Finite Group Databasebeta

← #30603366

Subgroups — #30603366

Order 15616.

870 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 870 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840742C211yesmaximal, characteristic
2#21462405C211yesmaximal, characteristic
3#21461506C212yesmaximal, complemented
4C2×#16840629C212yesmaximal, complemented
5#21448409C211yesmaximal, characteristic
6C2×D61⋊M16C2212yes—
7C2×#14519856C2212yes—
8#16840742—24no—
9C2×#14519835—22no—
10#16840736—24no—
11#16840732—22no—
12#16840635—24no—
13#16840629—24no—
14C22×#11758224C2211yescharacteristic
15C2×#14519734C2211yescharacteristic
16C2×C4⋊C4×D61C2211yescharacteristic
17#16839668C411yescharacteristic, complemented
18#16839667C411yescharacteristic, complemented
19#16839665C411yescharacteristic, complemented
20#16839638—22no—
21#16839620—22no—
22C61×#390C411yescharacteristic, complemented
23C2×#11758213—22no—
24D61⋊M16—22no—
25D61⋊M16—22no—
26D61⋊M16—22no—
27D61⋊M16—22no—
28C2×#11758217—22no—
29C2×D61⋊C8D414yes—
30#14519856—22no—
31#14519856—22no—
32#14519856—22no—
33#14519856D414yes—
34C2×C4⋊#6923—22no—
35C2×C22⋊#6923—22no—
36C2×C4×#6923—22no—
37#14519835—28no—
38C22⋊#92420—28no—
39C2×#11758224—22no—
40C2×#11758224—22no—
41C2×#11758224—22no—
42C2×#11758224—22no—
43C2×#11758224—22no—
44C2×#11758224—22no—
45C22×Dic122C2×C411yescharacteristic
46C22×C22⋊D61D412yes—
47C23×Dic61D412yes—
48C22×C4×D61—21no—
49C22×C4×D61C2311yescharacteristic
50#14519734—22no—
51#14519734—22no—
52C2×C4⋊Dic61C2×C411yescharacteristic
53C2×#11758196—21no—
54C2×#11758196—21no—
55C4⋊C4×D61—22no—
56C4⋊C4×D61—22no—
57C2×C22⋊Dic61—22no—
58C2×#11758198—21no—
59C2×C4×Dic61C2×C411yescharacteristic
60#14519684—22no—
61#14519684—22no—
62C22×D4×C61C2×C411yescharacteristic
63C2×C4⋊C4×C61C2×C411yescharacteristic
64C2×C22⋊C4×C61—22no—
65C2×C42×C61C2×C411yescharacteristic
66C24×C61C22⋊C412yes—
67C23×D61C2×D411yescharacteristic
68C22×#6923—42no—
69C2×D4×C61—21no—
70C2×D4×C61—21no—
71C2×D4×C61—22no—
72C2×D4×C61—22no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61C22×C411yescharacteristic
76C2×C22⋊D61—22no—
77C2×C22⋊D61—24no—
78C2×C22⋊D61—24no—
79C2×C22⋊D61—22no—
80C22×Dic61—22no—
81C22×Dic61—24no—
82C22×Dic61—24no—
83C22×Dic61—22no—
84C22×Dic61—21no—
85C22×Dic61—21no—
86C22×Dic61C22⋊C412yes—
87C22×Dic61C2×D411yescharacteristic
88#11758224—42no—
89#11758224—42no—
90#11758224—42no—
91#11758224—42no—
92#11758224—42no—
93#11758224—42no—
94#11758224—42no—
95#11758224—42no—
96C2×C4×D61—42no—
97C2×C4×D61—82no—
98C2×C4×D61C2×D412yes—
99C2×C4×D61C2×D412yes—
100C2×C4×D61#4112yes—
101C2×C4×D61#4112yes—
102C2×C4×D61#4112yes—
103C2×C4×D61#4112yes—
104C2×Dic122—21no—
105C2×Dic122—21no—
106C2×Dic122—22no—
107C2×Dic122—22no—
108C22⋊#6923—24no—
109C22⋊#6923—42no—
110#11758217—42no—
111#11758217—42no—
112C2×#92420—24no—
113C4⋊#6923—24no—
114C4⋊#6923—42no—
115C4×#6923—24no—
116C4×#6923—42no—
117#11758213—42no—
118#11758213—42no—
119D61⋊C8—24no—
120D61⋊C8—24no—
121C4⋊C4×C61—22no—
122C22⋊C4×C61—82no—
123C42×C61—22no—
124C22⋊Dic61—82no—
125#11758198—41no—
126#11758198—41no—
127C4⋊Dic61—22no—
128#11758196—41no—
129#11758196—42no—
130#11758196—41no—
