Finite Group Databasebeta

← #29579389

Subgroups — #29579389

Order 15616.

502 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456245C211yesmaximal, characteristic
2#21445981C211yesmaximal, characteristic, complemented
3#21444380C211yesmaximal, characteristic, complemented
4#21444379C211yesmaximal, characteristic, complemented
5#21444378C211yesmaximal, characteristic, complemented
6#21444103C211yesmaximal, characteristic, complemented
7C61×#1517C211yesmaximal, characteristic, complemented
8C2×#14519797—21no—
9C2×#14519797—21no—
10#16840312C2211yescharacteristic
11#16840144—21no—
12#16840144—21no—
13#16840141C411yescharacteristic, complemented
14#16840138C411yescharacteristic, complemented
15#16840136C411yescharacteristic, complemented
16#16840134C411yescharacteristic, complemented
17#16839815C2211yescharacteristic
18C2×#14519710—21no—
19C2×#14519708—21no—
20C2×C4×#92418C2211yescharacteristic
21#16839628C2211yescharacteristic
22#16839497—22no—
23#16839496—22no—
24#16839495—22no—
25#16839484—21no—
26C8×#92418—21no—
27C61×#532C2211yescharacteristic
28C61×#431C2211yescharacteristic
29C61×#382C2211yescharacteristic
30C61×#327—22no—
31C2×#11758222—21no—
32#14519797—42no—
33#14519797—42no—
34#14519797—42no—
35#14519797—42no—
36C2×#11758202—21no—
37C2×#11758202—21no—
38C2×#11758200—21no—
39C2×#11758200—21no—
40C2×#11758201—21no—
41C2×#11758201—21no—
42C2×#11758199—21no—
43C2×#11758199—21no—
44C22×#92418D411yescharacteristic
45C22×#92418D411yescharacteristic
46#14519777C2×C411yescharacteristic
47#14519774C2×C411yescharacteristic
48#14519765—21no—
49#14519751—21no—
50#14519751—21no—
51C4×D244—21no—
52C4×D244—21no—
53C2×C4⋊Dic61—21no—
54#14519710—42no—
55#14519708—42no—
56C4⋊#92418—21no—
57C22⋊#92418D412yes—
58C4×#92418—21no—
59C4×#92418C2×C411yescharacteristic
60C4×#92418C2×C411yescharacteristic
61C4×#92418D411yescharacteristic
62C4×#92418D411yescharacteristic
63#14519684—21no—
64#14519684—21no—
65C2×#41×C61—21no—
66C4⋊Q8×C61C2×C411yescharacteristic
67#127×C61C2×C411yescharacteristic
68#124×C61—21no—
69#121×C61—21no—
70#121×C61—21no—
71C4×D4×C61—21no—
72C4×D4×C61—21no—
73C2×C4⋊C4×C61—21no—
74C2×C42×C61C2311yescharacteristic
75C4⋊C8×C61—21no—
76C22⋊C8×C61D412yes—
77C4×C8×C61—21no—
78#95×C61—21no—
79#95×C61—21no—
80#41×C61—41no—
81#41×C61—41no—
82#41×C61—41no—
83#41×C61—41no—
84C2×Q8×C61—21no—
85C2×D4×C61—21no—
86C2×D4×C61—21no—
87C2×D4×C61—21no—
88C2×D4×C61—41no—
89C22×C4×C61—41no—
90C22×C4×C61—41no—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C2×C22⋊D61—41no—
95C22×Dic61—41no—
96#11758222—41no—
97#11758222—41no—
98#11758222—41no—
99#11758222—41no—
100C2×D244—21no—
101C2×D244—21no—
102C2×D244—21no—
103C2×C4×D61—41no—
104C2×Dic122—21no—
105C2×C8×C61—22no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—41no—
109C4⋊C4×C61—41no—
110C22⋊C4×C61—41no—
111C22⋊C4×C61—41no—
112C42×C61C22×C411yescharacteristic
113C42×C61C2×D411yescharacteristic
114C42×C61#4111yescharacteristic
115C42×C61#4111yescharacteristic
116#11758202—42no—
117#11758202—42no—
118#11758201—42no—
119#11758201—42no—
120#11758200—42no—
121#11758200—42no—
122#11758199—42no—
123#11758199—42no—
124#11758198—41no—
125#11758198—41no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128C4⋊Dic61—41no—
129C4⋊Dic61—41no—
130C2×#92418—22no—
131C2×#92418C22⋊C411yescharacteristic
132C2×#92418C22⋊C411yescharacteristic
