Finite Group Databasebeta

← #30616177

Subgroups — #30616177

Order 15616.

866 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840777C211yesmaximal, characteristic
2C22×#14519848C211yesmaximal, characteristic
3#21462462C211yesmaximal, characteristic
4#21462384C211yesmaximal, characteristic
5#21461935C214yesmaximal, complemented
6#21461933C214yesmaximal, complemented
7#21461513C211yesmaximal, characteristic
8#21461511C211yesmaximal, characteristic
9#424×D61C211yesmaximal, characteristic
10C22×#11758213C2211yescharacteristic
11C22×#11758213C2211yescharacteristic
12C2×D61⋊M16C2211yescharacteristic
13C2×D61⋊M16C2211yescharacteristic
14#16840777—24no—
15C2×#14519848—24no—
16C2×#14519848C2212yes—
17C2×#14519848C2214yes—
18C2×#14519847C2212yes—
19#16840769—22no—
20#16840768—22no—
21#16840767C2212yes—
22#16840766C2212yes—
23#16840727—22no—
24#16840726—22no—
25C4×#11758213C2214yes—
26#16840638C2212yes—
27#16840638C2212yes—
28#16840638C2214yes—
29C2×M16×D61C2211yescharacteristic
30C22×C8×D61C2211yescharacteristic
31#117×D61C2211yescharacteristic
32#16839790C411yescharacteristic
33#16839789C411yescharacteristic
34#16839788C411yescharacteristic
35#16839787C414yes—
36C4⋊C8×D61C2214yes—
37C61×#424C411yescharacteristic
38C2×#11758213—21no—
39C2×#11758213—22no—
40C2×#11758213—22no—
41C2×#11758213—22no—
42C2×#11758213—21no—
43C2×#11758213D414yes—
44C2×#11758213C2311yescharacteristic
45C2×#11758213C2312yes—
46C2×#11758213C2311yescharacteristic
47C2×#11758213C2312yes—
48C2×#11758213C2312yes—
49D61⋊M16—22no—
50D61⋊M16—22no—
51D61⋊M16—22no—
52D61⋊M16Q814yes—
53C2×#11758217—21no—
54C2×#11758217—21no—
55C2×D61⋊C8—21no—
56C2×D61⋊C8—21no—
57C22×#92420—21no—
58C22×#92420—21no—
59#14519848—28no—
60#14519848—24no—
61#14519847—24no—
62#14519844—24no—
63C4⋊#92420—24no—
64#14519832—24no—
65C4×#92420—24no—
66C22×C4×D61C2311yescharacteristic
67C2×#11758194C2×C411yescharacteristic
68C2×#11758189C2×C412yes—
69M16×D61—22no—
70M16×D61—22no—
71C22×#92418C2×C411yescharacteristic
72C2×C8×D61—22no—
73C2×C8×D61—22no—
74C2×C8×D61C2×C412yes—
75C2×C8×D61C2312yes—
76C2×C8×D61C2312yes—
77#14519736C2×C411yescharacteristic
78#14519735D412yes—
79#14519734C2×C412yes—
80#14519733D412yes—
81C4⋊C4×D61Q812yes—
82C22⋊C4×D61Q812yes—
83C42×D61C2312yes—
84C4⋊#92418C2×C414yes—
85#14519706C2×C414yes—
86#14519706C2×C414yes—
87C2×M16×C61C2×C411yescharacteristic
88C22×C8×C61C2×C411yescharacteristic
89#117×C61C2×C411yescharacteristic
90C4⋊C8×C61C2×C414yes—
91C23×D61C2×Q811yescharacteristic
92C22×C4×C61C22×C411yescharacteristic
93C22×Dic61C2×D411yescharacteristic
94C2×C4×D61C22×C411yescharacteristic
95C2×C4×D61C2×D412yes—
96C2×C4×D61C2×D411yescharacteristic
97C2×C4×D61C2×Q814yes—
98C2×C4×D61C2×Q812yes—
99C2×C4×D61C2×Q811yescharacteristic
100C2×C4×D61#4112yes—
101C2×C4×D61#4112yes—
102C2×C4×D61C2411yescharacteristic
103#11758217—24no—
104#11758217—42no—
105C2×#92420—24no—
106C2×#92420—22no—
107C2×#92420—22no—
108C2×#92420—21no—
109C2×#92420—22no—
110C2×#92420—21no—
111C2×#92420—42no—
112#11758213—24no—
113#11758213—22no—
114#11758213—22no—
115#11758213—21no—
116#11758213—21no—
117#11758213—22no—
118#11758213—21no—
119#11758213—21no—
120#11758213—22no—
121#11758213—42no—