131C4×Dic61—22no—
132C23×C61—22no—
133C23×C61—22no—
134C23×C61—22no—
135C23×C61—42no—
136C23×C61—42no—
137C23×C61C2×C22⋊C411yescharacteristic
138C22×D61—41no—
139C22×D61—41no—
140C22×D61#12011yescharacteristic
141C22×D61#12112yes—
142C22×D61#12312yes—
143C22×D61#12311yescharacteristic
144C2×#6923—44no—
145C2×#6923—82no—
146C2×#6923—82no—
147D4×C61—42no—
148D4×C61—42no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61#11711yescharacteristic
155C2×C4×C61#11711yescharacteristic
156C2×C4×C61C4×D412yes—
157C2×C4×C61C4×D412yes—
158C22⋊D61—44no—
159C22⋊D61—44no—
160C2×Dic61—22no—
161C2×Dic61—22no—
162C2×Dic61—22no—
163C2×Dic61—44no—
164C2×Dic61—44no—
165C2×Dic61—41no—
166C2×Dic61—42no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61#12011yescharacteristic
171C2×Dic61#12112yes—
172C2×Dic61#12312yes—
173C2×Dic61#12311yescharacteristic
174C4×D61—84no—
175C4×D61#12112yes—
176C4×D61#12112yes—
177C4×D61#12312yes—
178C4×D61#12312yes—
179C4×D61#12412yes—
180C4×D61#12412yes—
181C4×D61#12612yes—
182C4×D61#12612yes—
183Dic122—42no—
184Dic122—42no—
185#92420—44no—
186#11154—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—44no—
190C22×C61—42no—
191C22×C61—42no—
192C22×C61—42no—
193C22×C61—82no—
194C22×C61#38111yescharacteristic
195C22×C61#38611yescharacteristic
196C22×C61#38611yescharacteristic
197C2×D61—22no—
198C2×D61—22no—
199C2×D61—41no—
200C2×D61—42no—
201C2×D61—41no—
202C2×D61#39412yes—
203C2×D61#39712yes—
204#6923—84no—
205C4×C61—81no—
206C4×C61—81no—
207C4×C61#38812yes—
208C4×C61#38812yes—
209Dic61—42no—
210Dic61—81no—
211Dic61—81no—
212Dic61#39412yes—
213Dic61#39712yes—
214C2×#410—611no—
215#2879—611no—
216#2311—611no—
217#1736—612no—
218C2×H(3,ℤ/4)—612no—
219C2×C61—21no—
220C2×C61—21no—
221C2×C61—82no—
222C2×C61#144011yescharacteristic
223C2×C61#190911yescharacteristic
224C2×C61#191011yescharacteristic
225D61—22no—
226D61—22no—
227C22×#41—611no—
228C22×M16—612no—
229C2×#117—611no—
230C2×#117—612no—
231C22×C4⋊C4—611no—
232#410—1224no—
233C2×#99—1222no—
234#408—1224no—
235#407—1222no—
236#391—611no—
237#391—611no—
238#390—611no—
239#390#692311yescharacteristic, complemented
240#386—1222no—
241#380—1222no—
242#340—1224no—
243H(3,ℤ/4)—1224no—
244C61#1115411yescharacteristic, complemented, Sylow
245C2×#41—1222no—
246C2×#41—1222no—
247C2×#41—1222no—
248C2×#41—1222no—
249C2×#41—1222no—
250C2×#41—1222no—
251C22×Q8—611no—
252C22×D4—612no—
253C22×D4C2×#692311yescharacteristic
254C23×C4—611no—
255C23×C4—612no—
256C23×C4—1221no—
257C2×M16—1222no—
258C2×M16—1222no—
259C2×M16—1222no—
260C2×M16—1222no—
261C2×M16—1222no—
262C2×M16—1222no—
263C22×C8—614no—
264#117—614no—
265#117—1222no—
266#117—1222no—
267#117—1222no—
268#117—1222no—
269#117—1222no—
270C2×C4⋊C4—611no—
271C2×C4⋊C4—1222no—
272C2×C4⋊C4—1222no—
273C2×C4⋊C4—1222no—
274C2×C4⋊C4—1221no—
275C2×C4⋊C4—1221no—
276C2×C4⋊C4C2×#692311yescharacteristic
277C2×C22⋊C4—22no—
278C2×C22⋊C4—1222no—
279C2×C22⋊C4—1222no—
280C2×C22⋊C4—1221no—
281C2×C42—611no—
282C2×C42—1222no—
283C2×C42C2×#692311yescharacteristic
284#99—1228no—
285C22⋊C8—1228no—
286#95—1222no—
287#95—1222no—
288C24—611no—
289C24C22⋊#692312yes—
290#41—2442no—
