Finite Group Databasebeta

← #29663155

Subgroups — #29663155

Order 15616.

714 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#21456734C211yesmaximal, characteristic
2C2×#16840110C211yesmaximal, characteristic, complemented
3#21454531C211yesmaximal, characteristic, complemented
4#21445354C214yesmaximal, complemented
5#21445230C214yesmaximal, complemented
6#21444565C211yesmaximal, characteristic, complemented
7#21444500C211yesmaximal, characteristic, complemented
8#21444486C211yesmaximal, characteristic, complemented
9C61×#3227C211yesmaximal, characteristic, complemented
10C2×#14519784—21no—
11C2×#14519774C2211yescharacteristic
12#16840138—21no—
13#16840110—21no—
14#16840110—21no—
15#16840110—21no—
16#16840110C2212yes—
17#16840110C2212yes—
18#16839957C2214yes—
19#16839895—22no—
20#16839840—21no—
21#16839840C2212yescomplemented
22D4×#92418—21no—
23D4×#92418C2212yescomplemented
24#16839815C2211yescharacteristic
25#16839786C2211yescharacteristic
26C2×#14519697—21no—
27C2×#14519697—21no—
28#16839729—21no—
29C2×C4×#92418C2211yescharacteristic
30#16839514C2212yescomplemented
31#16839512C2212yescomplemented
32#16839511C2214yescomplemented
33#16839505C2214yes—
34C61×#532C2211yescharacteristic
35C2×C4⋊Q8×C61C2211yescharacteristic
36C61×#498—21no—
37C61×#486—21no—
38C61×#481—22no—
39C61×#476C2214yescomplemented
40C61×#431C2211yescharacteristic
41C22×D244—21no—
42C22×D244—21no—
43C22×D244—21no—
44C2×#11758199—24no—
45#14519784—44no—
46C2×#11758194—21no—
47C2×#11758194—22no—
48C22×#92418—22no—
49C22×#92418D411yescharacteristic
50C22×#92418D411yescharacteristic
51#14519774—21no—
52#14519774—21no—
53#14519774—21no—
54#14519774C2312yes—
55C4⋊#92418—21no—
56C4⋊#92418C2312yes—
57C4⋊#92418C2312yes—
58#14519697—24no—
59#14519697—22no—
60#14519697—22no—
61#14519697—22no—
62#14519697—22no—
63C22⋊#92418—21no—
64C22⋊#92418—22no—
65C22⋊#92418—22no—
66C4×#92418—21no—
67C4×#92418D411yescharacteristic
68C4×#92418D411yescharacteristic
69C4×#92418C2311yescharacteristic
70C4×#92418C2311yescharacteristic
71C2×#41×C61—21no—
72C22×Q8×C61—21no—
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61D411yescharacteristic
77C4⋊Q8×C61C2312yes—
78#127×C61D411yescharacteristic
79#124×C61—21no—
80#121×C61—21no—
81#121×C61D412yes—
82C4×D4×C61—21no—
83C4×D4×C61C2312yes—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C42×C61C2311yescharacteristic
87#106×C61—24no—
88C4⋊C8×C61C2312yes—
89#102×C61—24no—
90C22⋊C8×C61—21no—
91C4×C8×C61—21no—
92C23×D61—41no—
93#41×C61—42no—
94#41×C61—42no—
95C2×Q8×C61—21no—
96C2×Q8×C61—22no—
97C2×Q8×C61—41no—
98C2×Q8×C61—82no—
99C2×D4×C61—22no—
100C2×D4×C61—21no—
101C2×D4×C61—21no—
102C2×D4×C61C2×D412yes—
103C22×C4×C61—22no—
104C22×C4×C61—41no—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61#4111yescharacteristic
108C2×D244—22no—
109C2×D244—22no—
110C2×D244—22no—
111C2×D244—42no—
112C2×D244—42no—
113C2×D244—41no—
114C2×D244—41no—
115C2×D244—41no—
116C2×D244—82no—
117C2×D244—82no—
118C2×C8×C61—22no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—41no—
123C4⋊C4×C61—41no—
124C4⋊C4×C61C2×D412yes—
125C22⋊C4×C61—22no—
126C22⋊C4×C61—22no—
127C42×C61C2×D411yescharacteristic
128C42×C61C2×D411yescharacteristic
129C42×C61#4111yescharacteristic
130C42×C61C2411yescharacteristic