133C2×#92418C22⋊C411yescharacteristic
134C2×#92418C22⋊C411yescharacteristic
135C2×#92418D812yes—
136C2×#92418D812yes—
137C2×#92418SD1612yes—
138C2×#92418SD1612yes—
139C23×C61—41no—
140C23×C61#12711yescharacteristic
141C22×D61—41no—
142Q8×C61—41no—
143Q8×C61—41no—
144D4×C61—41no—
145D4×C61—41no—
146D4×C61—81no—
147D4×C61—81no—
148D4×C61—81no—
149D4×C61—81no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—81no—
154C2×C4×C61—81no—
155C2×C4×C61—81no—
156C2×C4×C61—81no—
157C2×C4×C61—81no—
158C2×C4×C61C2×C22⋊C411yescharacteristic
159C2×C4×C61C4×D411yescharacteristic
160C2×C4×C61C4×D411yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12411yescharacteristic
164C2×C4×C61C2×D811yescharacteristic
165C2×C4×C61C2×D811yescharacteristic
166C2×C4×C61C2×SD1611yescharacteristic
167C2×C4×C61C2×SD1611yescharacteristic
168C22⋊D61—81no—
169C22⋊D61—81no—
170C2×Dic61—41no—
171C2×Dic61—81no—
172C2×Dic61—81no—
173C2×Dic61—81no—
174D244—41no—
175D244—41no—
176D244—81no—
177D244—81no—
178C4×D61—81no—
179C4×D61—81no—
180Dic122—41no—
181Dic122—41no—
182C8×C61—42no—
183#92418—42no—
184#92418#10212yes—
185#92418#10212yes—
186#92418#10212yes—
187#92418#10212yes—
188#9756—611nomaximal, Sylow
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61—81no—
194C22×C61#39011yescharacteristic
195C22×C61#49211yescharacteristic
196C22×C61#49311yescharacteristic
197C2×D61—81no—
198C2×D61—81no—
199C2×D61—81no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—81no—
205C4×C61—161no—
206C4×C61C2×#10211yescharacteristic
207C4×C61C2×#10211yescharacteristic
208C4×C61#48611yescharacteristic
209C4×C61#48611yescharacteristic
210Dic61—81no—
211Dic61—161no—
212#3147—611no—
213#1846—611no—
214#1517—611no—
215#1517—611no—
216#1517—611no—
217#1517D6111yescharacteristic, complemented
218#1450—611no—
219C2×C61—21no—
220C2×C61—21no—
221C2×C61—161no—
222C2×C61#195711yescharacteristic
223C2×C61#197611yescharacteristic
224C2×C61#197611yescharacteristic
225D61—161no—
226C2×#135—1221no—
227C2×#135—1221no—
228#532—611no—
229#532C2×D6111yescharacteristic
230#495—1221no—
231#495—1221no—
232#494—611no—
233#493—611no—
234#492—611no—
235#492—611no—
236#431—611no—
237#431C2×D6111yescharacteristic
238C2×#107—1221no—
239C2×#106—1221no—
240C2×C4×C8—611no—
241#382—611no—
242#382C2×D6111yescharacteristic
243#327—22no—
244#327—1222no—
245#327—1222no—
246#327—1222no—
247#322—1221no—
248C82—1221no—
249C61#975611yescharacteristic, complemented, Sylow
250C2×#41—21no—
251C2×#41—1221no—
252#135—2442no—
253#135—2442no—
254#135—2442no—
255#135—2442no—
256C2×Q16—1221no—
257C2×Q16—1221no—
258C2×SD16—1221no—
259C2×SD16—1221no—
260C2×SD16—1221no—
261C2×SD16—1221no—
262C2×D8—1221no—
263C2×D8—1221no—
264C22×C8—611no—
265C22×C8—611no—
266C4⋊Q8—611no—
267C4⋊Q8C4×D6111yescharacteristic
268#127—611no—
269#127C4×D6111yescharacteristic
270#124—21no—
271#124—1221no—
272#121—21no—
273#121—21no—
274#121—1221no—
275#121—1221no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—1221no—
279C4×D4—1221no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—1221no—
282C2×C42C22×D6111yescharacteristic
283#107—2442no—
284#106—2442no—
285C4⋊C8—21no—
286C4⋊C8—1221no—
287C22⋊C8—612no—
288C22⋊C8C22⋊D6112yes—
289C4×C8—21no—
290C4×C8—611no—
291C4×C8—611no—
292C4×C8—611no—
293C4×C8—611no—
294C4×C8—1221no—
295#95—21no—
296#95—21no—
297#95—1221no—
298#95—1221no—
