Finite Group Databasebeta

← #29662733

Subgroups — #29662733

Order 15616.

596 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#21456405C211yesmaximal, characteristic, complemented
2#21455607C212yesmaximal
3#21445401C212yesmaximal, complemented
4#21445394C212yesmaximal, complemented
5#21445347C211yesmaximal, characteristic, complemented
6#21445342C211yesmaximal, characteristic, complemented
7#21445313C211yesmaximal, characteristic, complemented
8#21445308C211yesmaximal, characteristic, complemented
9C2×#16839513C211yesmaximal, characteristic, complemented
10#21444474C212yesmaximal, complemented
11C61×#2965C211yesmaximal, characteristic
12#16840319C2212yes—
13#16840167C2211yescharacteristic
14#16840142C2211yescharacteristic
15#16840113C2212yes—
16#16840069—22no—
17#16840062—22no—
18#16840057C2212yes—
19#16840052C2212yes—
20#16840047C2212yes—
21#16840042C2212yes—
22#16840003—21no—
23#16840003—21no—
24C4×Dic244—21no—
25C4×Dic244—21no—
26#16839798—22no—
27C2×C4⋊#92418C2212yes—
28#16839758—22no—
29#16839513C2212yes—
30#16839513C2211yescharacteristic
31#16839513C2211yescharacteristic
32#16839510C2212yes—
33#16839510C2212yes—
34#16839509C2212yes—
35#16839509C2212yes—
36C4×#41×C61C2211yescharacteristic
37C61×#446C2211yescharacteristic
38C61×#446C2211yescharacteristic
39C61×#436C2211yescharacteristic
40C61×#436C2211yescharacteristic
41C61×#435—21no—
42C61×#435—21no—
43C61×#431—21no—
44C61×#431C2211yescharacteristic
45C4×M16×C61—21no—
46C2×C4×C8×C61C2211yescharacteristic
47C2×Dic244—21no—
48C2×Dic244—21no—
49C22×#92418—22no—
50#14519777C2312yes—
51#14519777C2312yes—
52#14519769—22no—
53#14519756—22no—
54#14519756—22no—
55C4×Dic122—22no—
56C4×Dic122—22no—
57#14519736—22no—
58#14519711—21no—
59#14519711—21no—
60#14519710—22no—
61#14519710—22no—
62#14519708—22no—
63#14519708—22no—
64C4⋊#92418—22no—
65C4⋊#92418C2312yes—
66C4⋊#92418C2312yes—
67#14519702—22no—
68#14519702—22no—
69#14519701—22no—
70#14519701—22no—
71#14519698—22no—
72#14519698—22no—
73C2×#41×C61D411yescharacteristic
74C2×M16×C61—21no—
75C2×M16×C61—21no—
76C22×C8×C61D411yescharacteristic
77C22×C8×C61D411yescharacteristic
78C4×Q8×C61D411yescharacteristic
79C4×Q8×C61C2311yescharacteristic
80C4×D4×C61—21no—
81C4×D4×C61—21no—
82C4×D4×C61D411yescharacteristic
83C4×D4×C61C2311yescharacteristic
84#117×C61—21no—
85#117×C61D411yescharacteristic
86C2×C42×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
102#41×C61—24no—
103C2×Q8×C61C2×D411yescharacteristic
104C2×D4×C61—21no—
105C2×D4×C61C2×D411yescharacteristic
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C2×Dic122—22no—
113C2×Dic122—22no—
114M16×C61—42no—
115M16×C61—42no—
116C2×C8×C61—22no—
117C2×C8×C61—21no—
118C2×C8×C61—21no—
119C2×C8×C61—41no—
120C2×C8×C61C2×D411yescharacteristic
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61C2×D411yescharacteristic
123C2×C8×C61C2×D411yescharacteristic
124C4⋊C4×C61—21no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61C2×D411yescharacteristic
127C4⋊C4×C61C2×D411yescharacteristic
128C22⋊C4×C61—21no—
129C22⋊C4×C61—21no—
130C22⋊C4×C61—21no—
131C42×C61—21no—
132C42×C61—21no—
133C42×C61—21no—
134C42×C61C2×D411yescharacteristic
135C42×C61C2×D411yescharacteristic
136C42×C61#4111yescharacteristic
137C42×C61C2411yescharacteristic
138C22⋊Dic61—42no—
139C4⋊Dic61—22no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61—22no—
143#11758196—42no—
144#11758196—42no—
145C4×Dic61—42no—
146C2×#92418—22no—
147C2×#92418—22no—
148C2×#92418—44no—
149Dic244—42no—
150Dic244—42no—
151C23×C61—21no—
152C23×C61#12111yescharacteristic
153Q8×C61—22no—
154D4×C61—22no—
155D4×C61—42no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—41no—
162C2×C4×C61—42no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—42no—
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#13714yes—
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C22×D411yescharacteristic
172C2×C4×C61C2×#4111yescharacteristic
173C2×Dic61—42no—
174C2×Dic61—42no—
175Dic122—42no—
176Dic122—42no—
177Dic122—82no—
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#19158—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—41no—
190C22×C61—41no—
191C22×C61—41no—
192C22×C61#47012yes—
193C22×C61C2×#12111yescharacteristic
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—41no—
198C4×C61—41no—
199C4×C61—41no—
200C4×C61#47012yes—
201C4×C61C2×#13511yescharacteristic
202C4×C61C2×#13712yes—
203C4×C61#57511yescharacteristic
204Dic61—82no—
