Finite Group Databasebeta

← #29576472

Subgroups — #29576472

Order 15616.

566 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840080C211yesmaximal, characteristic
2#21448696C211yesmaximal, characteristic, complemented
3#21446654C211yesmaximal, characteristic
4#21444388C211yesmaximal, characteristic
5#21443686C211yesmaximal, characteristic, complemented
6#21443665C211yesmaximal, characteristic, complemented
7C61×#1824C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9#16840080—21no—
10#16840080—21no—
11#16840080—22no—
12C2×#14519708C412yescomplemented
13C2×#14519697C412yescomplemented
14C2×#14519691—21no—
15C2×#14519691—21no—
16C2×C22⋊#92418C2211yescharacteristic
17#16839661—21no—
18#16839648—22no—
19#16839622—21no—
20#16839622C2211yescharacteristic
21C4⋊C4×Dic61—21no—
22C4×C22⋊Dic61—21no—
23C4×C22⋊Dic61C2211yescharacteristic
24#16839497—22no—
25C22×C4⋊C4×C61C2211yescharacteristic
26C2×C22⋊C8×C61C2211yescharacteristic
27C61×#380C2211yescharacteristic
28C61×#340—22no—
29C22×D244C2×C411yescharacteristic
30C22×C22⋊D61—21no—
31C22×#92418C2×C412yes—
32#14519751—21no—
33#14519751—21no—
34#14519751—21no—
35#14519751—21no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61—22no—
38C2×C4⋊Dic61C2×C411yescharacteristic
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61D411yescharacteristic
41C2×C22⋊Dic61D411yescharacteristic
42C2×#11758198—21no—
43C2×C4×Dic61—21no—
44C2×C4×Dic61—22no—
45#14519708—22no—
46#14519708—22no—
47#14519697—22no—
48#14519697—22no—
49#14519691—42no—
50#14519691—42no—
51C22⋊#92418—21no—
52C22⋊#92418—21no—
53#14519684—22no—
54#14519684—22no—
55#14519684—21no—
56#14519684D412yes—
57C23×C4×C61—21no—
58C23×C4×C61C2311yescharacteristic
59C22×C8×C61—21no—
60C2×C4⋊C4×C61—21no—
61C2×C4⋊C4×C61—21no—
62C2×C4⋊C4×C61—21no—
63C2×C4⋊C4×C61—21no—
64C2×C4⋊C4×C61C2×C412yes—
65C2×C4⋊C4×C61D411yescharacteristic
66C2×C4⋊C4×C61D411yescharacteristic
67C2×C4⋊C4×C61D412yes—
68C2×C22⋊C4×C61—21no—
69C22⋊C8×C61—21no—
70C22⋊C8×C61—21no—
71#95×C61—21no—
72C24×C61C2×D411yescharacteristic
73C23×D61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—22no—
77C22×C4×C61—22no—
78C22×C4×C61—42no—
79C22×C4×C61—41no—
80C22×C4×C61—82no—
81C22×C4×C61C22×C411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61#4111yescharacteristic
86C22×C4×C61#4111yescharacteristic
87C2×C22⋊D61—41no—
88C2×C22⋊D61—41no—
89C2×C22⋊D61—41no—
90C2×C22⋊D61—41no—
91C2×C22⋊D61—41no—
92C2×C22⋊D61—41no—
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61—21no—
96C22×Dic61—22no—
97C2×D244—21no—
98C2×D244—21no—
99C2×D244—42no—
100C2×C8×C61—42no—
101C2×C8×C61—41no—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—42no—
107C4⋊C4×C61C22⋊C412yes—
108C4⋊C4×C61C22⋊C412yes—
109C22⋊C4×C61—41no—
110C22⋊C4×C61—41no—
111C22⋊Dic61—21no—
112C22⋊Dic61—21no—
113C22⋊Dic61—21no—
114C22⋊Dic61—41no—
115C22⋊Dic61—41no—
116C22⋊Dic61SD1612yes—
117#11758198—41no—
118#11758198—41no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61—42no—
122C4⋊Dic61—82no—
123C4×Dic61—42no—
124C4×Dic61—82no—
125C2×#92418—22no—
126C2×#92418—42no—
127C23×C61—41no—
128C23×C61—41no—
129C23×C61C4×D411yescharacteristic
