Finite Group Databasebeta

← #29576107

Subgroups — #29576107

Order 15616.

456 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455601C211yesmaximal, characteristic
2#21454780C211yesmaximal, characteristic
3#21445840C211yesmaximal, characteristic
4#21444274C211yesmaximal, characteristic, complemented
5#21443715C211yesmaximal, characteristic, complemented
6#21443660C211yesmaximal, characteristic, complemented
7C61×#1503C211yesmaximal, characteristic, complemented
8C4×#11758222C2211yescharacteristic
9#16840066—21no—
10#16840059—21no—
11#16840054C411yescharacteristic, complemented
12#16840049C411yescharacteristic, complemented
13#16840044C411yescharacteristic, complemented
14#16840039C411yescharacteristic, complemented
15#16839889—21no—
16#16839839—21no—
17#16839830—22no—
18#16839813—21no—
19#16839798—21no—
20#16839786C2211yescharacteristic
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519705—21no—
23#16839756—21no—
24C4×#11758189—21no—
25C2×C4×#92418C2211yescharacteristic
26#16839522—21no—
27Q8⋊#92418—21no—
28Q8⋊#92418—21no—
29C61×#533C2211yescharacteristic
30C61×#431C2211yescharacteristic
31C4×C4⋊C4×C61C2211yescharacteristic
32C61×#328—22no—
33C2×#11758222D411yescharacteristic
34C2×#11758194—21no—
35C2×#11758189—22no—
36C22×#92418—21no—
37C22×#92418—21no—
38C4×Dic122C2×C411yescharacteristic
39C4×Dic122D411yescharacteristic
40C4×D244C2×C411yescharacteristic
41C4×D244D411yescharacteristic
42C4×C22⋊D61—21no—
43C4×C22⋊D61—21no—
44#14519736D411yescharacteristic
45#14519734—21no—
46C42×D61—21no—
47#14519710—21no—
48#14519710—21no—
49#14519708—21no—
50#14519708—21no—
51C4⋊#92418—21no—
52C4⋊#92418—21no—
53C4⋊#92418—21no—
54C4⋊#92418C2×C411yescharacteristic
55C4⋊#92418C2×C411yescharacteristic
56#14519706—22no—
57#14519705—42no—
58#14519700—21no—
59#14519700—21no—
60#14519697—21no—
61#14519697—21no—
62#14519689—22no—
63#14519687—22no—
64C4×#92418—21no—
65C4×#92418D411yescharacteristic
66C4×#92418D411yescharacteristic
67C4⋊Q8×C61C2×C411yescharacteristic
68C4⋊Q8×C61C2×C411yescharacteristic
69#125×C61—21no—
70C22⋊Q8×C61—21no—
71C22⋊Q8×C61—21no—
72C4×Q8×C61—21no—
73C4×Q8×C61—21no—
74#117×C61—21no—
75C2×C4⋊C4×C61—21no—
76C2×C42×C61—21no—
77C2×C42×C61C2311yescharacteristic
78C4⋊C8×C61—21no—
79C22⋊C8×C61D412yes—
80C4×C8×C61—21no—
81#95×C61—21no—
82C2×Q8×C61—21no—
83C2×Q8×C61—21no—
84C22×C4×C61—41no—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61#4111yescharacteristic
88C2×C22⋊D61—21no—
89#11758222—21no—
90#11758222—21no—
91#11758222—21no—
92#11758222—21no—
93C2×D244C22⋊C411yescharacteristic
94C2×C4×D61—21no—
95C2×C4×D61—21no—
96C2×C4×D61—21no—
97C2×Dic122C22⋊C411yescharacteristic
98C2×C8×C61—22no—
99C4⋊C4×C61—22no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—41no—
105C4⋊C4×C61—41no—
106C4⋊C4×C61—41no—
107C22⋊C4×C61—41no—
108C42×C61—41no—
109C42×C61—41no—
110C42×C61—41no—
111C42×C61C22×C411yescharacteristic
112C42×C61C2×D411yescharacteristic
113C42×C61C2×D411yescharacteristic
114C42×C61#4111yescharacteristic
115C22⋊Dic61—21no—
116#11758198—21no—
117#11758198—21no—
118C4⋊Dic61C22⋊C411yescharacteristic
119C4⋊Dic61C22⋊C411yescharacteristic
120#11758196—21no—
121#11758196—21no—
122C4×Dic61—21no—
123C4×Dic61—21no—
124C4×Dic61—21no—