131#11758199—84no—
132#11758194—44no—
133#11758194—42no—
134C2×#92418—22no—
135C2×#92418—22no—
136C2×#92418—22no—
137C2×#92418—22no—
138C2×#92418—44no—
139C2×#92418C2×D411yescharacteristic
140C2×#92418C2×D411yescharacteristic
141C2×#92418C2×D411yescharacteristic
142C2×#92418C2×D411yescharacteristic
143C23×C61—22no—
144C23×C61#12111yescharacteristic
145C22×D61—42no—
146C22×D61—82no—
147C22×D61—82no—
148C22×D61—82no—
149Q8×C61—42no—
150Q8×C61—82no—
151Q8×C61—82no—
152D4×C61—24no—
153D4×C61—44no—
154D4×C61—42no—
155D4×C61—42no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—22no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61—42no—
162C2×C4×C61—82no—
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#13512yes—
167C2×C4×C61AGL(1,ℤ/8)12yes—
168C2×C4×C61C22×D411yescharacteristic
169C2×C4×C61C22×D411yescharacteristic
170C2×C4×C61C2×#4111yescharacteristic
171D244—42no—
172D244—42no—
173D244—44no—
174D244—82no—
175D244—82no—
176D244—82no—
177D244—82no—
178C8×C61—42no—
179#92418—21no—
180#92418—21no—
181#92418—21no—
182#92418—21no—
183#92418—42no—
184#92418—42no—
185#19196—611nomaximal, Sylow
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—42no—
189C22×C61—42no—
190C22×C61—42no—
191C22×C61#46312yes—
192C22×C61C2×#12111yescharacteristic
193C2×D61—82no—
194C2×D61—82no—
195C2×D61—82no—
196C2×D61—164no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—42no—
200C4×C61—82no—
201C4×C61#46312yes—
202C4×C61#47212yes—
203C4×C61C2×#13511yescharacteristic
204C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
205C4×C61#57712yes—
206#3227D6111yescharacteristic, complemented
207#3205—611no—
208C2×#486—611no—
209#2931—611no—
210#1697—614no—
211#1677—614no—
212#1564—611no—
213#1548—611no—
214#1545—611no—
215C2×C61—22no—
216C2×C61—82no—
217C2×C61C2×#46311yescharacteristic
218C2×C61#305711yescharacteristic
219C2×C61C2×#47211yescharacteristic
220D61—162no—
221C2×#131—1221no—
222#532C2×D6111yescharacteristic
223C2×C4⋊Q8C2×D6111yescharacteristic
224C2×#127—611no—
225#498—21no—
226#493—1221no—
227#486—21no—
228#486—612no—
229#486—612no—
230#486—1221no—
231#486—1221no—
232#486—1221no—
233#481—22no—
234#476C2×D6114yescomplemented
235#459—614no—
236#447—1222no—
237#436—612no—
238#436—1221no—
239C8×D4—612no—
240C8×D4—1221no—
241#431—611no—
242#431C2×D6111yescharacteristic
243#423—611no—
244C2×#102—1221no—
245C2×#102—1221no—
246#408—1221no—
247C2×C4×C8—611no—
248#335—612no—
249#334—612no—
250#333—614no—
251#331—614no—
252C61#1919611yescharacteristic, complemented, Sylow
253C2×#41—21no—
254C22×Q8—21no—
255C22×D4—1221no—
256C22×D4—1221no—
257C22×D4—1221no—
258C2×D8—1224no—
259#131—2444no—
260C2×M16—1222no—
261C2×M16—1221no—
262C22×C8—611no—
263C22×C8—611no—
264C22×C8—1222no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8C22⋊D6111yescharacteristic
269C4⋊Q8C22×D6112yes—
270#127—612no—
271#127—1221no—
272#127—1221no—
273#127—1221no—
274#127C22⋊D6111yescharacteristic
275#124—21no—
276#121—21no—
277#121C22⋊D6112yes—
278C4×D4—21no—
279C4×D4C22×D6112yes—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C42C22×D6111yescharacteristic
283#106—24no—
284C4⋊C8—612no—
285C4⋊C8—612no—
286C4⋊C8—1221no—
287C4⋊C8C22×D6112yes—
288#102—24no—
289#102—1222no—
290#102—1222no—
291#102—1222no—