130C23×C61#12011yescharacteristic
131C23×C61#12111yescharacteristic
132C23×C61C2×SD1611yescharacteristic
133C23×C61#13511yescharacteristic
134C23×C61AGL(1,ℤ/8)11yescharacteristic
135C23×C61#13711yescharacteristic
136C22×D61—41no—
137C22×D61—41no—
138C22×D61—41no—
139C22×D61—81no—
140C22×D61—81no—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—22no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61—82no—
152C2×C4×C61—82no—
153C2×C4×C61—81no—
154C2×C4×C61—81no—
155C2×C4×C61—81no—
156C2×C4×C61C2×C22⋊C411yescharacteristic
157C2×C4×C61C4×D411yescharacteristic
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61#12311yescharacteristic
160C22⋊D61—81no—
161C22⋊D61—81no—
162C22⋊D61—81no—
163C22⋊D61—81no—
164C2×Dic61—22no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—42no—
176D244—42no—
177D244—81no—
178C8×C61—81no—
179#92418—42no—
180#9261—611nomaximal, Sylow
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—41no—
188C22×C61—41no—
189C22×C61—41no—
190C22×C61—41no—
191C22×C61#38611yescharacteristic
192C22×C61C4×SD1611yescharacteristic
193C22×C61#44011yescharacteristic
194C22×C61#44811yescharacteristic
195C22×C61#44911yescharacteristic
196C22×C61#46011yescharacteristic
197C22×C61#46411yescharacteristic
198C2×D61—81no—
199C2×D61—81no—
200C2×D61—81no—
201C2×D61—161no—
202C2×D61—161no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—21no—
206C4×C61—42no—
207C4×C61—42no—
208C4×C61—161no—
209C4×C61#41012yes—
210Dic61—42no—
211Dic61—81no—
212Dic61—82no—
213Dic61—81no—
214C2×#481—611no—
215#2345—611no—
216#1964—611no—
217#1824D6111yescharacteristic, complemented
218#1519—611no—
219#1289—611no—
220#1282—611no—
221C2×C61—21no—
222C2×C61—21no—
223C2×C61—21no—
224C2×C61—21no—
225C2×C61—21no—
226C2×C61—21no—
227C2×C61#188011yescharacteristic
228C2×C61#189911yescharacteristic
229C2×C61#190411yescharacteristic
230D61—161no—
231C2×#121—611no—
232C22×C4⋊C4C2×D6111yescharacteristic
233#481—1221no—
234#481—1221no—
235#481—1222no—
236C2×#106—612no—
237C2×#102—612no—
238C2×#99—1221no—
239C2×#99—1221no—
240C2×C22⋊C8—611no—
241C2×C22⋊C8C2×D6111yescharacteristic
242#389—1221no—
243#387—1222no—
244#380—611no—
245#380—1221no—
246#380C2×D6111yescharacteristic
247C4×C4⋊C4—1221no—
248C4×C22⋊C4—611no—
249C4×C22⋊C4—1221no—
250#340—22no—
251#327—1222no—
252C61#926111yescharacteristic, complemented, Sylow
253C22×D4—611no—
254C22×D4—1221no—
255C23×C4—21no—
256C23×C4C22×D6111yescharacteristic
257C22×C8—21no—
258C22×C8—612no—
259#121—1221no—
260#121—1221no—
261#121—1221no—
262#121—1221no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—611no—
268C2×C4⋊C4—1222no—
269C2×C4⋊C4—1221no—
270C2×C4⋊C4C4×D6112yes—
271C2×C4⋊C4D24411yescharacteristic
272C2×C4⋊C4D24411yescharacteristic
273C2×C4⋊C4C22⋊D6112yes—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—611no—
276C2×C22⋊C4—611no—
277C2×C22⋊C4—1221no—
278C2×C22⋊C4—1221no—
279C2×C42—1222no—
280C2×C42—1221no—
281#106—1222no—
282#106—1222no—
283#102—1222no—
284#102—1222no—
285#99—2442no—
286#99—2442no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—1221no—
290C22⋊C8—1221no—
291#95—21no—
292#95—612no—
293#95—1222no—
294#95—1222no—
295#95—1221no—
