Finite Group Databasebeta

← #29833464

Subgroups — #29833464

Order 15616.

652 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839884C211yesmaximal, characteristic, complemented
2#21454620C211yesmaximal, characteristic, complemented
3#21447718C214yesmaximal, complemented
4#21447711C214yesmaximal, complemented
5#21447592C211yesmaximal, characteristic, complemented
6#21447568C211yesmaximal, characteristic, complemented
7#21447555C211yesmaximal, characteristic, complemented
8C2×#16839539C211yesmaximal, characteristic, complemented
9C61×#3414C211yesmaximal, characteristic
10C2×#14519784—21no—
11#41×Dic61—21no—
12C2×#14519717C2211yescharacteristic
13#16839884—22no—
14#16839884C2214yes—
15#16839874C2212yescomplemented
16#16839869C2212yescomplemented
17#16839863—22no—
18C2×#14519713C2211yescharacteristic
19#16839800C2211yescharacteristic
20C2×#14519704—21no—
21#16839758—21no—
22C2×#14519687C2211yescharacteristic
23#16839585C2214yescomplemented
24#16839578C2212yescomplemented
25#16839575C2212yescomplemented
26#16839572C2214yes—
27#16839539C2214yescomplemented
28C2×#135×C61C2211yescharacteristic
29C2×#135×C61C2211yescharacteristic
30C61×#510C412yescomplemented
31C61×#510C2212yes—
32C61×#509C412yescomplemented
33C61×#509C2212yes—
34C61×#508C414yescomplemented
35C2×M32×C61C2211yescharacteristic
36#14519784—24no—
37#14519784—42no—
38C2×#11758194—21no—
39C2×#11758194—41no—
40C22×#92418—41no—
41C22×#92418D411yescharacteristic
42Q8×Dic61—22no—
43D4×Dic61—22no—
44D4×Dic61—42no—
45#14519736—21no—
46#14519736—41no—
47C2×C4×Dic61—41no—
48C2×C4×Dic61D411yescharacteristic
49#14519717—21no—
50#14519717C2312yes—
51C2×#11758185—21no—
52#14519713—21no—
53#14519713C2312yes—
54#14519711—21no—
55#14519709C2312yes—
56#14519704—24no—
57#14519701—22no—
58#14519698—22no—
59#14519687D412yes—
60#14519687C2312yes—
61C2×#41×C61—21no—
62C2×#41×C61—21no—
63#135×C61—21no—
64#135×C61—21no—
65#135×C61—21no—
66#135×C61—21no—
67#135×C61C2×C412yes—
68#135×C61C2×C412yes—
69#135×C61C2×C412yescomplemented
70#135×C61C2312yes—
71C2×Q16×C61C2×C411yescharacteristic, complemented
72C2×Q16×C61C2311yescharacteristic
73C2×SD16×C61—21no—
74C2×SD16×C61—21no—
75C2×D8×C61C2×C411yescharacteristic, complemented
76C2×D8×C61C2311yescharacteristic
77C22×C8×C61C2311yescharacteristic
78Q32×C61C2×C414yescomplemented
79SD32×C61C2×C414yescomplemented
80SD32×C61C2×C414yescomplemented
81D16×C61C2×C414yescomplemented
82M32×C61C2×C412yescomplemented
83M32×C61C2312yes—
84C2×C16×C61C2×C412yescomplemented
85#41×C61—22no—
86#41×C61—22no—
87#41×C61—22no—
88#41×C61—22no—
89#41×C61—41no—
90#41×C61—41no—
91#41×C61—41no—
92#41×C61—41no—
93C2×Q8×C61—21no—
94C2×Q8×C61—21no—
95C2×D4×C61—21no—
96C2×D4×C61—21no—
97C2×D4×C61—41no—
98C2×D4×C61—41no—
99C22×C4×C61—41no—
100C22×C4×C61—41no—
101C22×C4×C61C2×D411yescharacteristic
102C22×Dic61—21no—
103C22×Dic61—42no—
104Q16×C61—21no—
105Q16×C61—21no—
106Q16×C61C22×C412yes—
107Q16×C61C22×C412yes—
108SD16×C61—22no—
109SD16×C61—22no—
110SD16×C61—22no—
111SD16×C61—22no—
112D8×C61—21no—
113D8×C61—21no—
114D8×C61C22×C412yes—
115D8×C61C22×C412yes—
116C2×C8×C61C22×C411yescharacteristic
117C2×C8×C61C22×C411yescharacteristic
118C2×C8×C61C2×D411yescharacteristic
119C2×C8×C61#4111yescharacteristic
120C2×C8×C61#4111yescharacteristic
121C2×C8×C61C2411yescharacteristic
122C22⋊Dic61—42no—
123C22⋊Dic61—41no—
124C4⋊Dic61—22no—
125C4⋊Dic61—42no—
126C4×Dic61—21no—
127C4×Dic61—42no—
128C4×Dic61—41no—
129C4×Dic61C2×D412yes—
130#11758194—22no—
131#11758194—44no—
132#11758194—41no—
133C2×#92418—21no—
134C2×#92418—21no—
135C2×#92418—44no—
136C2×#92418—41no—
137C2×#92418C2×D412yes—
138C16×C61C22×C414yes—
139#11758185—22no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61AGL(1,ℤ/8)11yescharacteristic
143Q8×C61—22no—
144Q8×C61—22no—
145Q8×C61—41no—
146Q8×C61—41no—
147D4×C61—22no—
148D4×C61—22no—
149D4×C61—42no—
150D4×C61—42no—
151D4×C61—42no—
152D4×C61—42no—
153D4×C61—41no—
154D4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61C4×D411yescharacteristic
162C2×C4×C61C4×D411yescharacteristic
163C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C61C22×D411yescharacteristic
167C2×Dic61—22no—
168C2×Dic61—42no—
169C2×Dic61—81no—
170C2×Dic61—84no—
171C8×C61C4×D411yescharacteristic
172C8×C61C4×D411yescharacteristic
173C8×C61C23×C411yescharacteristic
174C8×C61C2×#4111yescharacteristic
175#92418—22no—
176#92418—42no—
177#33468—611nomaximal, Sylow
178C22×C61—41no—
179C22×C61—42no—
180C22×C61—42no—
181C22×C61—82no—
182C22×C61—82no—
183C22×C61#44211yescharacteristic
184C22×C61#44211yescharacteristic
