61C2×C22⋊C411yescharacteristic
129C2×C4×C61C2×C4⋊C411yescharacteristic
130C2×C4×C61#11711yescharacteristic
131C2×C4×C61#12111yescharacteristic
132C2×C4×C61#12111yescharacteristic
133C2×C4×C61C22⋊Q811yescharacteristic
134C2×C4×C61C2×D811yescharacteristic
135C2×C4×C61C2×SD1611yescharacteristic
136C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C61#13711yescharacteristic
138C8×C61—81no—
139C8×C61—81no—
140#92418—21no—
141#92418—21no—
142#92418—42no—
143#92418—81no—
144#92418—81no—
145#92418#10212yes—
146#92418#10212yes—
147#8682—611nomaximal, Sylow
148C22×C61—21no—
149C22×C61—21no—
150C22×C61—42no—
151C22×C61—42no—
152C22×C61—42no—
153C22×C61#38011yescharacteristic
154C22×C61#47411yescharacteristic
155C22×C61#47611yescharacteristic
156C4×C61—21no—
157C4×C61—21no—
158C4×C61—21no—
159C4×C61—42no—
160C4×C61C2×#10211yescharacteristic
161C4×C61#41711yescharacteristic
162C4×C61#45911yescharacteristic
163C4×C61#46111yescharacteristic
164C4×C61#47111yescharacteristic
165C4×C61#47111yescharacteristic
166#2931—611no—
167#1848—611no—
168C2×#334—611no—
169#1547—611no—
170#1547D6111yescharacteristic
171#1292—611no—
172#1292—611no—
173C2×C61—21no—
174C2×C61—21no—
175C2×C61—82no—
176C2×C61#180211yescharacteristic
177C2×C61#181311yescharacteristic
178C2×C61#181511yescharacteristic
179C2×#131—1221no—
180#532C2×D6111yescharacteristic
181#436—612no—
182#436—1221no—
183C8×D4—612no—
184C8×D4—1221no—
185C2×#108—1221no—
186C2×#108—1221no—
187#423—611no—
188#423—611no—
189#423C2×D6111yescharacteristic
190C2×C4⋊C8—611no—
191C2×C4⋊C8C2×D6111yescharacteristic
192#408—1221no—
193C2×C4×C8—611no—
194#334—1221no—
195#334—1221no—
196#332—21no—
197#332—1221no—
198#331—21no—
199#331—1221no—
200C61#868211yescharacteristic, complemented, Sylow
201C2×#41—21no—
202#131—2442no—
203#131—2442no—
204C2×M16—21no—
205C2×M16—1222no—
206C2×M16—1221no—
207C2×M16—1221no—
208C22×C8—21no—
209C22×C8—611no—
210C22×C8—611no—
211C22×C8—1222no—
212C22×C8—1221no—
213C4⋊Q8C22⋊D6111yescharacteristic
214#127C22⋊D6111yescharacteristic
215#124—21no—
216#121—21no—
217#121Dic12212yes—
218C4×D4—21no—
219C4×D4C4×D6112yes—
220C2×C42C22×D6111yescharacteristic
221#108—2442no—
222#108—2442no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—612no—
228C4⋊C8—1221no—
229C4⋊C8—1221no—
230C4⋊C8—1221no—
231C4⋊C8—1221no—
232C4⋊C8—1221no—
233C22⋊C8—1222no—
234C22⋊C8—1222no—
235C4×C8—611no—
236C4×C8—611no—
237C4×C8—611no—
238C4×C8—611no—
239#41—41no—
240#41—41no—
241#41—41no—
242#41—41no—
243C2×Q8—21no—
244C2×D4—22no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4#1175819612yes—
248C22×C4—22no—
249C22×C4C2×Dic12211yescharacteristic
250C22×C4D4×D6111yescharacteristic
251C22×C4#1175822411yescharacteristic
252M16—41no—
253M16—41no—
254M16—2444no—
255M16—2441no—
256M16—2441no—
257M16—2441no—
258M16—2441no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—81no—
262C2×C8—611no—
263C2×C8—611no—
264C2×C8—612no—
265C2×C8—612no—
266C2×C8—611no—
267C2×C8—611no—
268C2×C8—1221no—
269C2×C8—1221no—
270C2×C8—1222no—
271C2×C8—4881no—
272C2×C8—2444no—
273C2×C8—2441no—
274C2×C8—2441no—
275C4⋊C4—21no—
276C4⋊C4#1175819612yes—
277C22⋊C4—22no—
278C22⋊C4—22no—
279C42C2×C4×D6111yescharacteristic
280C42D4×D6111yescharacteristic
281C42#1175822411yescharacteristic
282C42C2×C22⋊D6111yescharacteristic
283C23—22no—
284C23#1451975411yescharacteristic
285Q8—41no—
286Q8—41no—
287D4—42no—
288D4—42no—
289D4—42no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4—41no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4C22⋊C4×D6111yescharacteristic
301C2×C4C2×#1175819611yescharacteristic
302C2×C4#1451973311yescharacteristic
303C2×C4#1451974911yescharacteristic
304C2×C4#1451974911yescharacteristic
305C2×C4#1451975411yescharacteristic
306C2×C4D8×D6111yescharacteristic
307C2×C4SD16×D6111yescharacteristic
308C2×C4#1451980111yescharacteristic
309C2×C4#1451980611yescharacteristic
310C8—81no—
311C8—81no—
312C8—612no—
313C8—612no—
314C8—1221no—
315C8—1221no—
316C8—4881no—
317C8—4881no—
318C8—2442no—
319C22—21no—
320C22—21no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22#1683962011yescharacteristic
325C22#1684004011yescharacteristic
326C22#1684005011yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—42no—
331C4#102×D6111yescharacteristic
332C4#1683976011yescharacteristic
333C4#1683995511yescharacteristic
334C4#1683996511yescharacteristic
335C4#1684002311yescharacteristic
336C4#1684002711yescharacteristic
337C2—21no—
338C2—21no—
339C2—82no—
340C2#2144581711yescharacteristic
341C2#2144586711yescharacteristic
342C2#2144587611yescharacteristic
subgroupG/H
centerC22#16839620
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1547C2
socleC22×C61#380
prime radicalC61×#532C22
cosocleC2×C42×C61C23
hypercenter#1547D61
hypocenterC61#8682
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1547

Complements

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

NH ≅ G/Ntype
2. #21445990C2semidirect
3. C2×#16839512C2semidirect
5. #21443694C2semidirect
6. #21443693C2semidirect
200. C61#8682semidirect

Maximal subgroups

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

subgroupindex
#214545312
#214459902
C2×#168395122
#214444962
#214436942
#214436932
C61×#15472
#868261