Finite Group Databasebeta

← #29663471

Subgroups — #29663471

Order 15616.

584 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457156C212yesmaximal
2#21456405C211yesmaximal, characteristic, complemented
3#21445337C211yesmaximal, characteristic, complemented
4#21445332C211yesmaximal, characteristic, complemented
5#21445313C211yesmaximal, characteristic, complemented
6#21445308C211yesmaximal, characteristic, complemented
7#21445249C212yesmaximal, complemented
8#21445242C212yesmaximal, complemented
9C2×#16839513C211yesmaximal, characteristic, complemented
10#21444520C212yesmaximal, complemented
11C61×#2966C211yesmaximal, characteristic
12#16840319C2212yes—
13#16840167C2211yescharacteristic
14#16840142C2211yescharacteristic
15#16840113C2212yes—
16#16840066—22no—
17#16840047C2212yes—
18#16840042C2212yes—
19#16840003—21no—
20#16840003—21no—
21#16840001—22no—
22#16839989—22no—
23#16839981—22no—
24#16839972C2212yes—
25#16839967C2212yes—
26C4×Dic244—21no—
27C4×Dic244—21no—
28C2×C4⋊#92418C2212yes—
29#16839513C2212yes—
30#16839513C2211yescharacteristic
31#16839513C2211yescharacteristic
32#16839509C2212yes—
33#16839509C2212yes—
34#16839507C2212yes—
35#16839507C2212yes—
36C61×#529C2211yescharacteristic
37C61×#446C2211yescharacteristic
38C8×Q8×C61C2211yescharacteristic
39C61×#436C2211yescharacteristic
40C61×#435—21no—
41C8×D4×C61—21no—
42C8×D4×C61C2211yescharacteristic
43C61×#433—21no—
44C61×#432C2211yescharacteristic
45C61×#405—21no—
46C2×C4×C8×C61C2211yescharacteristic
47C2×Dic244—21no—
48C2×Dic244—21no—
49C2×#11758202—22no—
50C2×#11758200—22no—
51C22×#92418—22no—
52#14519777C2312yes—
53#14519777C2312yes—
54#14519769—22no—
55#14519756—22no—
56#14519756—22no—
57C4×Dic122—22no—
58C4×Dic122—22no—
59#14519736—22no—
60#14519711—21no—
61#14519711—21no—
62#14519710—22no—
63#14519708—22no—
64C4⋊#92418—22no—
65C4⋊#92418C2312yes—
66C4⋊#92418C2312yes—
67#14519702—22no—
68#14519702—22no—
69#14519701—22no—
70#14519700—22no—
71#14519700—22no—
72#14519698—22no—
73C2×M16×C61—21no—
74C22×C8×C61—21no—
75C22×C8×C61D411yescharacteristic
76C22×C8×C61D411yescharacteristic
77#126×C61—21no—
78#125×C61D411yescharacteristic
79#124×C61D411yescharacteristic
80#123×C61—21no—
81C22⋊Q8×C61D411yescharacteristic
82#121×C61D411yescharacteristic
83C4×Q8×C61C2311yescharacteristic
84C4×D4×C61—21no—
85C4×D4×C61C2311yescharacteristic
86#117×C61—21no—
87C2×C42×C61C2311yescharacteristic
88C4⋊C8×C61—21no—
89C4⋊C8×C61—21no—
90C4⋊C8×C61C2311yescharacteristic
91C4⋊C8×C61C2311yescharacteristic
92C22⋊C8×C61—21no—
93C22⋊C8×C61—21no—
94C22⋊C8×C61—21no—
95#97×C61—21no—
96#97×C61—21no—
97C4×C8×C61—21no—
98C4×C8×C61D411yescharacteristic
99C4×C8×C61D411yescharacteristic
100C4×C8×C61C2311yescharacteristic
101C4×C8×C61C2311yescharacteristic
102C2×Q8×C61C2×D411yescharacteristic
103C2×D4×C61—21no—
104C2×D4×C61C2×D411yescharacteristic
105C22×C4×C61—21no—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61#4111yescharacteristic
109C2×Dic122—22no—
110C2×Dic122—22no—
111M16×C61—42no—
112C2×C8×C61—22no—
113C2×C8×C61—21no—
114C2×C8×C61—21no—
115C2×C8×C61—41no—
116C2×C8×C61—42no—
117C2×C8×C61C2×D411yescharacteristic
118C2×C8×C61C2×D411yescharacteristic
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61C2×D411yescharacteristic
