Finite Group Databasebeta

← #30603440

Subgroups — #30603440

Order 15616.

850 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#21462299C211yesmaximal, characteristic
2#21461522C211yesmaximal, characteristic
3#21461506C212yesmaximal
4C2×#16840633C212yesmaximal
5#384×D61C211yesmaximal, characteristic
6C2×D61⋊M16C2212yes—
7C2×#14519856C2212yes—
8#16840713—24no—
9#16840704—22no—
10#16840642—22no—
11#16840635—24no—
12#16840633—24no—
13C2×C4⋊C4×D61—22no—
14C2×C4⋊C4×D61C2211yescharacteristic
15C2×C4⋊C4×D61C2211yescharacteristic
16C2×C42×D61C2211yescharacteristic
17#16839632C411yescharacteristic, complemented
18#16839631—22no—
19#16839631C411yescharacteristic, complemented
20#16839631C411yescharacteristic, complemented
21C61×#384C411yescharacteristic, complemented
22C2×#11758213—22no—
23D61⋊M16—22no—
24D61⋊M16—22no—
25D61⋊M16—22no—
26D61⋊M16Q814yes—
27C2×#11758217—22no—
28C2×D61⋊C8—22no—
29#14519856—22no—
30#14519856—22no—
31#14519856—22no—
32#14519856—22no—
33C2×C4⋊#6923—24no—
34C2×C4⋊#6923D414yes—
35C2×C22⋊#6923—22no—
36C2×C4×#6923—22no—
37(C2×#6923)⋊C4—24no—
38C22×C4×D61—21no—
39C22×C4×D61—21no—
40C22×C4×D61D412yes—
41C22×C4×D61C2311yescharacteristic
42C2×C4⋊Dic61—22no—
43C2×C4⋊Dic61C2×C411yescharacteristic
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×#11758196—22no—
46C2×#11758196—22no—
47C2×#11758196—21no—
48C2×#11758196D412yes—
49C4⋊C4×D61—22no—
50C4⋊C4×D61—22no—
51C4⋊C4×D61—22no—
52C4⋊C4×D61—22no—
53C4⋊C4×D61—48no—
54C2×C4×Dic61—21no—
55C2×C4×Dic61C2×C411yescharacteristic
56C42×D61—21no—
57C42×D61—21no—
58C42×D61D412yes—
59C42×D61Q812yes—
60C2×C4⋊C4×C61—22no—
61C2×C4⋊C4×C61C2×C411yescharacteristic
62C2×C4⋊C4×C61C2×C411yescharacteristic
63C2×C42×C61C2×C411yescharacteristic
64C23×D61C2×D411yescharacteristic
65C22×#6923—24no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61C22⋊C412yes—
69C22×C4×C61C22×C411yescharacteristic
70C22×Dic61—21no—
71C22×Dic61—21no—
72C22×Dic61C22⋊C412yes—
73C22×Dic61C2×D411yescharacteristic
74C2×C4×D61—24no—
75C2×C4×D61—24no—
76C2×C4×D61—22no—
77C2×C4×D61—22no—
78C2×C4×D61—22no—
79C2×C4×D61—22no—
80C2×C4×D61—44no—
81C2×C4×D61—44no—
82C2×C4×D61—41no—
83C2×C4×D61—41no—
84C2×C4×D61C2×D411yescharacteristic
85C2×C4×D61C2×D412yes—
86C2×C4×D61C2×Q812yes—
87C2×C4×D61C2×Q811yescharacteristic
88C2×C4×D61#4112yes—
89C2×C4×D61#4111yescharacteristic
90C2×C4×D61#4112yes—
91C2×C4×D61#4111yescharacteristic
92C22⋊#6923—42no—
93C22⋊#6923—42no—
94#11758217—24no—
95#11758217—42no—
96C2×#92420—42no—
97C4⋊#6923—24no—
98C4⋊#6923—24no—
99C4⋊#6923—84no—
100C4×#6923—42no—
101C4×#6923—42no—
102#11758213—24no—
103#11758213—42no—
104D61⋊C8—24no—
105D61⋊C8—42no—
106C4⋊C4×C61—22no—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—44no—
109C42×C61—21no—
110C42×C61C4⋊C412yescomplemented
111C4⋊Dic61—22no—
112C4⋊Dic61—22no—
113C4⋊Dic61—44no—
114#11758196—24no—
115#11758196—44no—
116#11758196—44no—
117#11758196—42no—
118C4×Dic61—21no—
119C4×Dic61—22no—
120C4×Dic61—22no—
121C4×Dic61C4⋊C412yescomplemented
122C23×C61C2×C22⋊C411yescharacteristic
123C22×D61—22no—
124C22×D61—22no—
125C22×D61#12011yescharacteristic