291#41—2442no—
292#41—2442no—
293#41—2442no—
294#41—2442no—
295#41—2442no—
296#41—2442no—
297#41—2442no—
298C2×Q8—1221no—
299C2×Q8—1221no—
300C2×Q8—1222no—
301C2×Q8—1222no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—1222no—
307C2×D4—1224no—
308C2×D4—1222no—
309C2×D4—1224no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—611no—
313C22×C4—612no—
314C22×C4—612no—
315C22×C4—612no—
316C22×C4—612no—
317C22×C4—612no—
318C22×C4—612no—
319C22×C4—612no—
320C22×C4—1222no—
321C22×C4—1224no—
322C22×C4—1224no—
323C22×C4—1222no—
324C22×C4—1221no—
325C22×C4—1221no—
326C22×C4—4882no—
327C22×C4—2442no—
328C22×C4—2442no—
329C22×C4C22×#692311yescharacteristic
330M16—2442no—
331M16—2442no—
332M16—2442no—
333M16—2442no—
334C2×C8—1224no—
335C2×C8—1224no—
336C2×C8—1224no—
337C4⋊C4—22no—
338C4⋊C4—1224no—
339C4⋊C4—1222no—
340C4⋊C4—2441no—
341C4⋊C4—2441no—
342C4⋊C4—2442no—
343C4⋊C4—2442no—
344C22⋊C4—82no—
345C22⋊C4—1224no—
346C22⋊C4—4882no—
347C22⋊C4—2441no—
348C22⋊C4—2441no—
349C22⋊C4—2442no—
350C42—22no—
351C42—1224no—
352C42—1222no—
353C42—2442no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—42no—
358C23—42no—
359C23—612no—
360C23—612no—
361C23—611no—
362C23—611no—
363C23—2441no—
364C23—2441no—
365C23C2×C22⋊#692311yescharacteristic
366Q8—2442no—
367Q8—2442no—
368D4—42no—
369D4—42no—
370D4—2444no—
371D4—2444no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—612no—
378C2×C4—612no—
379C2×C4—611no—
380C2×C4—611no—
381C2×C4—612no—
382C2×C4—612no—
383C2×C4—612no—
384C2×C4—612no—
385C2×C4—612no—
386C2×C4—612no—
387C2×C4—612no—
388C2×C4—612no—
389C2×C4—1222no—
390C2×C4—1222no—
391C2×C4—1222no—
392C2×C4—4884no—
393C2×C4—4882no—
394C2×C4—4882no—
395C2×C4—2444no—
396C2×C4—2444no—
397C2×C4—2441no—
398C2×C4—2442no—
399C2×C4—2441no—
400C2×C4—2444no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4#1451985611yescharacteristic
404C2×C4#1451985611yescharacteristic
405C2×C4D4×#692312yes—
406C2×C4D4×#692312yes—
407C8—2444no—
408C22—21no—
409C22—21no—
410C22—44no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—82no—
415C22—612no—
416C22—612no—
417C22—1222no—
418C22—1222no—
419C22—2441no—
420C22—2441no—
421C22—2442no—
422C22#1684070511yescharacteristic
423C22#1684071011yescharacteristic
424C22#1684071011yescharacteristic
425C4—81no—
426C4—81no—
427C4—612no—
428C4—612no—
429C4—4881no—
430C4—4884no—
431C4—4881no—
432C4—2442no—
433C4#1684071212yes—
434C4#1684071212yes—
435C2—21no—
436C2—21no—
437C2—82no—
438C2—1222no—
439C2—1222no—
440C2#2146100311yescharacteristic
441C2#2146186011yescharacteristic
442C2#2146186211yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC22×C61#386
prime radicalC22×#11758224C22
cosocleC22×C4×D61C23
hypercenter#390#6923
hypocenterC61#11154
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
3. #21461506C2semidirect
4. C2×#16840629C2semidirect
17. #16839668C4semidirect
18. #16839667C4semidirect
19. #16839665C4semidirect
22. C61×#390C4semidirect
239. #390#6923semidirect
244. C61#11154semidirect

Maximal subgroups

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

subgroupindex
C2×#168407422
#214624052
#214615062
C2×#168406292
#214484092
#1115461