Finite Group Databasebeta

← #29576473

Subgroups — #29576473

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×#16840081C211yesmaximal, characteristic
2#21448694C211yesmaximal, characteristic, complemented
3#21446655C211yesmaximal, characteristic
4#21445910C211yesmaximal, characteristic, complemented
5#21444388C211yesmaximal, characteristic
6#21443687C211yesmaximal, characteristic, complemented
7C61×#1282C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9C22×C4⋊Dic61C2211yescharacteristic
10#16840081—22no—
11#16840081—21no—
12#16840081—21no—
13C2×#14519709C412yescomplemented
14C2×#14519699C412yescomplemented
15C2×#14519691—21no—
16C2×#14519691—21no—
17#16839660—21no—
18#16839645—22no—
19#16839622C2211yescharacteristic
20#16839622C2211yescharacteristic
21#16839620—21no—
22C4×C4⋊Dic61—21no—
23C22⋊C4×Dic61—21no—
24#16839525—22no—
25#16839497—22no—
26C2×C22⋊C8×C61C2211yescharacteristic
27C2×C22⋊C8×C61C2211yescharacteristic
28C4×C22⋊C4×C61C2211yescharacteristic
29C22×D244C2×C411yescharacteristic
30C22×C22⋊D61—21no—
31C23×Dic61—21no—
32#14519751—21no—
33#14519751—21no—
34#14519751—21no—
35#14519751—21no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61C2×C411yescharacteristic
42C2×C4⋊Dic61C2×C412yes—
43C2×C4⋊Dic61D412yes—
44C2×C4⋊Dic61D411yescharacteristic
45C2×C4⋊Dic61D411yescharacteristic
46C2×#11758196—22no—
47C2×C22⋊Dic61—21no—
48C2×C22⋊Dic61—21no—
49C2×#11758198—21no—
50C2×C4×Dic61—22no—
51#14519709—22no—
52#14519709—22no—
53#14519699—22no—
54#14519699—22no—
55#14519691—42no—
56#14519691—42no—
57#14519684—22no—
58#14519684—22no—
59#14519684—21no—
60#14519684D412yes—
61C23×C4×C61C2311yescharacteristic
62C22×C8×C61—21no—
63C22×C8×C61C2×C412yes—
64C2×C22⋊C4×C61D411yescharacteristic
65C2×C22⋊C4×C61D411yescharacteristic
66C2×C42×C61—21no—
67C22⋊C8×C61—21no—
68C22⋊C8×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—42no—
77C22×C4×C61C22×C411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61#4111yescharacteristic
82C22×C4×C61#4111yescharacteristic
83C2×C22⋊D61—41no—
84C2×C22⋊D61—41no—
85C2×C22⋊D61—41no—
86C2×C22⋊D61—41no—
87C2×C22⋊D61—41no—
88C2×C22⋊D61—41no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61—22no—
92C22×Dic61—22no—
93C22×Dic61—21no—
94C22×Dic61—22no—
95C22×Dic61—41no—
96C22×Dic61—82no—
97C2×D244—21no—
98C2×D244—21no—
99C2×D244—42no—
100C2×C8×C61—22no—
101C2×C8×C61—42no—
102C2×C8×C61—42no—
103C2×C8×C61—41no—
104C22⋊C4×C61—21no—
105C22⋊C4×C61—21no—
106C22⋊C4×C61—21no—
107C22⋊C4×C61SD1612yes—
108C42×C61—42no—
109C22⋊Dic61—41no—
110C22⋊Dic61—41no—
111C22⋊Dic61—41no—
112C22⋊Dic61—41no—
113#11758198—41no—
114#11758198—41no—
115C4⋊Dic61—22no—
116C4⋊Dic61—22no—
117C4⋊Dic61—21no—
118C4⋊Dic61—21no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61—42no—
122C4⋊Dic61—42no—
123C4⋊Dic61C22⋊C412yes—
124C4⋊Dic61C22⋊C412yes—
125#11758196—82no—
126C4×Dic61—82no—
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—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61C2×C22⋊C411yescharacteristic
154C2×C4×C61C4×D411yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#12311yescharacteristic
157C22⋊D61—81no—
158C22⋊D61—81no—
159C22⋊D61—81no—
160C22⋊D61—81no—
161C2×Dic61—22no—
162C2×Dic61—22no—
163C2×Dic61—42no—
164C2×Dic61—42no—
165C2×Dic61—42no—
166C2×Dic61—42no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—42no—
171C2×Dic61—82no—
172C2×Dic61—82no—
173C2×Dic61—81no—
174C2×Dic61—81no—
175C2×Dic61—81no—
176D244—42no—
177D244—81no—
178C8×C61—42no—
179C8×C61—81no—
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—81no—
208C4×C61#41012yes—
209Dic61—42no—
210Dic61—42no—
211Dic61—82no—
212Dic61—81no—
213Dic61—161no—
214C2×#481—611no—
215#2345—611no—
216#1964—611no—
217#1824—611no—
218#1519—611no—
219#1289—611no—
220#1282D6111yescharacteristic, complemented
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⋊C4—611no—
233#481—1222no—
234#481—1221no—
235#481—1221no—
236C2×#106—612no—
237C2×#102—612no—
238C2×#99—1221no—
239C2×#99—1221no—
240C2×C22⋊C8C2×D6111yescharacteristic
