Finite Group Databasebeta

← #29662735

Subgroups — #29662735

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#21455609C212yesmaximal
3#21454785C211yesmaximal, characteristic
4#21445402C212yesmaximal, complemented
5#21445395C212yesmaximal, complemented
6#21445348C211yesmaximal, characteristic, complemented
7#21445343C211yesmaximal, characteristic, complemented
8#21445314C211yesmaximal, characteristic, complemented
9#21445309C211yesmaximal, characteristic, complemented
10#21444477C212yesmaximal, complemented
11C2×C61×#334C211yesmaximal, characteristic, complemented
12#16840319C2212yes—
13C4×#11758222C2211yescharacteristic
14#16840167C2211yescharacteristic
15#16840142C2211yescharacteristic
16#16840113C2212yes—
17#16840070—22no—
18#16840063—22no—
19#16840058C2212yes—
20#16840053C2212yes—
21#16840048C2212yes—
22#16840043C2212yes—
23#16840003—21no—
24#16840003—21no—
25#16839892C2211yescharacteristic
26#16839892C2211yescharacteristic
27C4×Dic244—21no—
28C4×Dic244—21no—
29#16839841C2211yescharacteristic
30#16839841C2211yescharacteristic
31#16839833—21no—
32#16839833—21no—
33#16839815C2211yescharacteristic
34#16839813—21no—
35#16839800—22no—
36#16839759—22no—
37C4×#11758189—21no—
38#16839510C2212yes—
39#16839510C2212yes—
40#16839507C2212yes—
41#16839507C2212yes—
42C2×C4⋊C8×C61C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C61×#334C2212yes—
45C61×#334C2211yescharacteristic
46C61×#334C2211yescharacteristic
47C2×#11758222D411yescharacteristic
48C2×Dic244—21no—
49C2×Dic244—21no—
50C2×#11758189—21no—
51C2×#11758189—21no—
52#14519777C2312yes—
53#14519777C2312yes—
54#14519769—22no—
55#14519756—22no—
56#14519756—22no—
57C4×Dic122—22no—
58C4×Dic122—22no—
59C4×Dic122D411yescharacteristic
60C4×Dic122C2311yescharacteristic
61C4×D244D411yescharacteristic
62C4×D244C2311yescharacteristic
63C4×C22⋊D61—21no—
64C4×C22⋊D61—21no—
65#14519736—22no—
66#14519736D411yescharacteristic
67#14519734—21no—
68C42×D61—21no—
69#14519711—21no—
70#14519711—21no—
71#14519711—22no—
72#14519711—22no—
73#14519709—22no—
74#14519709—22no—
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2311yescharacteristic
77#14519706—21no—
78#14519706—21no—
79#14519702—22no—
80#14519702—22no—
81#14519702—22no—
82#14519702—22no—
83#14519699—22no—
84#14519699—22no—
85C22⋊#92418—21no—
86#14519689—21no—
87#14519689—21no—
88#14519687—21no—
89#14519687—21no—
90C4×#92418—21no—
91C22×C8×C61—22no—
92C22×C8×C61D411yescharacteristic
93C22×C8×C61D411yescharacteristic
94C2×C42×C61C2311yescharacteristic
95C4⋊C8×C61—22no—
96C4⋊C8×C61C2312yes—
97C4⋊C8×C61C2312yes—
98C4×C8×C61D411yescharacteristic
99C4×C8×C61D411yescharacteristic
100C4×C8×C61C2311yescharacteristic
101C4×C8×C61C2311yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61#4111yescharacteristic
105C2×C22⋊D61—21no—
106#11758222—24no—
107C2×D244C2×D411yescharacteristic
108C2×C4×D61—21no—
109C2×C4×D61—21no—
110C2×C4×D61—21no—
111C2×Dic122—22no—
112C2×Dic122—22no—
113C2×Dic122C2×D411yescharacteristic
114C2×C8×C61—22no—
115C2×C8×C61—22no—
116C2×C8×C61—22no—
117C2×C8×C61—41no—
118C2×C8×C61—44no—
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61C2×D411yescharacteristic
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61C2×D411yescharacteristic
123C42×C61C2×D411yescharacteristic
124C42×C61C2×D411yescharacteristic
125C42×C61#4111yescharacteristic