299#41—41no—
300#41—41no—
301#41—41no—
302#41—41no—
303#41—2441no—
304#41—2441no—
305#41—2441no—
306#41—2441no—
307C2×Q8—21no—
308C2×Q8—1221no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—41no—
313C2×D4—1221no—
314C2×D4—1221no—
315C2×D4—1221no—
316C2×D4—2441no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—2441no—
320C22×C4—2441no—
321C22×C4D4×D6111yescharacteristic
322C22×C4D4×D6111yescharacteristic
323C22×C4C2×C22⋊D6111yescharacteristic
324Q16—2442no—
325Q16—2442no—
326SD16—2442no—
327SD16—2442no—
328SD16—2442no—
329SD16—2442no—
330D8—2442no—
331D8—2442no—
332C2×C8—22no—
333C2×C8—611no—
334C2×C8—611no—
335C2×C8—611no—
336C2×C8—611no—
337C2×C8—612no—
338C2×C8—612no—
339C2×C8—612no—
340C2×C8—612no—
341C2×C8—1222no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—1221no—
347C4⋊C4—1221no—
348C4⋊C4—2441no—
349C4⋊C4—2441no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—2441no—
353C22⋊C4—2441no—
354C42C2×C4×D6111yescharacteristic
355C42#1175822211yescharacteristic
356C42D4×D6111yescharacteristic
357C42#1175822411yescharacteristic
358C23—41no—
359C23—2441no—
360C23#1451977311yescharacteristic
361Q8—41no—
362Q8—41no—
363Q8—2441no—
364Q8—2441no—
365D4—41no—
366D4—41no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—81no—
371D4—4881no—
372D4—4881no—
373D4—4881no—
374D4—4881no—
375D4—2441no—
376D4—2441no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—4881no—
386C2×C4—4881no—
387C2×C4—4881no—
388C2×C4—4881no—
389C2×C4—4881no—
390C2×C4—2441no—
391C2×C4C22⋊C4×D6111yescharacteristic
392C2×C4#1451973711yescharacteristic
393C2×C4C4×C22⋊D6111yescharacteristic
394C2×C4#1451974911yescharacteristic
395C2×C4#1451975011yescharacteristic
396C2×C4#1451976311yescharacteristic
397C2×C4D8×D6111yescharacteristic
398C2×C4D8×D6111yescharacteristic
399C2×C4SD16×D6111yescharacteristic
400C2×C4SD16×D6111yescharacteristic
401C8—42no—
402C8—612no—
403C8—612no—
404C8—612no—
405C8—612no—
406C8—2442no—
407C22—21no—
408C22—21no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—4881no—
413C22—4881no—
414C22—4881no—
415C22#1683966211yescharacteristic
416C22#1684013311yescharacteristic
417C22#1684013711yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—81no—
423C4—161no—
424C4—9761no—
425C4—4881no—
426C4#102×D6111yescharacteristic
427C4#102×D6111yescharacteristic
428C4#1684010711yescharacteristic
429C4#1684010711yescharacteristic
430C2—21no—
431C2—21no—
432C2—161no—
433C2—9761no—
434C2#2144662311yescharacteristic
435C2#2144671411yescharacteristic
436C2#2144671611yescharacteristic
subgroupG/H
centerC22#16839662
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1517C2
socleC22×C61#390
prime radical#21456245C2
cosocleC2×C42×C61C23
hypercenter#1517D61
hypocenterC61#9756
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1517

Complements

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

NH ≅ G/Ntype
2. #21445981C2semidirect
3. #21444380C2semidirect
4. #21444379C2semidirect
5. #21444378C2semidirect
6. #21444103C2semidirect
7. C61×#1517C2semidirect
13. #16840141C4semidirect
14. #16840138C4semidirect
15. #16840136C4semidirect
16. #16840134C4semidirect
217. #1517D61semidirect
249. C61#9756semidirect

Maximal subgroups

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

subgroupindex
#214562452
#214459812
#214443802
#214443792
#214443782
#214441032
C61×#15172
#975661