292#102—1222no—
293#102—1224no—
294C22⋊C8—21no—
295C22⋊C8—1222no—
296C22⋊C8—1222no—
297C22⋊C8—1221no—
298C4×C8—21no—
299C4×C8—611no—
300C4×C8—611no—
301C4×C8—611no—
302C4×C8—611no—
303C4×C8—1221no—
304C24—2441no—
305#41—42no—
306#41—42no—
307C2×Q8—21no—
308C2×Q8—22no—
309C2×Q8—41no—
310C2×Q8—82no—
311C2×D4—22no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—1222no—
315C2×D4—1222no—
316C2×D4—1222no—
317C2×D4—4882no—
318C2×D4—4882no—
319C2×D4—2441no—
320C2×D4—2441no—
321C2×D4—2441no—
322C2×D4—2442no—
323C2×D4—2442no—
324C2×D4C2×C22⋊D6112yes—
325C22×C4—22no—
326C22×C4—41no—
327C22×C4D4×D6111yescharacteristic
328C22×C4#1175822411yescharacteristic
329C22×C4C2×C22⋊D6111yescharacteristic
330D8—4884no—
331M16—2444no—
332M16—2442no—
333C2×C8—22no—
334C2×C8—611no—
335C2×C8—611no—
336C2×C8—611no—
337C2×C8—611no—
338C2×C8—1222no—
339C2×C8—1222no—
340C2×C8—1222no—
341C2×C8—1222no—
342C2×C8—2444no—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4C2×C22⋊D6112yes—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C42D4×D6111yescharacteristic
352C42#1175822411yescharacteristic
353C42C2×C22⋊D6111yescharacteristic
354C42C23×D6111yescharacteristic
355C23—22no—
356C23—4882no—
357C23—4882no—
358C23—4882no—
359C23—2442no—
360C23#1451974911yescharacteristic
361Q8—42no—
362Q8—82no—
363Q8—82no—
364D4—24no—
365D4—44no—
366D4—42no—
367D4—42no—
368D4—4882no—
369D4—4882no—
370D4—4882no—
371D4—4882no—
372D4—2442no—
373D4—2442no—
374D4—2444no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—82no—
382C2×C4#1451974911yescharacteristic
383C2×C4#1451974911yescharacteristic
384C2×C4#1451974911yescharacteristic
385C2×C4#1451979812yes—
386C2×C4#1451980412yes—
387C2×C4C2×D4×D6111yescharacteristic
388C2×C4C22×C22⋊D6111yescharacteristic
389C2×C4C2×#1175822411yescharacteristic
390C8—42no—
391C8—1221no—
392C8—1221no—
393C8—1221no—
394C8—1221no—
395C8—2442no—
396C8—2442no—
397C22—21no—
398C22—21no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—9764no—
403C22—4882no—
404C22—4882no—
405C22—4882no—
406C22#1683997912yes—
407C22C2×#1451974911yescharacteristic
408C4—21no—
409C4—21no—
410C4—42no—
411C4—82no—
412C4#1683997912yes—
413C4#1684003012yes—
414C4C2×#1451979811yescharacteristic
415C4C2×#1451980411yescharacteristic
416C4#1684057012yes—
417C2—22no—
418C2—82no—
419C2—9762no—
420C2C2×#1683997911yescharacteristic
421C2#2145546011yescharacteristic
422C2C2×#1684003011yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3227C2
socleC22×C61C2×#121
prime radical#21456734C2
hypercenter#3227D61
hypocenterC61#19196
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
2. C2×#16840110C2semidirect
3. #21454531C2semidirect
4. #21445354C2semidirect
5. #21445230C2semidirect
6. #21444565C2semidirect
7. #21444500C2semidirect
8. #21444486C2semidirect
9. C61×#3227C2semidirect
21. #16839840C22semidirect
23. D4×#92418C22semidirect
30. #16839514C22semidirect
31. #16839512C22semidirect
32. #16839511C22semidirect
39. C61×#476C22semidirect
206. #3227D61semidirect
234. #476C2×D61semidirect
252. C61#19196semidirect

Maximal subgroups

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

subgroupindex
#214567342
C2×#168401102
#214545312
#214453542
#214452302
#214445652
#214445002
#214444862
C61×#32272
#1919661