122#11758213C2×D414yes—
123#11758213C2×Q814yes—
124D61⋊C8—24no—
125D61⋊C8—42no—
126M16×C61—22no—
127C2×C8×C61—22no—
128C2×C8×C61C22×C412yes—
129C2×C8×C61C22×C412yes—
130C4⋊C4×C61C4⋊C412yes—
131C22⋊C4×C61C4⋊C412yes—
132C42×C61C22×C412yes—
133C22⋊Dic61C4⋊C412yes—
134#11758198C4⋊C414yes—
135C4⋊Dic61C4⋊C412yes—
136#11758196C4⋊C414yes—
137C4×Dic61C22×C412yes—
138C4×Dic61C2×D414yes—
139#11758194—22no—
140C2×#92418—22no—
141C2×#92418C22×C412yes—
142C2×#92418C22×C412yes—
143#11758189—24no—
144C8×D61—24no—
145C8×D61—22no—
146C8×D61—22no—
147C8×D61—22no—
148C8×D61—22no—
149C23×C61C2×C4⋊C411yescharacteristic
150C22×D61—42no—
151C22×D61C2×C4⋊C411yescharacteristic
152C22×D61#12512yes—
153C22×D61C4⋊Q812yes—
154C22×D61C22×Q811yescharacteristic
155C2×C4×C61C2×C4⋊C414yes—
156C2×C4×C61C2×C4⋊C411yescharacteristic
157C2×C4×C61C4×D412yes—
158C2×C4×C61C4×Q812yes—
159C2×C4×C61C23×C411yescharacteristic
160C2×Dic61C2×C4⋊C414yes—
161C2×Dic61C2×C4⋊C411yescharacteristic
162C2×Dic61#12112yes—
163C2×Dic61C22⋊Q812yes—
164C2×Dic61C22×D411yescharacteristic
165C4×D61—44no—
166C4×D61C4×D412yes—
167C4×D61C4×Q812yes—
168C4×D61#12112yes—
169C4×D61C22⋊Q812yes—
170C4×D61#12512yes—
171C4×D61C4⋊Q812yes—
172C4×D61C22×D411yescharacteristic
173C4×D61C22×Q811yescharacteristic
174C4×D61C2×#4111yescharacteristic
175C4×D61C2×#4111yescharacteristic
176#92420—24no—
177#92420—24no—
178#92420—41no—
179#92420—41no—
180#92420—42no—
181C8×C61—22no—
182C8×C61—22no—
183#92418—22no—
184#92418—22no—
185#21178—611nomaximal, Sylow
186C22×C61—22no—
187C22×C61#38912yes—
188C22×C61C22×C4⋊C411yescharacteristic
189C2×D61—22no—
190C2×D61—22no—
191C2×D61—42no—
192C2×D61—42no—
193C2×D61#38912yes—
194C2×D61C2×#12511yescharacteristic
195C2×D61C2×C4⋊Q811yescharacteristic
196C4×C61—24no—
197C4×C61#38912yes—
198C4×C61C2×C4×D411yescharacteristic
199C4×C61C2×C4×Q811yescharacteristic
200Dic61—24no—
201Dic61#38912yes—
202Dic61C2×#12111yescharacteristic
203Dic61C2×C22⋊Q811yescharacteristic
204C2×#430—611no—
205C22×#108—611no—
206#2905—611no—
207C2×#424—611no—
208#2873—611no—
209#1950—614no—
210#1950—614no—
211#1738—611no—
212C2×#341—611no—
213C2×C61—22no—
214C2×C61C2×#38911yescharacteristic
215C2×C61#291612yes—
216D61—21no—
217D61—21no—
218D61—22no—
219C22×M16—611no—
220C22×M16—611no—
221C22×M16—611no—
222C22×M16—611no—
223C22×M16—611no—
224C23×C8—611no—
225C2×#117—611no—
226#430—1224no—
227C2×#108—612no—
228C2×#108—614no—
229C2×#108—612no—
230C2×#108—1224no—
231#424—611no—
232#424—611no—
233#424—611no—
234#424—614no—
235#424—1222no—
236#424—1222no—
237#424#692311yescharacteristic
238#423—612no—
239#423—612no—
240C2×C4⋊C8—614no—
241#405—1222no—
242#405—1222no—
243C4×M16—614no—
244#341—612no—
245#341—612no—
246#341—614no—
247C61#2117811yescharacteristic, complemented, Sylow
248C23×C4—611no—
249C2×M16—611no—
250C2×M16—612no—
251C2×M16—614no—
252C2×M16—614no—
253C2×M16—612no—
254C2×M16—612no—
255C2×M16—612no—
256C2×M16—611no—
257C2×M16—611no—
258C2×M16—1222no—
259C2×M16—1222no—
260C2×M16—1221no—
261C2×M16—1222no—
262C2×M16—1222no—
263C2×M16—1221no—
264C2×M16—1221no—
265C2×M16—1221no—
266C2×M16—1222no—
267C2×M16—1222no—
268C2×M16—1222no—
269C2×M16—1222no—
270C2×M16C2×#692311yescharacteristic