241C2×C22⋊C8C2×D6111yescharacteristic
242#389—1221no—
243#387—1222no—
244#380—611no—
245#380—611no—
246#380—1221no—
247C4×C4⋊C4—1221no—
248C4×C22⋊C4—1221no—
249C4×C22⋊C4C2×D6111yescharacteristic
250#340—1222no—
251#327—1222no—
252C61#926111yescharacteristic, complemented, Sylow
253C22×D4—611no—
254C22×D4—1221no—
255C23×C4—1221no—
256C23×C4C22×D6111yescharacteristic
257C22×C8—21no—
258C22×C8C4×D6112yes—
259#121—1221no—
260#121—1221no—
261#121—1221no—
262#121—1221no—
263C2×C4⋊C4—612no—
264C2×C4⋊C4—612no—
265C2×C4⋊C4—611no—
266C2×C4⋊C4—611no—
267C2×C4⋊C4—611no—
268C2×C4⋊C4—1221no—
269C2×C4⋊C4—1221no—
270C2×C4⋊C4—1221no—
271C2×C4⋊C4—1222no—
272C2×C4⋊C4—1221no—
273C2×C4⋊C4—1221no—
274C2×C22⋊C4—1221no—
275C2×C22⋊C4—1221no—
276C2×C22⋊C4—1221no—
277C2×C22⋊C4D24411yescharacteristic
278C2×C22⋊C4D24411yescharacteristic
279C2×C42—21no—
280C2×C42—1222no—
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—21no—
290C22⋊C8—21no—
291#95—21no—
292#95—612no—
293#95—1222no—
294#95—1222no—
295#95—1221no—
296C24—1221no—
297C24C2×D24411yescharacteristic
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—42no—
310C22×C4—1222no—
311C22×C4—1222no—
312C22×C4—1221no—
313C22×C4—1221no—
314C22×C4—1221no—
315C22×C4—1222no—
316C22×C4—4882no—
317C22×C4—2441no—
318C22×C4C2×C4×D6111yescharacteristic
319C22×C4#1175822211yescharacteristic
320C22×C4D4×D6111yescharacteristic
321C22×C4D4×D6111yescharacteristic
322C22×C4D4×D6111yescharacteristic
323C22×C4#1175822611yescharacteristic
324C2×C8—22no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—41no—
328C4⋊C4—612no—
329C4⋊C4—612no—
330C4⋊C4—1222no—
331C4⋊C4—1222no—
332C4⋊C4—1221no—
333C4⋊C4—1221no—
334C4⋊C4—1221no—
335C4⋊C4—1221no—
336C4⋊C4—4882no—
337C4⋊C4—2442no—
338C4⋊C4—2442no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—2441no—
343C22⋊C4—2441no—
344C22⋊C4—2441no—
345C22⋊C4—2441no—
346C22⋊C4—2441no—
347C22⋊C4—2441no—
348C22⋊C4#1175819012yes—
349C42—42no—
350C42—4882no—
351C23—41no—
352C23—41no—
353C23—4881no—
354C23—4881no—
355C23—2441no—
356C23—2441no—
357C23—2441no—
358C23C4×D24411yescharacteristic
359C23#1451974711yescharacteristic
360C23#1451975211yescharacteristic
361C23C2×#1175819011yescharacteristic
362C23#1451979911yescharacteristic
363C23#1451980511yescharacteristic
364C23#1451981011yescharacteristic
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—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—1222no—
384C2×C4—1222no—
385C2×C4—4881no—
386C2×C4—4881no—
387C2×C4—4881no—
388C2×C4—4882no—
389C2×C4—4882no—
390C2×C4—2442no—
391C2×C4—2442no—
392C2×C4—2441no—
393C2×C4—2441no—
394C2×C4—2441no—
395C2×C4—2442no—
396C2×C4—2442no—
397C2×C4—2442no—
398C2×C4C22⋊C4×D6111yescharacteristic
399C2×C4#1451973911yescharacteristic
400C2×C4#1451975011yescharacteristic
401C2×C4#1451976211yescharacteristic
402C8—42no—
403C8—81no—
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#1683964011yescharacteristic
420C22C4×#1175819011yescharacteristic
421C22#1683986411yescharacteristic
422C22#1683990311yescharacteristic
423C22#1683990911yescharacteristic
424C22#1683996311yescharacteristic
425C22#1683998511yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—81no—
431C4—9761no—
432C4—4882no—
433C4—4881no—
434C4—2442no—
435C4—2442no—
436C4#1683973412yes—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—9761no—
444C2#2144617311yescharacteristic
445C2#2144627611yescharacteristic
446C2#2144630511yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1282C2
socleC22×C61#386
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1282D61
hypocenterC61#9261
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
2. #21448694C2semidirect
4. #21445910C2semidirect
6. #21443687C2semidirect
7. C61×#1282C2semidirect
13. C2×#14519709C4semidirect
14. C2×#14519699C4semidirect
220. #1282D61semidirect
252. C61#9261semidirect

Maximal subgroups

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

subgroupindex
C2×#168400812
#214486942
#214466552
#214459102
#214443882
#214436872
C61×#12822
#926161