Finite Group Databasebeta

← #30597462

Subgroups — #30597462

Order 15616.

698 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×#6923C211yesmaximal, characteristic
2#21461547C212yesmaximal
3D61⋊#324C211yesmaximal, characteristic
4#21460656C212yesmaximal
5#324×D61C211yesmaximal, characteristic
6C22×C4⋊#6923C411yescharacteristic
7C22×C4×#6923C411yescharacteristic
8C22×C4×#6923C411yescharacteristic
9C22×C4×#6923C2211yescharacteristic
10D61⋊(C22⋊C8)C2212yes—
11C2×#14519831—22no—
12C4⋊C4×#6923—22no—
13C4⋊C4×#6923—22no—
14C2×(C2×#6923)⋊C4C411yescharacteristic
15C2×(C2×#6923)⋊C4C2211yescharacteristic
16(C2×#6923)⋊C8—22no—
17#16840608C412yescomplemented
18#16840607C412yescomplemented
19C2×C4⋊C4×D61C2211yescharacteristic
20C22⋊C8×D61C2212yes—
21#16839488C411yescharacteristic, complemented
22#16839487C412yescomplemented
23C61×#324C411yescharacteristic, complemented
24C23×#6923C2×C411yescharacteristic
25C23×#6923C2×C411yescharacteristic
26C23×#6923C2×C411yescharacteristic
27C23×#6923C2×C411yescharacteristic
28C2×D61⋊C8—22no—
29C2×C4⋊#6923—22no—
30C2×C4⋊#6923—22no—
31C2×C4⋊#6923—22no—
32C2×C4×#6923—21no—
33C2×C4×#6923—21no—
34C2×C4×#6923—22no—
35C2×C4×#6923—22no—
36C2×C4×#6923—21no—
37C2×C4×#6923—21no—
38C2×C4×#6923—42no—
39C2×C4×#6923—42no—
40C2×C4×#6923D412yes—
41C2×C4×#6923D412yes—
42#14519834—22no—
43C22⋊#92420C2×C414yes—
44#14519831—44no—
45(C2×#6923)⋊C4—22no—
46(C2×#6923)⋊C4—22no—
47C22×C4×D61C2×C411yescharacteristic
48C22×C4×D61C2×C411yescharacteristic
49C22×C4×D61C2311yescharacteristic
50C2×C8×D61—22no—
51C2×C4⋊Dic61C2×C411yescharacteristic
52C2×#11758196C2×C411yescharacteristic
53C2×#11758196C2×C411yescharacteristic
54C4⋊C4×D61—22no—
55C4⋊C4×D61—22no—
56C22⋊#92418C2×C412yes—
57#14519689—22no—
58C2×C4⋊C4×C61C2×C411yescharacteristic
59C22⋊C8×C61C2×C412yes—
60C23×D61C22×C411yescharacteristic
61C22×#6923—21no—
62C22×#6923—21no—
63C22×#6923—22no—
64C22×#6923—22no—
65C22×#6923—21no—
66C22×#6923—21no—
67C22×#6923—21no—
68C22×#6923—21no—
69C22×#6923—21no—
70C22×#6923—21no—
71C22×#6923—41no—
72C22×#6923—41no—
73C22×#6923—41no—
74C22×#6923—41no—
75C22×#6923—41no—
76C22×#6923—41no—
77C22×#6923C22⋊C412yes—
78C22×#6923C22⋊C411yescharacteristic
79C22×#6923C22⋊C411yescharacteristic
80C22×#6923M1612yes—
81C22×#6923M1612yes—
82C22×C4×C61C4211yescharacteristic
83C22×C4×C61C4211yescharacteristic
84C22×C4×C61C22×C411yescharacteristic
85C22×Dic61C4211yescharacteristic
86C22×Dic61C4211yescharacteristic
87C22×Dic61C22×C411yescharacteristic
88C2×C4×D61—21no—
89C2×C4×D61—21no—
90C2×C4×D61—21no—
91C2×C4×D61—21no—
92C2×C4×D61—21no—
93C2×C4×D61—21no—
94C2×C4×D61—41no—
95C2×C4×D61—41no—
96C2×C4×D61—41no—
97C2×C4×D61—41no—
98C2×C4×D61M1612yes—
99C2×C4×D61M1612yes—
100C2×C4×D61C2×D412yes—
101C2×C4×D61#4112yes—
102C2×#92420—24no—
103C4⋊#6923—28no—
104C4×#6923—24no—
105C4×#6923—24no—
106C4×#6923—24no—
107C4×#6923—42no—
108C4×#6923—42no—
109C4×#6923—42no—
110D61⋊C8—42no—
111D61⋊C8—42no—
112C2×C8×C61—22no—
113C4⋊C4×C61—22no—
114C4⋊Dic61—22no—
115#11758196—22no—
116#11758196—22no—
117C2×#92418—22no—
118C8×D61—42no—
119C8×D61—42no—
120C23×C61C2×C4211yescharacteristic
121C22×D61—21no—
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—21no—
125C22×D61C2×C22⋊C411yescharacteristic
126C22×D61#11711yescharacteristic
127C22×D61C2×M1611yescharacteristic