126C42×C61C2411yescharacteristic
127C22⋊Dic61—21no—
128C22⋊Dic61—42no—
129#11758198—21no—
130#11758198—21no—
131C4⋊Dic61—22no—
132C4⋊Dic61—22no—
133C4⋊Dic61—22no—
134C4⋊Dic61—22no—
135C4⋊Dic61C2×D411yescharacteristic
136C4⋊Dic61C2×D411yescharacteristic
137#11758196—21no—
138#11758196—21no—
139#11758196—42no—
140#11758196—42no—
141C4×Dic61—21no—
142C4×Dic61—21no—
143C4×Dic61—21no—
144C4×Dic61—42no—
145C2×#92418—21no—
146C2×#92418—21no—
147Dic244—42no—
148Dic244—42no—
149#11758189—42no—
150#11758189—42no—
151C23×C61#12111yescharacteristic
152C22×D61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—41no—
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#12111yescharacteristic
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61#13714yes—
160C2×C4×C61C22×D411yescharacteristic
161C2×C4×C61C22×D411yescharacteristic
162C2×C4×C61C2×#4111yescharacteristic
163C22⋊D61—42no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—21no—
167C2×Dic61—42no—
168C2×Dic61—42no—
169D244—22no—
170C4×D61—42no—
171C4×D61—41no—
172C4×D61—41no—
173C4×D61—42no—
174Dic122—22no—
175Dic122—42no—
176Dic122—42no—
177Dic122—82no—
178C8×C61—21no—
179C8×C61—21no—
180C8×C61—21no—
181C8×C61—42no—
182C8×C61—42no—
183C8×C61#13512yes—
184#92418—41no—
185#92418—41no—
186#19158—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61#47012yes—
190C22×C61C2×#12111yescharacteristic
191C2×D61—41no—
192C2×D61—41no—
193C2×D61—41no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61#47012yes—
198C4×C61C2×#13511yescharacteristic
199C4×C61C2×#13712yes—
200C4×C61#57511yescharacteristic
201Dic61—41no—
202Dic61—41no—
203Dic61—41no—
204Dic61—82no—
205Dic61—82no—
206#3170—611no—
207#3076—612no—
208#2965—611no—
209#1704—612no—
210#1703—612no—
211#1696—611no—
212#1695—611no—
213#1690—611no—
214#1689—611no—
215C2×#334D6111yescharacteristic, complemented
216#1542—612no—
217C2×C61—22no—
218C2×C61#303511yescharacteristic
219C2×C61C2×#47011yescharacteristic
220C2×C61#306711yescharacteristic
221D61—81no—
222#533—612no—
223C4×#41—611no—
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#446—611no—
236#446—611no—
237C4×Q16—1221no—
238C4×Q16—1221no—
239#436—611no—
240#436—611no—
241#435—1221no—
242#435—1221no—
243#431—611no—
244#431—1221no—
245#427—1222no—
246C2×C4⋊C8C2×D6112yes—
247#416—1222no—
248C4×M16—1221no—
249C2×C4×C8C2×D6111yescharacteristic
250#334C2×D6111yescharacteristic
251#334C2×D6111yescharacteristic
252#334C2×D6112yes—
253#333—612no—
254#333—612no—
255#332—612no—
256#332—612no—
257C61#1915811yescharacteristic, complemented, Sylow
258C2×#41—611no—
259C2×Q16—1221no—
260C2×Q16—1221no—
261C2×M16—1221no—
262C2×M16—1221no—
263C22×C8—22no—
264C22×C8D24411yescharacteristic
265C22×C8D24411yescharacteristic
266C4⋊Q8—612no—
267C4⋊Q8—612no—
268#125—1222no—
269C22⋊Q8—1222no—
270C22⋊Q8—1222no—
271C4×Q8—611no—
272C4×Q8—611no—
273C4×Q8—1222no—
274C4×Q8—1222no—
275C4×D4—611no—
276C4×D4—611no—
277C4×D4—1221no—
278C4×D4—1221no—
279#117—611no—
280#117—1221no—
281#117—1222no—
282C2×C42—1221no—
283C2×C42C22×D6111yescharacteristic
284#107—1222no—
285#107—1222no—
286#107—1221no—
287#107—1221no—
288#106—1222no—
289#106—1222no—
290C4⋊C8—22no—
291C4⋊C8—611no—
292C4⋊C8—611no—
293C4⋊C8—1221no—
294C4⋊C8—1221no—
295C4⋊C8C22×D6112yes—
296C4⋊C8C22×D6112yes—
297#103—1222no—
298#103—1222no—