126C22×D61#12111yescharacteristic
127C22×D61C22⋊Q811yescharacteristic
128C22×D61#12311yescharacteristic
129C22×D61#12411yescharacteristic
130C22×D61C4⋊Q811yescharacteristic
131C2×#6923—44no—
132C2×#6923—44no—
133C2×#6923—44no—
134C2×C4×C61—22no—
135C2×C4×C61—22no—
136C2×C4×C61—22no—
137C2×C4×C61—22no—
138C2×C4×C61—42no—
139C2×C4×C61—42no—
140C2×C4×C61—41no—
141C2×C4×C61C2×C4⋊C411yescharacteristic
142C2×C4×C61#11711yescharacteristic
143C2×C4×C61C4×D412yes—
144C2×C4×C61C4×Q812yes—
145C2×Dic61—22no—
146C2×Dic61—22no—
147C2×Dic61—22no—
148C2×Dic61—22no—
149C2×Dic61—42no—
150C2×Dic61—42no—
151C2×Dic61—41no—
152C2×Dic61#12111yescharacteristic
153C2×Dic61#12112yes—
154C2×Dic61C22⋊Q812yes—
155C2×Dic61C22⋊Q811yescharacteristic
156C4×D61—28no—
157C4×D61—28no—
158C4×D61—44no—
159C4×D61—82no—
160C4×D61—82no—
161C4×D61—82no—
162C4×D61#12112yes—
163C4×D61C22⋊Q812yes—
164C4×D61C22⋊Q812yes—
165C4×D61#12312yes—
166C4×D61#12512yes—
167C4×D61#12512yes—
168C4×D61#12612yes—
169C4×D61C4⋊Q812yes—
170#92420—44no—
171#11210—611nomaximal, Sylow
172C22×C61—21no—
173C22×C61—21no—
174C22×C61#38011yescharacteristic
175C22×C61#38611yescharacteristic
176C22×C61#39111yescharacteristic
177C2×D61—21no—
178C2×D61—21no—
179C2×D61—22no—
180C2×D61—22no—
181C2×D61—22no—
182C2×D61—22no—
183C2×D61#39411yescharacteristic
184C2×D61#39511yescharacteristic
185C2×D61#39811yescharacteristic
186C2×D61#40011yescharacteristic
187C2×D61#45312yes—
188C2×D61#45612yes—
189#6923—84no—
190C4×C61—24no—
191C4×C61—42no—
192C4×C61—81no—
193C4×C61#38712yes—
194C4×C61#38912yes—
195Dic61—24no—
196Dic61—42no—
197Dic61—81no—
198Dic61#39612yes—
199Dic61#39812yes—
200#2316—611no—
201C2×#384—611no—
202C2×#343—611no—
203#1736—612no—
204C2×#339—612no—
205C2×C61—21no—
206C2×C61—21no—
207C2×C61#144111yescharacteristic
208C2×C61#188611yescharacteristic
209C2×C61#188711yescharacteristic
210D61—21no—
211D61—21no—
212D61#203912yes—
213D61#206012yes—
214C22×M16—612no—
215C2×#117—612no—
216C22×C4⋊C4—611no—
217C22×C4⋊C4—611no—
218C22×C4⋊C4—1222no—
219C22×C42—611no—
220#389—1224no—
221#384—611no—
222#384—611no—
223#384—611no—
224#384—1222no—
225#384#692311yescharacteristic, complemented
226#380—1222no—
227#343—1222no—
228#340—1224no—
229#339—1224no—
230C61#1121011yescharacteristic, complemented, Sylow
231C23×C4—611no—
232C23×C4—612no—
233C23×C4—1221no—
234C23×C4—1221no—
235C2×M16—614no—
236C2×M16—1222no—
237C2×M16—1222no—
238C2×M16—1222no—
239C2×M16—1222no—
240C2×M16—1222no—
241C22×C8—1222no—
242#117—1222no—
243#117—1222no—
244#117—1222no—
245#117—1222no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—612no—
248C2×C4⋊C4—611no—
249C2×C4⋊C4—614no—
250C2×C4⋊C4—611no—
251C2×C4⋊C4—1222no—
252C2×C4⋊C4—1222no—
253C2×C4⋊C4—1222no—
254C2×C4⋊C4—1222no—
255C2×C4⋊C4—1224no—
256C2×C4⋊C4—1221no—
257C2×C4⋊C4—1222no—
258C2×C4⋊C4—1222no—
259C2×C4⋊C4—1222no—
260C2×C4⋊C4—2448no—
261C2×C4⋊C4C2×#692311yescharacteristic
262C2×C4⋊C4C2×#692311yescharacteristic
263C2×C22⋊C4—1222no—
264C2×C42—611no—
265C2×C42—612no—
266C2×C42—612no—
267C2×C42—1221no—
268C2×C42—1221no—
269C2×C42—1221no—
270C2×C42—1222no—
271C2×C42C2×#692311yescharacteristic