125#11758194—41no—
126#11758194—41no—
127C2×#92418—21no—
128C2×#92418—21no—
129C2×#92418—21no—
130C2×#92418—21no—
131C2×#92418—42no—
132C2×#92418—41no—
133C2×#92418—41no—
134C2×#92418—41no—
135#11758189—44no—
136C23×C61#12111yescharacteristic
137C22×D61—21no—
138Q8×C61—41no—
139Q8×C61—41no—
140Q8×C61—81no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—41no—
144C2×C4×C61—42no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—81no—
148C2×C4×C61—81no—
149C2×C4×C61C2×C22⋊C411yescharacteristic
150C2×C4×C61C4×D411yescharacteristic
151C2×C4×C61C4×D411yescharacteristic
152C2×C4×C61#12011yescharacteristic
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61#12311yescharacteristic
155C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C61#13711yescharacteristic
157C2×C4×C61#13711yescharacteristic
158C2×C4×C61#13711yescharacteristic
159C22⋊D61—41no—
160C22⋊D61—41no—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164D244—21no—
165D244—21no—
166C4×D61—42no—
167C4×D61—41no—
168C4×D61—41no—
169C4×D61—41no—
170C4×D61—41no—
171Dic122—21no—
172Dic122—21no—
173C8×C61—42no—
174#92418—22no—
175#92418—41no—
176#92418—41no—
177#92418—41no—
178#92418—81no—
179#9209—611nomaximal, Sylow
180C22×C61—21no—
181C22×C61—21no—
182C22×C61#38611yescharacteristic
183C22×C61#46811yescharacteristic
184C22×C61#47011yescharacteristic
185C2×D61—41no—
186C2×D61—41no—
187C2×D61—41no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—82no—
192C4×C61—81no—
193C4×C61—81no—
194C4×C61#41711yescharacteristic
195C4×C61#41911yescharacteristic
196C4×C61#44311yescharacteristic
197C4×C61#44411yescharacteristic
198C4×C61#46811yescharacteristic
199C4×C61#47011yescharacteristic
200Dic61—41no—
201Dic61—41no—
202Dic61—41no—
203#3076—611no—
204#2965—611no—
205#1806—611no—
206#1503D6111yescharacteristic, complemented
207#1493—611no—
208#1301—611no—
209#1279—611no—
210C2×C61—21no—
211C2×C61—21no—
212C2×C61#187611yescharacteristic
213C2×C61#189411yescharacteristic
214C2×C61#190611yescharacteristic
215D61—81no—
216#533C2×D6111yescharacteristic
217C4×#41—611no—
218#479—1221no—
219#478—1221no—
220#477—611no—
221#476—611no—
222#475—611no—
223#474—611no—
224#446—1221no—
225#436—1221no—
226#435—1222no—
227#431—1221no—
228#431C2×D6111yescharacteristic
229#427—1221no—
230#423—611no—
231C2×C4⋊C8—611no—
232C2×#104—1221no—
233#416—1221no—
234C4×M16—1221no—
235C2×C4×C8—611no—
236C4×C4⋊C4C2×D6111yescharacteristic
237#339—1221no—
238#328—22no—
239Q8⋊C8—1221no—
240Q8⋊C8—1221no—
241C61#920911yescharacteristic, complemented, Sylow
242C2×#41—611no—
243C2×M16—1222no—
244C2×M16—1221no—
245C22×C8—1221no—
246C22×C8—1221no—
247C4⋊Q8C4×D6111yescharacteristic
248C4⋊Q8C4×D6111yescharacteristic
249#125—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—611no—
255C4×Q8—611no—
256C4×D4—611no—
257C4×D4—611no—
258C4×D4—1221no—
259C4×D4—1221no—
260#117—21no—
261#117—611no—
262#117—1221no—
263C2×C4⋊C4—21no—
264C2×C42—21no—
265C2×C42—1221no—
266C2×C42C22×D6111yescharacteristic
267#107—1221no—
268#107—1221no—
269#106—1221no—
270#106—1221no—
271C4⋊C8—21no—
272C4⋊C8—611no—
273C4⋊C8—611no—
274C4⋊C8—1222no—
275C4⋊C8—1221no—
276C4⋊C8—1221no—
277C4⋊C8—1221no—
278#104—2442no—
279#103—1221no—
280#103—1221no—