296C24—1221no—
297C24D4×D6111yescharacteristic
298C2×D4—1221no—
299C2×D4—1221no—
300C2×D4—2442no—
301C2×D4—2441no—
302C2×D4—2441no—
303C2×D4—2441no—
304C2×D4—2441no—
305C2×D4—2441no—
306C2×D4—2441no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—82no—
314C22×C4—1221no—
315C22×C4—1221no—
316C22×C4—1222no—
317C22×C4—1221no—
318C22×C4C2×C4×D6111yescharacteristic
319C22×C4C2×D24411yescharacteristic
320C22×C4D4×D6111yescharacteristic
321C22×C4#1175822411yescharacteristic
322C22×C4#1175822411yescharacteristic
323C22×C4C2×C22⋊D6111yescharacteristic
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—1222no—
327C2×C8—2442no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—42no—
333C4⋊C4—1221no—
334C4⋊C4—1221no—
335C4⋊C4—4882no—
336C4⋊C4—2442no—
337C4⋊C4#1175819812yes—
338C4⋊C4#1175819812yes—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—612no—
342C22⋊C4—1221no—
343C22⋊C4—1221no—
344C22⋊C4—1221no—
345C22⋊C4—2441no—
346C22⋊C4—2441no—
347C22⋊C4—2441no—
348C22⋊C4—2441no—
349C42—4882no—
350C42—2442no—
351C23—41no—
352C23—41no—
353C23—4881no—
354C23—4881no—
355C23—2441no—
356C23—2441no—
357C23—2441no—
358C23#1451973711yescharacteristic
359C23#1451974711yescharacteristic
360C23#1451974911yescharacteristic
361C23SD16×D6111yescharacteristic
362C23#1451980011yescharacteristic
363C23#1451980411yescharacteristic
364C23#1451980911yescharacteristic
365D4—4881no—
366D4—4881no—
367D4—4881no—
368D4—4881no—
369D4—4881no—
370D4—2442no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—1222no—
387C2×C4—2442no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4—2442no—
391C2×C4—2441no—
392C2×C4—2441no—
393C2×C4—2441no—
394C2×C4—2441no—
395C2×C4—2442no—
396C2×C4—2441no—
397C2×C4—2441no—
398C2×C4C2×#1175819811yescharacteristic
399C2×C4#1451973711yescharacteristic
400C2×C4#1451974911yescharacteristic
401C2×C4#1451976111yescharacteristic
402C8—81no—
403C8—2442no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—9761no—
415C22—9761no—
416C22—4881no—
417C22—4881no—
418C22—4881no—
419C22#1683963911yescharacteristic
420C22#1683984711yescharacteristic
421C22#1683986511yescharacteristic
422C22#1683990211yescharacteristic
423C22#1683991011yescharacteristic
424C22#1683996011yescharacteristic
425C22#1683998611yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—42no—
431C4—161no—
432C4—4882no—
433C4—4881no—
434C4—4881no—
435C4—2442no—
436C4#1683973512yes—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—9761no—
444C2#2144617211yescharacteristic
445C2#2144627511yescharacteristic
446C2#2144630411yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1824C2
socleC22×C61#386
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1824D61
hypocenterC61#9261
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
2. #21448696C2semidirect
5. #21443686C2semidirect
6. #21443665C2semidirect
7. C61×#1824C2semidirect
12. C2×#14519708C4semidirect
13. C2×#14519697C4semidirect
217. #1824D61semidirect
252. C61#9261semidirect

Maximal subgroups

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

subgroupindex
C2×#168400802
#214486962
#214466542
#214443882
#214436862
#214436652
C61×#18242
#926161