299#103—1222no—
300#103—1222no—
301#102—1222no—
302#102—1222no—
303C22⋊C8—1221no—
304C22⋊C8—1221no—
305C22⋊C8—1221no—
306#97—1221no—
307#97—1221no—
308C4×C8—1221no—
309C4×C8D24411yescharacteristic
310C4×C8D24411yescharacteristic
311C4×C8C22×D6111yescharacteristic
312C4×C8C22×D6111yescharacteristic
313#41—1224no—
314C2×Q8—611no—
315C2×Q8—1222no—
316C2×Q8—1222no—
317C2×D4—611no—
318C2×D4—1221no—
319C22×C4—1221no—
320C22×C4—1221no—
321C22×C4—1221no—
322C22×C4C2×D24411yescharacteristic
323C22×C4D4×D6111yescharacteristic
324C22×C4#1175822611yescharacteristic
325Q16—2442no—
326Q16—2442no—
327M16—2442no—
328M16—2442no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—41no—
333C2×C8—44no—
334C2×C8—1221no—
335C2×C8—1221no—
336C2×C8C2×D24411yescharacteristic
337C2×C8C2×D24411yescharacteristic
338C2×C8C2×D24411yescharacteristic
339C2×C8C2×D24411yescharacteristic
340C4⋊C4—611no—
341C4⋊C4—611no—
342C4⋊C4—1221no—
343C4⋊C4—1221no—
344C4⋊C4—1222no—
345C4⋊C4—1222no—
346C4⋊C4—1222no—
347C4⋊C4—1222no—
348C4⋊C4—2442no—
349C4⋊C4—2442no—
350C22⋊C4—1221no—
351C22⋊C4—1221no—
352C22⋊C4—1221no—
353C22⋊C4—2442no—
354C42—1221no—
355C42—1221no—
356C42—1221no—
357C42—2442no—
358C42C2×D24411yescharacteristic
359C42D4×D6111yescharacteristic
360C42#1175822611yescharacteristic
361C42C23×D6111yescharacteristic
362C23—1221no—
363C23#1451975211yescharacteristic
364Q8—1222no—
365Q8—4882no—
366Q8—2442no—
367Q8—2442no—
368D4—1222no—
369D4—2442no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—41no—
373C2×C4—1221no—
374C2×C4—1221no—
375C2×C4—1221no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4—2442no—
379C2×C4—2442no—
380C2×C4—2441no—
381C2×C4—2441no—
382C2×C4#1451975211yescharacteristic
383C2×C4#1451975211yescharacteristic
384C2×C4#1451975211yescharacteristic
385C2×C4#1451980614yes—
386C2×C4C2×D4×D6111yescharacteristic
387C2×C4C22×D24411yescharacteristic
388C2×C4C2×#1175822611yescharacteristic
389C8—21no—
390C8—21no—
391C8—21no—
392C8—42no—
393C8—42no—
394C8—2441no—
395C8—2441no—
396C8#1451979912yes—
397C22—21no—
398C22—21no—
399C22—2441no—
400C22—2441no—
401C22—2441no—
402C22#1684002112yes—
403C22C2×#1451975211yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—4882no—
408C4—4882no—
409C4—2441no—
410C4—2441no—
411C4—2441no—
412C4#1684002112yes—
413C4C2×#1451979911yescharacteristic
414C4C2×#1451980612yes—
415C4#1684056211yescharacteristic
416C2—22no—
417C2—4881no—
418C2#2145527611yescharacteristic
419C2C2×#1684002111yescharacteristic
420C2#2145554011yescharacteristic
subgroupG/H
centerC2×C4#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#334C2
socleC22×C61C2×#121
prime radicalC2×#11758222D4
hypercenterC2×#334D61
hypocenterC61#19158
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#334

Complements

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

NH ≅ G/Ntype
1. #21456405C2semidirect
4. #21445402C2semidirect
5. #21445395C2semidirect
6. #21445348C2semidirect
7. #21445343C2semidirect
8. #21445314C2semidirect
9. #21445309C2semidirect
10. #21444477C2semidirect
11. C2×C61×#334C2semidirect
215. C2×#334D61semidirect
257. C61#19158semidirect

Maximal subgroups

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

subgroupindex
#214564052
#214556092
#214547852
#214454022
#214453952
#214453482
#214453432
#214453142
#214453092
#214444772
C2×C61×#3342
#1915861