281#102—1221no—
282#102—1221no—
283C22⋊C8—1222no—
284C22⋊C8C22⋊D6112yes—
285#97—1222no—
286C4×C8—21no—
287C4×C8—611no—
288C4×C8—611no—
289C4×C8—1221no—
290#95—21no—
291#41—1221no—
292#41—1221no—
293#41—1221no—
294#41—1221no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—611no—
298C2×D4—611no—
299C2×D4—1221no—
300C22×C4—41no—
301C22×C4—1221no—
302C22×C4—1221no—
303C22×C4—1221no—
304C22×C4D4×D6111yescharacteristic
305C22×C4#1175822411yescharacteristic
306C22×C4C2×C22⋊D6111yescharacteristic
307M16—2444no—
308M16—2441no—
309M16—2441no—
310C2×C8—22no—
311C2×C8—1221no—
312C2×C8—1221no—
313C2×C8—1221no—
314C2×C8—1221no—
315C2×C8—2442no—
316C2×C8—2441no—
317C2×C8—2441no—
318C2×C8—2441no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—611no—
328C4⋊C4—611no—
329C4⋊C4—1221no—
330C4⋊C4—1221no—
331C22⋊C4—41no—
332C22⋊C4—1221no—
333C22⋊C4—1221no—
334C22⋊C4—1221no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—1221no—
339C42—1221no—
340C42—1221no—
341C42C2×C4×D6111yescharacteristic
342C42#1175822211yescharacteristic
343C42D4×D6111yescharacteristic
344C42D4×D6111yescharacteristic
345C23—1221no—
346C23#1451974811yescharacteristic
347Q8—41no—
348Q8—41no—
349Q8—81no—
350Q8—1221no—
351Q8—1221no—
352D4—1221no—
353D4—1221no—
354D4—2441no—
355D4—2441no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—1221no—
365C2×C4—1221no—
366C2×C4—1221no—
367C2×C4—2442no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4C22⋊C4×D6111yescharacteristic
373C2×C4#1451973711yescharacteristic
374C2×C4C4×C22⋊D6111yescharacteristic
375C2×C4#1451974511yescharacteristic
376C2×C4#1451975011yescharacteristic
377C2×C4#1451975911yescharacteristic
378C2×C4#1451980411yescharacteristic
379C2×C4#1451980611yescharacteristic
380C2×C4#1451980911yescharacteristic
381C2×C4#1451980911yescharacteristic
382C8—42no—
383C8—1222no—
384C8—4881no—
385C8—2441no—
386C8—2441no—
387C8—2441no—
388C22—21no—
389C22—21no—
390C22—2441no—
391C22—2441no—
392C22—2441no—
393C22#1683963811yescharacteristic
394C22#1684000511yescharacteristic
395C22#1684001711yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—82no—
400C4—81no—
401C4—81no—
402C4—2441no—
403C4—2441no—
404C4—2441no—
405C4#1683976211yescharacteristic
406C4#1683976911yescharacteristic
407C4#1683987611yescharacteristic
408C4#1683988111yescharacteristic
409C4#1684001111yescharacteristic
410C4#1684001911yescharacteristic
411C2—21no—
412C2—21no—
413C2—4881no—
414C2#2144614311yescharacteristic
415C2#2144624911yescharacteristic
416C2#2144631711yescharacteristic
subgroupG/H
centerC2×C4#14519750
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1503C2
socleC22×C61#386
prime radicalC2×#11758222D4
cosocleC2×C42×C61C23
hypercenter#1503D61
hypocenterC61#9209
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
4. #21444274C2semidirect
5. #21443715C2semidirect
6. #21443660C2semidirect
7. C61×#1503C2semidirect
11. #16840054C4semidirect
12. #16840049C4semidirect
13. #16840044C4semidirect
14. #16840039C4semidirect
206. #1503D61semidirect
241. C61#9209semidirect

Maximal subgroups

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

subgroupindex
#214556012
#214547802
#214458402
#214442742
#214437152
#214436602
C61×#15032
#920961