121C4⋊C4×C61—21no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61—21no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61C2×D411yescharacteristic
126C4⋊C4×C61C2×D411yescharacteristic
127C22⋊C4×C61—21no—
128C22⋊C4×C61—21no—
129C22⋊C4×C61—21no—
130C22⋊C4×C61—21no—
131C22⋊C4×C61—21no—
132C42×C61—21no—
133C42×C61C2×D411yescharacteristic
134C42×C61C2×D411yescharacteristic
135C42×C61#4111yescharacteristic
136C42×C61C2411yescharacteristic
137C22⋊Dic61—42no—
138#11758202—82no—
139#11758200—82no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61—22no—
143C4⋊Dic61—22no—
144#11758196—42no—
145#11758196—42no—
146C4×Dic61—42no—
147C2×#92418—22no—
148C2×#92418—22no—
149C2×#92418—82no—
150Dic244—42no—
151Dic244—42no—
152C23×C61—21no—
153C23×C61#12111yescharacteristic
154Q8×C61—22no—
155D4×C61—22no—
156D4×C61—42no—
157C2×C4×C61—22no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—22no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61C22×D411yescharacteristic
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C2×#4111yescharacteristic
172C2×Dic61—42no—
173C2×Dic61—42no—
174Dic122—42no—
175Dic122—42no—
176Dic122—42no—
177Dic122—42no—
178C8×C61—21no—
179C8×C61—21no—
180C8×C61—21no—
181C8×C61—41no—
182C8×C61—41no—
183C8×C61#13512yes—
184#92418—42no—
185#92418—42no—
186#19228—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—41no—
190C22×C61—41no—
191C22×C61—41no—
192C22×C61#47312yes—
193C22×C61C2×#12111yescharacteristic
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—41no—
199C4×C61—41no—
200C4×C61—41no—
201C4×C61C2×#13511yescharacteristic
202C4×C61#57511yescharacteristic
203C4×C61#57712yes—
204Dic61—82no—
205Dic61—82no—
206#3248—612no—
207#3170—611no—
208#2966D6111yescharacteristic
209#1694—611no—
210#1693—611no—
211#1690—611no—
212#1689—611no—
213#1680—612no—
214#1679—612no—
215C2×#334—611no—
216#1552—612no—
217C2×C61—22no—
218C2×C61—81no—
219C2×C61#303511yescharacteristic
220C2×C61#306011yescharacteristic
221C2×C61C2×#47311yescharacteristic
222#533—612no—
223#529C2×D6111yescharacteristic
224#500—611no—
225#494—611no—
226#487—612no—
227#479—1222no—
228#475—612no—
229#474—612no—
230#467—1221no—
231#467—1221no—
232#467—1222no—
233#465—1222no—
234#464—1222no—
235#462—612no—
236#461—612no—
237#446C2×D6111yescharacteristic
238C8×Q8C2×D6111yescharacteristic
239C4×Q16—1221no—
240C4×Q16—1221no—
241#436C2×D6111yescharacteristic
242#435—21no—
243C8×D4—21no—
244C8×D4C2×D6111yescharacteristic
245#433—21no—
246#432C2×D6111yescharacteristic
247C2×C4⋊C8—612no—
248#405—21no—
249C2×C4×C8C2×D6111yescharacteristic
250#334—612no—
251#334—611no—
252#334—611no—
253#332—612no—
254#332—612no—
255#332—612no—
256#332—612no—
257C61#1922811yescharacteristic, complemented, Sylow
258C2×Q16—1222no—
259C2×Q16—1221no—
260C2×Q16—1221no—
261C2×SD16—1222no—
262C2×M16—21no—
263C22×C8—21no—
264C22×C8—1222no—
265C22×C8C22⋊D6111yescharacteristic
266C22×C8C22⋊D6111yescharacteristic
267C4⋊Q8—612no—
268C4⋊Q8—612no—
269#126—21no—
270#125—1222no—
271#125D24411yescharacteristic
272#124D24411yescharacteristic
273#123—21no—
274C22⋊Q8—1222no—
275C22⋊Q8—1222no—
276C22⋊Q8D24411yescharacteristic
277#121D24411yescharacteristic
278C4×Q8—1222no—
279C4×Q8—1222no—
280C4×Q8C22×D6111yescharacteristic
281C4×D4—21no—
282C4×D4C22×D6111yescharacteristic
283#117—21no—
284#117—1222no—
285C2×C42C22×D6111yescharacteristic
286#107—1221no—