128C22×D61C2×M1611yescharacteristic
129C22×D61C2×M1611yescharacteristic
130C22×D61C2×M1611yescharacteristic
131C2×#6923—24no—
132C2×#6923—24no—
133C2×#6923—21no—
134C2×#6923—21no—
135C2×#6923—21no—
136C2×#6923—21no—
137C2×#6923—22no—
138C2×#6923—22no—
139C2×#6923—22no—
140C2×#6923—22no—
141C2×#6923—22no—
142C2×#6923—42no—
143C2×#6923—42no—
144C2×#6923—42no—
145C2×#6923—42no—
146C2×#6923—41no—
147C2×#6923—41no—
148C2×#6923—41no—
149C2×#6923—41no—
150C2×#6923—42no—
151C2×#6923—42no—
152C2×#6923#9912yes—
153C2×#6923#10112yes—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61#9712yescomplemented
160C2×C4×C61C4×D412yes—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61#9712yescomplemented
167C2×Dic61C4×D412yes—
168C4×D61—22no—
169C4×D61—22no—
170C4×D61—22no—
171C4×D61—22no—
172C4×D61—22no—
173C4×D61—22no—
174C4×D61—42no—
175C4×D61—42no—
176C4×D61—41no—
177C4×D61—41no—
178#92420—44no—
179C8×C61—42no—
180#92418—42no—
181#8153—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61C4×C22⋊C411yescharacteristic
185C22×C61C2×#9711yescharacteristic
186C22×C61C4×M1611yescharacteristic
187C2×D61—21no—
188C2×D61—21no—
189C2×D61—21no—
190C2×D61—21no—
191C2×D61—21no—
192C2×D61—21no—
193C2×D61—21no—
194C2×D61—21no—
195C2×D61—21no—
196C2×D61—21no—
197C2×D61#40711yescharacteristic
198C2×D61#40711yescharacteristic
199C2×D61C2×#9911yescharacteristic
200C2×D61#41011yescharacteristic
201C2×D61C2×#10111yescharacteristic
202C2×D61#41311yescharacteristic
203C2×D61#43211yescharacteristic
204C2×D61#43211yescharacteristic
205#6923—22no—
206#6923—22no—
207#6923—42no—
208#6923—42no—
209#6923—42no—
210C4×C61—22no—
211C4×C61—22no—
212C4×C61—41no—
213Dic61—22no—
214Dic61—22no—
215Dic61—41no—
216C2×C4×C4⋊C4—611no—
217#1749—612no—
218C2×#324—611no—
219C2×#324—611no—
220#1280—612no—
221C2×C61—21no—
222C2×C61—21no—
223C2×C61#174911yescharacteristic
224C2×C61C4×#9911yescharacteristic
225C2×C61C4×#10111yescharacteristic
226D61—21no—
227D61—21no—
228D61#146311yescharacteristic
229D61#146411yescharacteristic
230D61#146511yescharacteristic
231D61#146711yescharacteristic
232C22×C4⋊C4—611no—
233C22×C4⋊C4—611no—
234C22×C42—611no—
235C22×C42—611no—
236C22×C42—611no—
237C2×C22⋊C8—612no—
238C2×C22⋊C8—612no—
239C2×#97—1222no—
240C4×C4⋊C4—1222no—
241C4×C4⋊C4—1222no—
242C2×#95—611no—
243C2×#95—611no—
244#336—1222no—
245#324—612no—
246#324—612no—
247#324—611no—
248#324—612no—
249#324#692311yescharacteristic, complemented
250C61#815311yescharacteristic, complemented, Sylow
251C23×C4—611no—
252C23×C4—611no—
253C23×C4—611no—
254C23×C4—611no—
255C23×C4—611no—
256C23×C4—611no—
257C23×C4—611no—
258C22×C8—1222no—
259C22×C8—1222no—
260C2×C4⋊C4—611no—
261C2×C4⋊C4—611no—
262C2×C4⋊C4—611no—
263C2×C4⋊C4—1222no—
264C2×C4⋊C4—1222no—
265C2×C4⋊C4—1222no—
266C2×C4⋊C4—1222no—
267C2×C4⋊C4—1222no—
268C2×C4⋊C4C2×#692311yescharacteristic
269C2×C42—612no—
270C2×C42—612no—
271C2×C42—1221no—
272C2×C42—1221no—
273C2×C42—1222no—
274C2×C42—1222no—
275C2×C42—1221no—
276C2×C42—1221no—
277C2×C42—2442no—
278C2×C42—2442no—
279C22⋊C8—614no—
280C22⋊C8—612no—
281C22⋊C8—1222no—
282C22⋊C8—1222no—
283C22⋊C8C2×#692312yes—
284#97—2444no—
285#95—1222no—
286#95—1222no—
287C24—611no—
288C22×C4—611no—
289C22×C4—611no—
290C22×C4—611no—
291C22×C4—611no—
292C22×C4—612no—
293C22×C4—612no—
