Finite Group Databasebeta

← #29579407

Subgroups — #29579407

Order 15616.

478 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456245C211yesmaximal, characteristic
2#21445949C211yesmaximal, characteristic, complemented
3#21444380C211yesmaximal, characteristic, complemented
4#21444379C211yesmaximal, characteristic, complemented
5#21444378C211yesmaximal, characteristic, complemented
6#21444112C211yesmaximal, 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×C4×#92418C2211yescharacteristic
19#16839628C2211yescharacteristic
20#16839524—21no—
21#16839524—21no—
22#16839497—22no—
23#16839496—22no—
24#16839495—22no—
25#16839484—21no—
26#16839483—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—
54C4⋊#92418—21no—
55C22⋊#92418D412yes—
56C4×#92418—21no—
57C4×#92418C2×C411yescharacteristic
58C4×#92418C2×C411yescharacteristic
59C4×#92418D411yescharacteristic
60C4×#92418D411yescharacteristic
61#14519684—21no—
62#14519684—21no—
63C2×#41×C61—21no—
64C4⋊Q8×C61C2×C411yescharacteristic
65#127×C61C2×C411yescharacteristic
66#124×C61—21no—
67#121×C61—21no—
68#121×C61—21no—
69C4×D4×C61—21no—
70C4×D4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C42×C61C2311yescharacteristic
73C4⋊C8×C61—21no—
74C22⋊C8×C61D412yes—
75C4×C8×C61—21no—
76#95×C61—21no—
77#95×C61—21no—
78#41×C61—41no—
79#41×C61—41no—
80#41×C61—41no—
81#41×C61—41no—
82C2×Q8×C61—21no—
83C2×D4×C61—21no—
84C2×D4×C61—21no—
85C2×D4×C61—21no—
86C2×D4×C61—41no—
87C22×C4×C61—41no—
88C22×C4×C61—41no—
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C2×C22⋊D61—41no—
93C22×Dic61—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—22no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—41no—
107C4⋊C4×C61—41no—
108C22⋊C4×C61—41no—
109C22⋊C4×C61—41no—
110C42×C61C22×C411yescharacteristic
111C42×C61C2×D411yescharacteristic
112C42×C61#4111yescharacteristic
113C42×C61#4111yescharacteristic
114#11758202—42no—
115#11758202—42no—
116#11758201—42no—
117#11758201—42no—
118#11758200—42no—
119#11758200—42no—
120#11758199—42no—
121#11758199—42no—
122#11758198—41no—
123#11758198—41no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61—41no—
127C4⋊Dic61—41no—
128C2×#92418—21no—
129C2×#92418—21no—
130C2×#92418—21no—
131C2×#92418—21no—
132C2×#92418—22no—
133C2×#92418C22⋊C411yescharacteristic
134C2×#92418C22⋊C411yescharacteristic
135C2×#92418C22⋊C411yescharacteristic
136C2×#92418C22⋊C411yescharacteristic
137C23×C61—41no—
138C23×C61#12711yescharacteristic
139C22×D61—41no—
140Q8×C61—41no—
141Q8×C61—41no—
142D4×C61—41no—
143D4×C61—41no—
144D4×C61—81no—
145D4×C61—81no—
146D4×C61—81no—
147D4×C61—81no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—81no—
152C2×C4×C61—81no—
153C2×C4×C61—81no—
154C2×C4×C61—81no—
155C2×C4×C61—81no—
156C2×C4×C61C2×C22⋊C411yescharacteristic
157C2×C4×C61C4×D411yescharacteristic
158C2×C4×C61C4×D411yescharacteristic
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12411yescharacteristic
162C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
166C22⋊D61—81no—
167C22⋊D61—81no—
168C2×Dic61—41no—
169C2×Dic61—81no—
170C2×Dic61—81no—
171C2×Dic61—81no—
172D244—41no—
173D244—41no—
174D244—81no—
175D244—81no—
176C4×D61—81no—
177C4×D61—81no—
178Dic122—41no—
179Dic122—41no—
180C8×C61—42no—
181#92418—21no—
182#92418—21no—
183#92418—21no—
184#92418—21no—
185#92418—42no—
186#9760—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—81no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61#39011yescharacteristic
193C22×C61#49611yescharacteristic
194C22×C61#49611yescharacteristic
195C2×D61—81no—
196C2×D61—81no—
197C2×D61—81no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—81no—
203C4×C61—161no—
204C4×C61#41811yescharacteristic
205C4×C61#41811yescharacteristic
206C4×C61#49011yescharacteristic
207C4×C61#49011yescharacteristic
208Dic61—81no—
209Dic61—161no—
210#3147—611no—
211#1835—611no—
212#1517—611no—
213#1517—611no—
214#1517—611no—
215#1517D6111yescharacteristic, complemented
216#1452—611no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—161no—
220C2×C61#196111yescharacteristic