185C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
186C4×C61—42no—
187C4×C61—42no—
188C4×C61#44211yescharacteristic
189C4×C61#44211yescharacteristic
190C4×C61C2×C4×D411yescharacteristic
191C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
192Dic61—42no—
193Dic61—82no—
194#3414D6111yescharacteristic
195C2×#444—611no—
196#2943—611no—
197AGL(1,ℤ/16)—614no—
198#2180—614no—
199#2155—611no—
200#2148—611no—
201#2145—611no—
202C2×#347—611no—
203C2×C61—21no—
204C2×C61—21no—
205C2×C61—21no—
206C2×C61—82no—
207C2×C61—82no—
208C2×C61C2×#44211yescharacteristic
209C2×#135C2×D6111yescharacteristic
210C2×#135C2×D6111yescharacteristic
211C2×#131—1221no—
212C4×#41—1221no—
213#510Dic6112yescomplemented
214#510C2×D6112yes—
215#509Dic6112yescomplemented
216#509C2×D6112yes—
217#508Dic6114yescomplemented
218C2×M32—611no—
219C2×M32C2×D6111yescharacteristic
220#444—614no—
221#444—1222no—
222#442—612no—
223#441—612no—
224#440—1222no—
225C2×#108—611no—
226#427—611no—
227C2×#104—1221no—
228#416—1221no—
229C2×#97—611no—
230#365—614no—
231#362—612no—
232#361—612no—
233#360—614no—
234#347—614no—
235C61#3346811yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C2×#41—21no—
238#135—21no—
239#135—21no—
240#135—21no—
241#135—21no—
242#135C2×Dic6112yes—
243#135C2×Dic6112yes—
244#135C2×Dic6112yescomplemented
245#135C22×D6112yes—
246C2×Q16C2×Dic6111yescharacteristic, complemented
247C2×Q16C22×D6111yescharacteristic
248C2×SD16—21no—
249C2×SD16—21no—
250C2×D8C2×Dic6111yescharacteristic, complemented
251C2×D8C22×D6111yescharacteristic
252#131—1224no—
253#131—2442no—
254C2×M16—1221no—
255C2×M16—2441no—
256C22×C8—611no—
257C22×C8—2441no—
258C22×C8C22×D6111yescharacteristic
259C4×Q8—1222no—
260C4×D4—1222no—
261C4×D4—2442no—
262#117—1221no—
263#117—2441no—
264C2×C42—611no—
265C2×C42—2441no—
266Q32C2×Dic6114yescomplemented
267SD32C2×Dic6114yescomplemented
268SD32C2×Dic6114yescomplemented
269D16C2×Dic6114yescomplemented
270M32—612no—
271M32—1221no—
272M32C2×Dic6112yescomplemented
273M32C22×D6112yes—
274C2×C16—1221no—
275C2×C16C2×Dic6112yescomplemented
276#108—612no—
277#108—1221no—
278#107—1221no—
279#106—612no—
280#104—1224no—
281#103—1222no—
282#102—1222no—
283#97—612no—
284#97—612no—
285#41—22no—
286#41—22no—
287#41—22no—
288#41—22no—
289#41—41no—
290#41—41no—
291#41—41no—
292#41—41no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—1221no—
302C22×C4—2442no—
303C22×C4D4×D6111yescharacteristic
304Q16—21no—
305Q16—21no—
306Q16C22×Dic6112yes—
307Q16C22×Dic6112yes—
308SD16—22no—
309SD16—22no—
310SD16—22no—
311SD16—22no—
312D8—21no—
313D8—21no—
314D8C22×Dic6112yes—
315D8C22×Dic6112yes—
316M16—1222no—
317M16—2444no—
318M16—2441no—
319C2×C8—612no—
320C2×C8—1221no—
321C2×C8—1221no—
322C2×C8—2444no—
323C2×C8—2441no—
324C2×C8D4×D6111yescharacteristic
325C2×C8#1175822411yescharacteristic
326C2×C8#1175822411yescharacteristic
327C2×C8C22×Dic6111yescharacteristic
328C2×C8C22×Dic6111yescharacteristic
329C2×C8C23×D6111yescharacteristic
330C4⋊C4—1222no—
331C4⋊C4—2442no—
332C22⋊C4—2442no—
333C22⋊C4—2441no—
334C42—612no—
335C42—1221no—
336C42—2442no—
337C42—2441no—
338C16—1222no—
339C16C22×Dic6114yes—
340C23—41no—
341C23—41no—
342C23#1451980111yescharacteristic
343Q8—22no—
344Q8—22no—
345Q8—41no—
346Q8—41no—
347D4—22no—
348D4—22no—
349D4—42no—
350D4—42no—
351D4—42no—
352D4—42no—
353D4—41no—
354D4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—1222no—
362C2×C4—4884no—
363C2×C4—4881no—
364C2×C4—2442no—
365C2×C4D4×Dic6111yescharacteristic
366C2×C4D4×Dic6111yescharacteristic
367C2×C4#1451980111yescharacteristic
368C2×C4#1451980111yescharacteristic
369C2×C4#1451980111yescharacteristic
370C2×C4C2×D4×D6111yescharacteristic
371C8—1222no—
372C8—2442no—
373C8D4×Dic6111yescharacteristic
374C8D4×Dic6111yescharacteristic
375C8C23×Dic6111yescharacteristic
376C8C2×#1175822411yescharacteristic
377C22—41no—
378C22—42no—
379C22—42no—
380C22—82no—
381C22—82no—
382C22#1683987411yescharacteristic
383C22#1683987411yescharacteristic
384C22C2×#1451980111yescharacteristic
385C4—42no—
386C4—42no—
387C4—4882no—
388C4—2442no—
389C4#1683987411yescharacteristic
390C4#1683987411yescharacteristic
391C4C2×D4×Dic6111yescharacteristic
392C4C2×#1451980111yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—82no—
398C2C2×#1683987411yescharacteristic
subgroupG/H
centerC4#16839874
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
Fitting, prime radicalC61×#3414C2
socleC2×C61C2×#442
cosocleC2×C8×C61C24
hypercenter#3414D61
hypocenterC61#33468
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #3414

