Finite Group Databasebeta

← #29579373

Subgroups — #29579373

Order 15616.

502 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456245C211yesmaximal, characteristic
2#21445688C211yesmaximal, characteristic, complemented
3#21444565C211yesmaximal, characteristic, complemented
4#21444313C211yesmaximal, characteristic, complemented
5#21444219C211yesmaximal, characteristic, complemented
6#21444218C211yesmaximal, characteristic, complemented
7C61×#1502C211yesmaximal, 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×C4⋊#92418C2211yescharacteristic
19C2×#14519688—21no—
20C2×C4×#92418C2211yescharacteristic
21#16839517—21no—
22#16839514—21no—
23#16839512—21no—
24#16839495—22no—
25#16839491—22no—
26D4⋊#92418—22no—
27C61×#532C2211yescharacteristic
28C61×#431C2211yescharacteristic
29C4×C4⋊C4×C61C2211yescharacteristic
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×#92418—21no—
45C22×#92418D411yescharacteristic
46C22×#92418D411yescharacteristic
47#14519777C2×C411yescharacteristic
48#14519774C2×C411yescharacteristic
49#14519765—21no—
50#14519751—21no—
51#14519751—21no—
52C4×D244—21no—
53C4×D244—21no—
54C4⋊#92418—21no—
55C4⋊#92418—21no—
56C4⋊#92418D412yes—
57C22⋊#92418—21no—
58#14519688—42no—
59C4×#92418—21no—
60C4×#92418C2×C411yescharacteristic
61C4×#92418C2×C411yescharacteristic
62C4×#92418D411yescharacteristic
63C4×#92418D411yescharacteristic
64C2×#41×C61—21no—
65C4⋊Q8×C61C2×C411yescharacteristic
66#127×C61C2×C411yescharacteristic
67#124×C61—21no—
68#121×C61—21no—
69#121×C61—21no—
70C4×D4×C61—21no—
71C4×D4×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C42×C61—21no—
74C2×C42×C61C2311yescharacteristic
75C4⋊C8×C61—21no—
76C22⋊C8×C61—21no—
77C4×C8×C61—21no—
78#95×C61D412yes—
79#41×C61—41no—
80#41×C61—41no—
81#41×C61—41no—
82#41×C61—41no—
83C2×Q8×C61—21no—
84C2×D4×C61—21no—
85C2×D4×C61—21no—
86C2×D4×C61—21no—
87C2×D4×C61—41no—
88C22×C4×C61—22no—
89C22×C4×C61—41no—
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61#4111yescharacteristic
93C2×C22⋊D61—41no—
94#11758222—41no—
95#11758222—41no—
96#11758222—41no—
97#11758222—41no—
98C2×D244—21no—
99C2×D244—21no—
100C2×D244—21no—
101C2×C4×D61—41no—
102C2×Dic122—21no—
103C2×C8×C61—41no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—42no—
107C22⋊C4×C61—41no—
108C22⋊C4×C61—41no—
109C42×C61—41no—
110C42×C61—41no—
111C42×C61C22×C411yescharacteristic
112C42×C61C2×D411yescharacteristic
113C42×C61C2×D411yescharacteristic
114C42×C61#4111yescharacteristic
115#11758202—42no—
116#11758202—42no—
117#11758201—42no—
118#11758201—42no—
119#11758200—42no—
120#11758200—42no—
121#11758199—42no—
122#11758199—42no—
123#11758198—41no—
124#11758198—41no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127C2×#92418—21no—
128C2×#92418—21no—
129C2×#92418—22no—
130C2×#92418—41no—
131C2×#92418—41no—
132C2×#92418—41no—
133C2×#92418C22⋊C411yescharacteristic
134C2×#92418C22⋊C411yescharacteristic
135C2×#92418C22⋊C411yescharacteristic
136C2×#92418C22⋊C411yescharacteristic
137C2×#92418D812yes—
138C2×#92418SD1612yes—
139C23×C61—41no—
140C23×C61#12111yescharacteristic
141C22×D61—41no—
142Q8×C61—41no—