287#107—1221no—
288#107—1222no—
289#106—1222no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—612no—
293C4⋊C8—612no—
294C4⋊C8—1222no—
295C4⋊C8C22×D6111yescharacteristic
296C4⋊C8C22×D6111yescharacteristic
297#103—1222no—
298#103—1222no—
299#103—1222no—
300#103—1222no—
301#103—1222no—
302#102—1222no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306#97—21no—
307#97—21no—
308C4×C8—21no—
309C4×C8C22⋊D6111yescharacteristic
310C4×C8C22⋊D6111yescharacteristic
311C4×C8C22×D6111yescharacteristic
312C4×C8C22×D6111yescharacteristic
313C2×Q8—1222no—
314C2×Q8—1222no—
315C2×Q8C2×D24411yescharacteristic
316C2×D4—21no—
317C2×D4C2×D24411yescharacteristic
318C22×C4—21no—
319C22×C4C2×D24411yescharacteristic
320C22×C4#1175822211yescharacteristic
321C22×C4C2×C22⋊D6111yescharacteristic
322Q16—4882no—
323Q16—2442no—
324Q16—2442no—
325SD16—4882no—
326M16—42no—
327C2×C8—22no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—1222no—
333C2×C8—1222no—
334C2×C8—4882no—
335C2×C8C2×C22⋊D6111yescharacteristic
336C2×C8C2×C22⋊D6111yescharacteristic
337C2×C8C2×C22⋊D6111yescharacteristic
338C2×C8C2×C22⋊D6111yescharacteristic
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—1222no—
344C4⋊C4—1222no—
345C4⋊C4—1222no—
346C4⋊C4—1222no—
347C4⋊C4—2442no—
348C4⋊C4—2442no—
349C4⋊C4C2×D24411yescharacteristic
350C4⋊C4C2×D24411yescharacteristic
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—21no—
354C22⋊C4—21no—
355C22⋊C4—21no—
356C22⋊C4—2442no—
357C42—21no—
358C42—2442no—
359C42C2×D24411yescharacteristic
360C42#1175822211yescharacteristic
361C42C2×C22⋊D6111yescharacteristic
362C42C23×D6111yescharacteristic
363C23—21no—
364C23#1451975111yescharacteristic
365Q8—22no—
366Q8—2442no—
367Q8—2442no—
368Q8—2442no—
369Q8—2442no—
370D4—22no—
371D4—42no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—2442no—
382C2×C4—2442no—
383C2×C4#1451975111yescharacteristic
384C2×C4#1451975111yescharacteristic
385C2×C4#1451975111yescharacteristic
386C2×C4C22×C22⋊D6111yescharacteristic
387C2×C4C22×D24411yescharacteristic
388C2×C4C2×#1175822211yescharacteristic
389C8—21no—
390C8—21no—
391C8—21no—
392C8—41no—
393C8—41no—
394C8—2442no—
395C8—2442no—
396C8#1451979712yes—
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22#1684003712yes—
403C22C2×#1451975111yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—4882no—
412C4—4882no—
413C4C2×#1451979711yescharacteristic
414C4#1684056011yescharacteristic
415C4#1684057412yes—
416C2—22no—
417C2—81no—
418C2#2145527411yescharacteristic
419C2#2145548111yescharacteristic
420C2C2×#1684003711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2966C2
socleC22×C61C2×#121
prime radical#121×C61D4
hypercenter#2966D61
hypocenterC61#19228
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2966

Complements

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

NH ≅ G/Ntype
2. #21456405C2semidirect
3. #21445337C2semidirect
4. #21445332C2semidirect
5. #21445313C2semidirect
6. #21445308C2semidirect
7. #21445249C2semidirect
8. #21445242C2semidirect
9. C2×#16839513C2semidirect
10. #21444520C2semidirect
257. C61#19228semidirect

Maximal subgroups

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

subgroupindex
#214571562
#214564052
#214453372
#214453322
#214453132
#214453082
#214452492
#214452422
C2×#168395132
#214445202
C61×#29662
#1922861