205Dic61—82no—
206#3170—611no—
207#3076—612no—
208#2965D6111yescharacteristic
209#1704—612no—
210#1703—612no—
211#1696—611no—
212#1695—611no—
213#1690—611no—
214#1689—611no—
215C2×#334—611no—
216#1542—612no—
217C2×C61—22no—
218C2×C61—81no—
219C2×C61#303511yescharacteristic
220C2×C61C2×#47011yescharacteristic
221C2×C61#306711yescharacteristic
222#533—612no—
223C4×#41C2×D6111yescharacteristic
224#500—611no—
225#494—611no—
226#487—612no—
227#479—1222no—
228#478—1222no—
229#477—612no—
230#476—612no—
231#475—612no—
232#474—612no—
233#467—1221no—
234#467—1221no—
235#446C2×D6111yescharacteristic
236#446C2×D6111yescharacteristic
237C4×Q16—1221no—
238C4×Q16—1221no—
239#436C2×D6111yescharacteristic
240#436C2×D6111yescharacteristic
241#435—21no—
242#435—21no—
243#431—21no—
244#431C2×D6111yescharacteristic
245#427—1222no—
246C2×C4⋊C8—612no—
247#416—1222no—
248C4×M16—21no—
249C2×C4×C8C2×D6111yescharacteristic
250#334—611no—
251#334—611no—
252#334—612no—
253#333—612no—
254#333—612no—
255#332—612no—
256#332—612no—
257C61#1915811yescharacteristic, complemented, Sylow
258C2×#41D24411yescharacteristic
259C2×Q16—1221no—
260C2×Q16—1221no—
261C2×M16—21no—
262C2×M16—21no—
263C22×C8—1222no—
264C22×C8C22⋊D6111yescharacteristic
265C22×C8C22⋊D6111yescharacteristic
266C4⋊Q8—612no—
267C4⋊Q8—612no—
268#125—1222no—
269C22⋊Q8—1222no—
270C22⋊Q8—1222no—
271C4×Q8—1222no—
272C4×Q8—1222no—
273C4×Q8D24411yescharacteristic
274C4×Q8C22×D6111yescharacteristic
275C4×D4—21no—
276C4×D4—21no—
277C4×D4D24411yescharacteristic
278C4×D4C22×D6111yescharacteristic
279#117—21no—
280#117—1222no—
281#117D24411yescharacteristic
282C2×C42—21no—
283C2×C42C22×D6111yescharacteristic
284#107—1221no—
285#107—1221no—
286#107—1222no—
287#107—1222no—
288#106—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#102—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
313#41—24no—
314C2×Q8—1222no—
315C2×Q8—1222no—
316C2×Q8C2×D24411yescharacteristic
317C2×D4—21no—
318C2×D4C2×D24411yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4C2×D24411yescharacteristic
323C22×C4#1175822211yescharacteristic
324C22×C4C2×C22⋊D6111yescharacteristic
325Q16—2442no—
326Q16—2442no—
327M16—42no—
328M16—42no—
329C2×C8—22no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—41no—
333C2×C8—1222no—
334C2×C8—1222no—
335C2×C8—2444no—
336C2×C8C2×C22⋊D6111yescharacteristic
337C2×C8C2×C22⋊D6111yescharacteristic
338C2×C8C2×C22⋊D6111yescharacteristic
339C2×C8C2×C22⋊D6111yescharacteristic
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—1222no—
343C4⋊C4—1222no—
344C4⋊C4—1222no—
345C4⋊C4—1222no—
346C4⋊C4—2442no—
347C4⋊C4—2442no—
348C4⋊C4C2×D24411yescharacteristic
349C4⋊C4C2×D24411yescharacteristic
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—2442no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—2442no—
358C42C2×D24411yescharacteristic
359C42#1175822211yescharacteristic
360C42C2×C22⋊D6111yescharacteristic
361C42C23×D6111yescharacteristic
362C23—21no—
363C23#1451975111yescharacteristic
364Q8—22no—
365Q8—4882no—
366Q8—2442no—
367Q8—2442no—
368D4—22no—
369D4—42no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4#1451975111yescharacteristic
383C2×C4#1451975111yescharacteristic
384C2×C4#1451975111yescharacteristic
385C2×C4#1451981014yes—
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#1684002012yes—
403C22C2×#1451975111yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—4882no—
411C4—4882no—
412C4#1684002012yes—
413C4C2×#1451979711yescharacteristic
414C4C2×#1451981012yes—
415C4#1684056011yescharacteristic
416C2—22no—
417C2—81no—
418C2#2145527411yescharacteristic
419C2C2×#1684002011yescharacteristic
420C2#2145553911yescharacteristic
subgroupG/H
centerC2×C4#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2965C2
socleC22×C61C2×#121
prime radicalC2×#41×C61D4
hypercenter#2965D61
hypocenterC61#19158
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2965

Complements

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

NH ≅ G/Ntype
1. #21456405C2semidirect
3. #21445401C2semidirect
4. #21445394C2semidirect
5. #21445347C2semidirect
6. #21445342C2semidirect
7. #21445313C2semidirect
8. #21445308C2semidirect
9. C2×#16839513C2semidirect
10. #21444474C2semidirect
257. C61#19158semidirect

Maximal subgroups

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

subgroupindex
#214564052
#214556072
#214454012
#214453942
#214453472
#214453422
#214453132
#214453082
C2×#168395132
#214444742
C61×#29652
#1915861