143Q8×C61—41no—
144D4×C61—41no—
145D4×C61—41no—
146D4×C61—42no—
147D4×C61—81no—
148D4×C61—81no—
149D4×C61—81no—
150C2×C4×C61—22no—
151C2×C4×C61—41no—
152C2×C4×C61—42no—
153C2×C4×C61—42no—
154C2×C4×C61—41no—
155C2×C4×C61—42no—
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#12411yescharacteristic
163C2×C4×C61#12711yescharacteristic
164C2×C4×C61C2×D811yescharacteristic
165C2×C4×C61C2×SD1611yescharacteristic
166C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
168C22⋊D61—81no—
169C22⋊D61—81no—
170C2×Dic61—41no—
171D244—41no—
172D244—41no—
173D244—42no—
174D244—81no—
175C4×D61—81no—
176C4×D61—81no—
177Dic122—41no—
178Dic122—41no—
179C8×C61—81no—
180#92418—21no—
181#92418—21no—
182#92418—42no—
183#92418—81no—
184#92418#10212yes—
185#92418#10212yes—
186#9753—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—81no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61#39011yescharacteristic
193C22×C61#45911yescharacteristic
194C22×C61#46011yescharacteristic
195C2×D61—81no—
196C2×D61—81no—
197C2×D61—81no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—82no—
202C4×C61—81no—
203C4×C61C2×#10211yescharacteristic
204C4×C61#41811yescharacteristic
205C4×C61#44412yes—
206C4×C61#45911yescharacteristic
207C4×C61#46011yescharacteristic
208Dic61—81no—
209#3147—611no—
210#1770—611no—
211#1564—611no—
212#1502—611no—
213#1502D6111yescharacteristic, complemented
214#1480—611no—
215#1480—611no—
216C2×C61—21no—
217C2×C61—21no—
218C2×C61—161no—
219C2×C61#197511yescharacteristic
220C2×C61#197611yescharacteristic
221C2×C61#198211yescharacteristic
222D61—161no—
223C2×#135—1221no—
224C2×#135—1221no—
225#532—611no—
226#532C2×D6111yescharacteristic
227#495—1221no—
228#495—1221no—
229#494—611no—
230#493—611no—
231#492—611no—
232#492—611no—
233#431—611no—
234#431C2×D6111yescharacteristic
235C2×C4⋊C8—611no—
236C2×#97—1221no—
237C2×C4×C8—611no—
238C4×C4⋊C4C2×D6111yescharacteristic
239#336—1221no—
240#335—1221no—
241#334—1221no—
242#327—22no—
243#327—1222no—
244D4⋊C8—1222no—
245D4⋊C8—1222no—
246C61#975311yescharacteristic, complemented, Sylow
247C2×#41—21no—
248C2×#41—1221no—
249#135—2442no—
250#135—2442no—
251#135—2442no—
252#135—2442no—
253C2×Q16—1221no—
254C2×Q16—1221no—
255C2×SD16—1221no—
256C2×SD16—1221no—
257C2×SD16—1221no—
258C2×SD16—1221no—
259C2×D8—1221no—
260C2×D8—1221no—
261C22×C8—611no—
262C22×C8—611no—
263C22×C8—1221no—
264C4⋊Q8—611no—
265C4⋊Q8C4×D6111yescharacteristic
266#127—611no—
267#127C4×D6111yescharacteristic
268#124—21no—
269#124—1221no—
270#121—21no—
271#121—21no—
272#121—1221no—
273#121—1221no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—1221no—
277C4×D4—1221no—
278C2×C4⋊C4—21no—
279C2×C42—21no—
280C2×C42C22×D6111yescharacteristic
281C4⋊C8—21no—
282C4⋊C8—612no—
283C4⋊C8—1221no—
284C4⋊C8—1221no—
285C22⋊C8—21no—
286C22⋊C8—1221no—
287#97—2442no—
288C4×C8—21no—
289C4×C8—611no—
290C4×C8—611no—
291C4×C8—611no—
292C4×C8—611no—
293C4×C8—1221no—
294#95C22⋊D6112yes—
295#41—41no—
296#41—41no—
297#41—41no—
298#41—41no—
299#41—2441no—
300#41—2441no—
301#41—2441no—
302#41—2441no—
303C2×Q8—21no—
304C2×Q8—1221no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—41no—