272#95—1224no—
273C24—611no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—611no—
277C22×C4—611no—
278C22×C4—611no—
279C22×C4—611no—
280C22×C4—611no—
281C22×C4—612no—
282C22×C4—612no—
283C22×C4—612no—
284C22×C4—612no—
285C22×C4—612no—
286C22×C4—1222no—
287C22×C4—1222no—
288C22×C4—1224no—
289C22×C4—1224no—
290C22×C4—1222no—
291C22×C4—1222no—
292C22×C4—1221no—
293C22×C4—1224no—
294C22×C4—1221no—
295C22×C4—2441no—
296C22×C4—2441no—
297C22×C4—2444no—
298C22×C4—2444no—
299C22×C4C22⋊#692312yes—
300C22×C4C22×#692311yescharacteristic
301M16—1224no—
302M16—1224no—
303M16—2442no—
304M16—2442no—
305C2×C8—1224no—
306C2×C8—2442no—
307C2×C8—2442no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—44no—
311C4⋊C4—1224no—
312C4⋊C4—1224no—
313C4⋊C4—1224no—
314C4⋊C4—1222no—
315C4⋊C4—1222no—
316C4⋊C4—4884no—
317C4⋊C4—2444no—
318C4⋊C4—2442no—
319C4⋊C4—2444no—
320C4⋊C4—2444no—
321C22⋊C4—2442no—
322C22⋊C4—2442no—
323C42—21no—
324C42—612no—
325C42—1222no—
326C42—1222no—
327C42—1221no—
328C42—2442no—
329C42—2442no—
330C42C4⋊#692312yescomplemented
331C23—611no—
332C23—611no—
333C23—611no—
334C23—611no—
335C23—611no—
336C23—611no—
337C23—1222no—
338C23—1222no—
339C23C2×C22⋊#692311yescharacteristic
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—611no—
348C2×C4—611no—
349C2×C4—612no—
350C2×C4—612no—
351C2×C4—612no—
352C2×C4—612no—
353C2×C4—612no—
354C2×C4—612no—
355C2×C4—612no—
356C2×C4—612no—
357C2×C4—612no—
358C2×C4—612no—
359C2×C4—1228no—
360C2×C4—1228no—
361C2×C4—1222no—
362C2×C4—1222no—
363C2×C4—1222no—
364C2×C4—1222no—
365C2×C4—4882no—
366C2×C4—4882no—
367C2×C4—4882no—
368C2×C4—2444no—
369C2×C4—2442no—
370C2×C4—2444no—
371C2×C4—2444no—
372C2×C4—2444no—
373C2×C4—2441no—
374C2×C4—2442no—
375C2×C4C2×C4⋊#692311yescharacteristic
376C2×C4#1451985611yescharacteristic
377C2×C4D4×#692312yes—
378C2×C4Q8×#692312yes—
379C8—2444no—
380C22—21no—
381C22—21no—
382C22—611no—
383C22—611no—
384C22—612no—
385C22—612no—
386C22—611no—
387C22—611no—
388C22—1222no—
389C22—1222no—
390C22—1222no—
391C22—1222no—
392C22—1221no—
393C22—1221no—
394C22#1684070411yescharacteristic
395C22#1684071011yescharacteristic
396C22(C2×#6923)⋊Q811yescharacteristic
397C4—24no—
398C4—42no—
399C4—81no—
400C4—612no—
401C4—612no—
402C4—1224no—
403C4—4884no—
404C4—4881no—
405C4—2442no—
406C4#1684071112yes—
407C4#1684071312yes—
408C2—21no—
409C2—21no—
410C2—612no—
411C2—612no—
412C2—1221no—
413C2—1221no—
414C2#2146100411yescharacteristic
415C2#2146182511yescharacteristic
416C2#2146182611yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#384C4
socleC22×C61#386
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#384#6923
hypocenterC61#11210
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
17. #16839632C4semidirect
19. #16839631C4semidirect
20. #16839631C4semidirect
21. C61×#384C4semidirect
110. C42×C61C4⋊C4semidirect
121. C4×Dic61C4⋊C4semidirect
225. #384#6923semidirect
230. C61#11210semidirect
330. C42C4⋊#6923semidirect

Maximal subgroups

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

subgroupindex
#214622992
#214615222
#214615062
C2×#168406332
#384×D612
#1121061