271C22×C8—611no—
272C22×C8—612no—
273C22×C8—612no—
274C22×C8—612no—
275C22×C8—1222no—
276C22×C8—1222no—
277C22×C8—1221no—
278C22×C8—1221no—
279C22×C8—1221no—
280C22×C8—1221no—
281C22×C8C2×#692311yescharacteristic
282#117—611no—
283#117—612no—
284#117—612no—
285#117—612no—
286#117C2×#692311yescharacteristic
287C2×C4⋊C4—612no—
288C2×C22⋊C4—612no—
289C2×C42—612no—
290#108—1224no—
291#108—1228no—
292#108—1224no—
293C4⋊C8—614no—
294C4⋊C8—614no—
295C4⋊C8—614no—
296C4⋊C8—1224no—
297C4⋊C8—1224no—
298C4⋊C8C2×#692314yes—
299#97—1224no—
300C4×C8—1224no—
301C24—611no—
302C22×C4—611no—
303C22×C4—611no—
304C22×C4—611no—
305C22×C4—612no—
306C22×C4—612no—
307C22×C4—612no—
308C22×C4—612no—
309C22×C4—611no—
310C22×C4—611no—
311C22×C4—614no—
312C22×C4C22×#692311yescharacteristic
313M16—22no—
314M16—614no—
315M16—614no—
316M16—1224no—
317M16—1224no—
318M16—1222no—
319M16—1222no—
320M16—1224no—
321M16—1222no—
322M16—1221no—
323M16—1222no—
324M16—1221no—
325M16—1221no—
326M16—1221no—
327M16—1222no—
328M16—2442no—
329M16—2442no—
330C2×C8—22no—
331C2×C8—612no—
332C2×C8—612no—
333C2×C8—1222no—
334C2×C8—1222no—
335C2×C8—1224no—
336C2×C8—1222no—
337C2×C8—1222no—
338C2×C8—1222no—
339C2×C8—1222no—
340C2×C8—1224no—
341C2×C8—1224no—
342C2×C8—1222no—
343C2×C8—1221no—
344C2×C8—1222no—
345C2×C8—1221no—
346C2×C8—2442no—
347C2×C8—2442no—
348C2×C8C22×#692312yes—
349C2×C8C22×#692312yes—
350C4⋊C4—614no—
351C4⋊C4—612no—
352C4⋊C4C4⋊#692312yes—
353C22⋊C4—612no—
354C22⋊C4—614no—
355C22⋊C4C4⋊#692312yes—
356C42—614no—
357C42—612no—
358C42C22×#692312yes—
359C23—611no—
360C23—611no—
361C23—612no—
362C23—612no—
363C23—2442no—
364C23C2×C4⋊#692311yescharacteristic
365C2×C4—612no—
366C2×C4—612no—
367C2×C4—611no—
368C2×C4—611no—
369C2×C4—612no—
370C2×C4—612no—
371C2×C4—612no—
372C2×C4—612no—
373C2×C4—612no—
374C2×C4—611no—
375C2×C4—611no—
376C2×C4—612no—
377C2×C4—611no—
378C2×C4—611no—
379C2×C4—614no—
380C2×C4—2444no—
381C2×C4C2×C4⋊#692314yes—
382C2×C4C2×C4⋊#692311yescharacteristic
383C2×C4D4×#692312yes—
384C2×C4Q8×#692312yes—
385C2×C4C23×#692311yescharacteristic
386C8—22no—
387C8—22no—
388C8—1224no—
389C8—1224no—
390C8—1222no—
391C8—1222no—
392C8—2441no—
393C8—2441no—
394C8—2442no—
395C22—22no—
396C22—611no—
397C22—611no—
398C22—612no—
399C22—1222no—
400C22—1222no—
401C22—2442no—
402C22—2442no—
403C22#1684071312yes—
404C22C22×C4⋊#692311yescharacteristic
405C4—24no—
406C4—611no—
407C4—611no—
408C4—612no—
409C4—1224no—
410C4#1684071312yes—
411C4C2×D4×#692311yescharacteristic
412C4C2×Q8×#692311yescharacteristic
413C2—22no—
414C2—1221no—
415C2—1221no—
416C2—1222no—
417C2C2×#1684071311yescharacteristic
418C2#2146248112yes—
subgroupG/H
centerC22C22×C4⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#424C4
socleC22×C61C22×C4⋊C4
prime radicalC23×D61C2×Q8
cosocleC2×C4×D61C24
hypercenter#424#6923
hypocenterC61#21178
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
5. #21461935C2semidirect
6. #21461933C2semidirect
247. C61#21178semidirect

Maximal subgroups

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

subgroupindex
C2×#168407772
C22×#145198482
#214624622
#214623842
#214619352
#214619332
#214615132
#214615112
#424×D612
#2117861