309C2×D4—1221no—
310C2×D4—1221no—
311C2×D4—1221no—
312C2×D4—2441no—
313C22×C4—22no—
314C22×C4—41no—
315C22×C4—2441no—
316C22×C4D4×D6111yescharacteristic
317C22×C4#1175822411yescharacteristic
318C22×C4C2×C22⋊D6111yescharacteristic
319Q16—2442no—
320Q16—2442no—
321SD16—2442no—
322SD16—2442no—
323SD16—2442no—
324SD16—2442no—
325D8—2442no—
326D8—2442no—
327C2×C8—41no—
328C2×C8—611no—
329C2×C8—611no—
330C2×C8—612no—
331C2×C8—612no—
332C2×C8—611no—
333C2×C8—611no—
334C2×C8—1221no—
335C2×C8—1221no—
336C2×C8—1222no—
337C2×C8—2441no—
338C2×C8—2441no—
339C2×C8—2441no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—42no—
343C4⋊C4—1221no—
344C4⋊C4—1221no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—2441no—
348C22⋊C4—2441no—
349C42—41no—
350C42—41no—
351C42C2×C4×D6111yescharacteristic
352C42#1175822211yescharacteristic
353C42D4×D6111yescharacteristic
354C42D4×D6111yescharacteristic
355C23—41no—
356C23—2441no—
357C23#1451974911yescharacteristic
358Q8—41no—
359Q8—41no—
360Q8—2441no—
361Q8—2441no—
362D4—41no—
363D4—41no—
364D4—42no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—4881no—
369D4—4881no—
370D4—4881no—
371D4—2441no—
372D4—2441no—
373D4—2442no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—4881no—
383C2×C4—4881no—
384C2×C4—2441no—
385C2×C4C22⋊C4×D6111yescharacteristic
386C2×C4#1451973711yescharacteristic
387C2×C4C4×C22⋊D6111yescharacteristic
388C2×C4#1451975011yescharacteristic
389C2×C4#1451976311yescharacteristic
390C2×C4#1451977311yescharacteristic
391C2×C4D8×D6111yescharacteristic
392C2×C4SD16×D6111yescharacteristic
393C2×C4#1451980111yescharacteristic
394C2×C4#1451980411yescharacteristic
395C8—81no—
396C8—612no—
397C8—612no—
398C8—1221no—
399C8—1221no—
400C8—4881no—
401C8—2442no—
402C22—21no—
403C22—21no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—4881no—
408C22—4881no—
409C22—4881no—
410C22#1683966211yescharacteristic
411C22#1683995511yescharacteristic
412C22#1683996011yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—82no—
417C4—81no—
418C4—4881no—
419C4#102×D6111yescharacteristic
420C4#1683976511yescharacteristic
421C4#1683988112yes—
422C4#1683995611yescharacteristic
423C4#1683996111yescharacteristic
424C2—21no—
425C2—21no—
426C2—161no—
427C2—9761no—
428C2#2144670911yescharacteristic
429C2#2144671411yescharacteristic
430C2#2144674411yescharacteristic
subgroupG/H
centerC2×C4#14519750
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1502C2
socleC22×C61#390
prime radical#21456245C2
cosocleC2×C42×C61C23
hypercenter#1502D61
hypocenterC61#9753
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. #21445688C2semidirect
3. #21444565C2semidirect
4. #21444313C2semidirect
5. #21444219C2semidirect
6. #21444218C2semidirect
7. C61×#1502C2semidirect
13. #16840141C4semidirect
14. #16840138C4semidirect
15. #16840136C4semidirect
16. #16840134C4semidirect
213. #1502D61semidirect
246. C61#9753semidirect

Maximal subgroups

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

subgroupindex
#214562452
#214456882
#214445652
#214443132
#214442192
#214442182
C61×#15022
#975361