Complements

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

NH ≅ G/Ntype
1. C2×#16839884C2semidirect
2. #21454620C2semidirect
3. #21447718C2semidirect
4. #21447711C2semidirect
5. #21447592C2semidirect
6. #21447568C2semidirect
7. #21447555C2semidirect
8. C2×#16839539C2semidirect
15. #16839874C22semidirect
16. #16839869C22semidirect
23. #16839585C22semidirect
24. #16839578C22semidirect
25. #16839575C22semidirect
27. #16839539C22semidirect
30. C61×#510C4semidirect
32. C61×#509C4semidirect
34. C61×#508C4semidirect
69. #135×C61C2×C4semidirect
71. C2×Q16×C61C2×C4semidirect
75. C2×D8×C61C2×C4semidirect
78. Q32×C61C2×C4semidirect
79. SD32×C61C2×C4semidirect
80. SD32×C61C2×C4semidirect
81. D16×C61C2×C4semidirect
82. M32×C61C2×C4semidirect
84. C2×C16×C61C2×C4semidirect
213. #510Dic61semidirect
215. #509Dic61semidirect
217. #508Dic61semidirect
235. C61#33468semidirect
244. #135C2×Dic61semidirect
246. C2×Q16C2×Dic61semidirect
250. C2×D8C2×Dic61semidirect
266. Q32C2×Dic61semidirect
267. SD32C2×Dic61semidirect
268. SD32C2×Dic61semidirect
269. D16C2×Dic61semidirect
272. M32C2×Dic61semidirect
275. C2×C16C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398842
#214546202
#214477182
#214477112
#214475922
#214475682
#214475552
C2×#168395392
C61×#34142
#3346861