221C2×C61#197611yescharacteristic
222C2×C61#197611yescharacteristic
223D61—161no—
224C2×#135—1221no—
225C2×#135—1221no—
226#532—611no—
227#532C2×D6111yescharacteristic
228#495—1221no—
229#495—1221no—
230#494—611no—
231#493—611no—
232#492—611no—
233#492—611no—
234#431—611no—
235#431C2×D6111yescharacteristic
236C2×C4×C8—611no—
237#382—611no—
238#382C2×D6111yescharacteristic
239#340—1221no—
240#340—1221no—
241#327—22no—
242#327—1222no—
243#327—1222no—
244#327—1222no—
245#322—1221no—
246#322—1221no—
247C61#976011yescharacteristic, complemented, Sylow
248C2×#41—21no—
249C2×#41—1221no—
250#135—2442no—
251#135—2442no—
252#135—2442no—
253#135—2442no—
254C2×Q16—1221no—
255C2×Q16—1221no—
256C2×SD16—1221no—
257C2×SD16—1221no—
258C2×SD16—1221no—
259C2×SD16—1221no—
260C2×D8—1221no—
261C2×D8—1221no—
262C22×C8—611no—
263C22×C8—611no—
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×C4⋊C4—1221no—
280C2×C42C22×D6111yescharacteristic
281C4⋊C8—21no—
282C4⋊C8—1221no—
283C22⋊C8—612no—
284C22⋊C8C22⋊D6112yes—
285C4×C8—21no—
286C4×C8—611no—
287C4×C8—611no—
288C4×C8—611no—
289C4×C8—611no—
290C4×C8—1221no—
291#95—21no—
292#95—21no—
293#95—1221no—
294#95—1221no—
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—41no—
314C22×C4—41no—
315C22×C4—2441no—
316C22×C4—2441no—
317C22×C4D4×D6111yescharacteristic
318C22×C4D4×D6111yescharacteristic
319C22×C4C2×C22⋊D6111yescharacteristic
320Q16—2442no—
321Q16—2442no—
322SD16—2442no—
323SD16—2442no—
324SD16—2442no—
325SD16—2442no—
326D8—2442no—
327D8—2442no—
328C2×C8—22no—
329C2×C8—611no—
330C2×C8—611no—
331C2×C8—611no—
332C2×C8—611no—
333C2×C8—1221no—
334C2×C8—1221no—
335C2×C8—1221no—
336C2×C8—1221no—
337C2×C8—1222no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—1221no—
343C4⋊C4—1221no—
344C4⋊C4—2441no—
345C4⋊C4—2441no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—2441no—
349C22⋊C4—2441no—
350C42C2×C4×D6111yescharacteristic
351C42#1175822211yescharacteristic
352C42D4×D6111yescharacteristic
353C42#1175822411yescharacteristic
354C23—41no—
355C23—2441no—
356C23#1451977311yescharacteristic
357Q8—41no—
358Q8—41no—
359Q8—2441no—
360Q8—2441no—
361D4—41no—
362D4—41no—
363D4—81no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—4881no—
368D4—4881no—
369D4—4881no—
370D4—4881no—
371D4—2441no—
372D4—2441no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—4881no—
382C2×C4—4881no—
383C2×C4—4881no—
384C2×C4—4881no—
385C2×C4—4881no—
386C2×C4—2441no—
387C2×C4C22⋊C4×D6111yescharacteristic
388C2×C4#1451973711yescharacteristic
389C2×C4C4×C22⋊D6111yescharacteristic
390C2×C4#1451974911yescharacteristic
391C2×C4#1451975011yescharacteristic
392C2×C4#1451976311yescharacteristic
393C2×C4#1451980111yescharacteristic
394C2×C4#1451980111yescharacteristic
395C2×C4#1451980411yescharacteristic
396C2×C4#1451980411yescharacteristic
397C8—42no—
398C8—1221no—
399C8—1221no—
400C8—1221no—
401C8—1221no—
402C8—2442no—
403C22—21no—
404C22—21no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—4881no—
409C22—4881no—
410C22—4881no—
411C22#1683966211yescharacteristic
412C22#1684014811yescharacteristic
413C22#1684015011yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—81no—
419C4—161no—
420C4—9761no—
421C4—4881no—
422C4#1683976511yescharacteristic
423C4#1683976511yescharacteristic
424C4#1684012511yescharacteristic
425C4#1684012511yescharacteristic
426C2—21no—
427C2—21no—
428C2—161no—
429C2—9761no—
430C2#2144664111yescharacteristic
431C2#2144671411yescharacteristic
432C2#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#9760
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. #21445949C2semidirect
3. #21444380C2semidirect
4. #21444379C2semidirect
5. #21444378C2semidirect
6. #21444112C2semidirect
7. C61×#1517C2semidirect
13. #16840141C4semidirect
14. #16840138C4semidirect
15. #16840136C4semidirect
16. #16840134C4semidirect
215. #1517D61semidirect
247. C61#9760semidirect

Maximal subgroups

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

subgroupindex
#214562452
#214459492
#214443802
#214443792
#214443782
#214441122
C61×#15172
#976061