Finite Group Databasebeta

← #30598378

Subgroups — #30598378

Order 15616.

780 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 104 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
1#21461722C212yesmaximal, complemented
2#21461679C211yesmaximal, characteristic
3#21461258C212yesmaximal, complemented
4#21461054C211yesmaximal, characteristic
5C2×#16840451C211yesmaximal, characteristic
6C2×#14519835—21no—
7D61⋊(C22⋊C8)C2212yes—
8#16840704C2212yes—
9#16840640—21no—
10#16840635—24no—
11#16840615—24no—
12#16840608—21no—
13#16840605—21no—
14#16840451—24no—
15#16840451—24no—
16C2×#14519770—21no—
17C2×#14519769—21no—
18C2×#14519768C411yescharacteristic, complemented
19C2×#14519759—21no—
20C2×#14519757C2211yescharacteristic
21C2×#14519756C411yescharacteristic, complemented
22C2×#14519742C411yescharacteristic, complemented
23C2×C4×C22⋊D61C2211yescharacteristic
24C2×#14519734—21no—
25C2×C4⋊C4×D61C2211yescharacteristic
26C2×#117×C61C411yescharacteristic, complemented
27C2×D61⋊C8—22no—
28C2×C4⋊#6923—22no—
29C2×C22⋊#6923—22no—
30#14519835—42no—
31#14519834—22no—
32C22⋊#92420—22no—
33(C2×#6923)⋊C4—22no—
34C22×Dic122C2×C411yescharacteristic
35C22×C22⋊D61D411yescharacteristic
36C22×C4×D61—21no—
37C22×C4×D61C2311yescharacteristic
38#14519770—28no—
39#14519769—28no—
40#14519768—22no—
41#14519768—22no—
42#14519759—28no—
43#14519757—22no—
44#14519757D414yes—
45#14519756—22no—
46#14519756—22no—
47#14519742—22no—
48#14519742—22no—
49C4×C22⋊D61—22no—
50C4×C22⋊D61—22no—
51#14519734—28no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2×C411yescharacteristic
55C2×#11758196—21no—
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×#11758196—21no—
59C2×#11758196D411yescharacteristic
60C4⋊C4×D61—22no—
61C4⋊C4×D61D414yes—
62C2×C22⋊Dic61—21no—
63C2×C22⋊Dic61D411yescharacteristic
64C2×#11758198—21no—
65C2×#11758198D411yescharacteristic
66C2×C4×Dic61—21no—
67C2×C4×Dic61C2×C411yescharacteristic
68C23×C4×C61C2×C411yescharacteristic
69#117×C61—22no—
70#117×C61—22no—
71C2×C4⋊C4×C61C2×C411yescharacteristic
72C2×C4⋊C4×C61C2×C411yescharacteristic
73C2×C22⋊C4×C61—21no—
74C2×C42×C61—21no—
75C24×C61C22⋊C411yescharacteristic
76C23×D61C2×D411yescharacteristic
77C22×#6923—42no—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—42no—
83C22×C4×C61C22⋊C411yescharacteristic
84C22×C4×C61C22×C411yescharacteristic
85C2×C22⋊D61—22no—
86C2×C22⋊D61—22no—
87C2×C22⋊D61—21no—
88C2×C22⋊D61—21no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61C22⋊C411yescharacteristic
92C22×Dic61C22⋊C411yescharacteristic
93C22×Dic61C2×D411yescharacteristic
94C2×C4×D61—24no—
95C2×C4×D61—42no—
96C2×C4×D61—42no—
97C2×C4×D61—41no—
98C2×C4×D61—41no—
99C2×C4×D61C2×D412yes—
100C2×C4×D61C2×D412yes—
101C2×Dic122—21no—
102C2×Dic122—42no—
103C2×Dic122C22⋊C412yes—
104C22⋊#6923—42no—
105C22⋊#6923—42no—
106C2×#92420—42no—
107C4⋊#6923—44no—
108D61⋊C8—44no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61C22⋊C412yes—
112C4⋊C4×C61C22⋊C412yes—
113C22⋊C4×C61—24no—