294C22×C4—612no—
295C22×C4—612no—
296C22×C4—612no—
297C22×C4—612no—
298C22×C4—612no—
299C22×C4—611no—
300C22×C4—1221no—
301C22×C4—1221no—
302C22×C4—1221no—
303C22×C4—1221no—
304C22×C4—1221no—
305C22×C4—1221no—
306C22×C4—1222no—
307C22×C4—1222no—
308C22×C4—1221no—
309C22×C4—1221no—
310C22×C4—1221no—
311C22×C4—1221no—
312C22×C4—1221no—
313C22×C4—1221no—
314C22×C4—1221no—
315C22×C4—1221no—
316C22×C4—2441no—
317C22×C4—2441no—
318C22×C4—2441no—
319C22×C4—2441no—
320C22×C4—2441no—
321C22×C4—2441no—
322C22×C4—2441no—
323C22×C4—2441no—
324C22×C4—2441no—
325C22×C4—2441no—
326C22×C4C4×#692311yescharacteristic
327C22×C4C4×#692311yescharacteristic
328C22×C4C22×#692311yescharacteristic
329C2×C8—22no—
330C2×C8—1224no—
331C2×C8—1222no—
332C2×C8—2442no—
333C2×C8—2442no—
334C2×C8—2442no—
335C2×C8—2442no—
336C4⋊C4—22no—
337C4⋊C4—1222no—
338C4⋊C4—1222no—
339C4⋊C4—1228no—
340C4⋊C4—1222no—
341C42—1224no—
342C42—1224no—
343C42—1224no—
344C42—2442no—
345C42—2442no—
346C42—2442no—
347C23—611no—
348C23—611no—
349C23—611no—
350C23—611no—
351C23—611no—
352C23—611no—
353C23—1221no—
354C23—1221no—
355C23—1221no—
356C23—1221no—
357C23C2×C4×#692311yescharacteristic
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—612no—
364C2×C4—612no—
365C2×C4—612no—
366C2×C4—612no—
367C2×C4—1222no—
368C2×C4—1222no—
369C2×C4—1222no—
370C2×C4—1222no—
371C2×C4—1222no—
372C2×C4—1222no—
373C2×C4—1221no—
374C2×C4—1221no—
375C2×C4—1221no—
376C2×C4—1221no—
377C2×C4—1224no—
378C2×C4—1224no—
379C2×C4—1221no—
380C2×C4—1221no—
381C2×C4—1222no—
382C2×C4—1222no—
383C2×C4—1222no—
384C2×C4—1222no—
385C2×C4—1222no—
386C2×C4—1221no—
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—2441no—
396C2×C4—2442no—
397C2×C4—2442no—
398C2×C4—2442no—
399C2×C4—2442no—
400C2×C4—2441no—
401C2×C4—2442no—
402C2×C4—2442no—
403C2×C4#1451983112yescomplemented
404C2×C4D4×#692312yes—
405C8—42no—
406C8—2442no—
407C8—2444no—
408C22—21no—
409C22—21no—
410C22—611no—
411C22—611no—
412C22—611no—
413C22—611no—
414C22—611no—
415C22—611no—
416C22—611no—
417C22—611no—
418C22—1221no—
419C22—1221no—
420C22—1221no—
421C22—1221no—
422C22—1221no—
423C22—1221no—
424C22—1221no—
425C22—1221no—
426C22—1221no—
427C22—1221no—
428C22C22⋊C4×#692311yescharacteristic
429C22C2×#1451983111yescharacteristic
430C22M16×#692311yescharacteristic
431C4—22no—
432C4—22no—
433C4—41no—
434C4—1222no—
435C4—1222no—
436C4—1222no—
437C4—1222no—
438C4—2442no—
439C4—2442no—
440C4—2442no—
441C4—2441no—
442C2—21no—
443C2—21no—
444C2—611no—
445C2—611no—
446C2—611no—
447C2—611no—
448C2—1221no—
449C2—1221no—
450C2#2146154711yescharacteristic
451C2#99×#692311yescharacteristic
452C2#101×#692311yescharacteristic
subgroupG/H
centerC22C22⋊C4×#6923
commutatorC23×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#324C4
socleC22×C61C4×C22⋊C4
prime radicalC23×D61C22×C4
cosocleC22×C4×D61C23
hypercenter#324#6923
hypocenterC61#8153
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
17. #16840608C4semidirect
18. #16840607C4semidirect
21. #16839488C4semidirect
22. #16839487C4semidirect
23. C61×#324C4semidirect
159. C2×C4×C61#97semidirect
166. C2×Dic61#97semidirect
249. #324#6923semidirect
250. C61#8153semidirect
403. C2×C4#14519831semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×#69232
#214615472
D61⋊#3242
#214606562
#324×D612
#815361