114C42×C61—24no—
115C22⋊Dic61—24no—
116C22⋊Dic61—22no—
117#11758198—24no—
118#11758198—22no—
119C4⋊Dic61—24no—
120C4⋊Dic61—21no—
121C4⋊Dic61—24no—
122C4⋊Dic61C22⋊C412yes—
123#11758196—24no—
124#11758196—24no—
125#11758196—24no—
126#11758196—24no—
127#11758196—22no—
128C4×Dic61—24no—
129C4×Dic61—21no—
130C4×Dic61—21no—
131C23×C61—21no—
132C23×C61—21no—
133C23×C61—21no—
134C23×C61—41no—
135C23×C61—41no—
136C23×C61C2×C22⋊C411yescharacteristic
137C22×D61—41no—
138C22×D61—41no—
139C22×D61#12012yes—
140C22×D61#13512yes—
141C22×D61AGL(1,ℤ/8)11yescharacteristic
142C22×D61#13711yescharacteristic
143C2×#6923—82no—
144C2×#6923—82no—
145C2×#6923—82no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—21no—
149C2×C4×C61—22no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—42no—
155C2×C4×C61—41no—
156C2×C4×C61C2×C22⋊C411yescharacteristic
157C2×C4×C61C2×C22⋊C411yescharacteristic
158C22⋊D61—42no—
159C22⋊D61—42no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—22no—
163C2×Dic61—22no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—41no—
167C2×Dic61—42no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61#12012yes—
172C4×D61—24no—
173C4×D61—42no—
174C4×D61—44no—
175C4×D61—44no—
176Dic122—42no—
177Dic122—42no—
178#92420—82no—
179#8382—611nomaximal, Sylow
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—41no—
183C22×C61—41no—
184C22×C61—41no—
185C22×C61—81no—
186C22×C61—81no—
187C22×C61#37911yescharacteristic
188C22×C61#41611yescharacteristic
189C22×C61#41711yescharacteristic
190C2×D61—22no—
191C2×D61—22no—
192C2×D61—42no—
193C2×D61—41no—
194C2×D61—41no—
195C2×D61#44912yes—
196C2×D61#45212yes—
197#6923—162no—
198C4×C61—21no—
199C4×C61—42no—
200C4×C61—41no—
201C4×C61—42no—
202C4×C61#41012yes—
203Dic61—22no—
204Dic61—41no—
205Dic61—42no—
206Dic61—42no—
207Dic61—41no—
208C2×#552—611no—
209#1823—612no—
210#1799—611no—
211#1520—612no—
212#1459—611no—
213C2×C61—21no—
214C2×C61—21no—
215C2×C61—81no—
216C2×C61#178211yescharacteristic
217C2×C61#178811yescharacteristic
218C2×C61#179611yescharacteristic
219D61—22no—
220D61—22no—
221#552—1224no—
222#552—1224no—
223C2×#126—1221no—
224C2×#125—611no—
225C2×#125—1221no—
226C2×#123—1221no—
227C2×C22⋊Q8—611no—
228C2×C22⋊Q8—611no—
229C2×C4×Q8—611no—
230C2×C4×D4—611no—
231C2×#117—1221no—
232C2×#117#692311yescharacteristic, complemented
233C22×C4⋊C4—611no—
234C2×#99—1221no—
235C2×C22⋊C8—612no—
236#380—612no—
237#342—1221no—
238#340—1224no—
239#328—1224no—
240#324—1221no—
241#323—1221no—
242C61#838211yescharacteristic, complemented, Sylow
243C22×Q8—611no—
244C22×D4—611no—
245C23×C4—611no—
246C23×C4—1221no—
247C23×C4C2×#692311yescharacteristic
248C22×C8—1222no—
249#126—1228no—
250#125—1228no—
251#125—1222no—
252#125—1222no—
253#123—1228no—
254C22⋊Q8—614no—
255C22⋊Q8—1222no—
256C22⋊Q8—1222no—
257C22⋊Q8—1222no—
258C4×Q8—1222no—
259C4×Q8—1222no—
260C4×D4—1222no—
261C4×D4—1222no—
262#117—22no—
263#117—22no—
264#117—1228no—
265C2×C4⋊C4—611no—
266C2×C4⋊C4—614no—
267C2×C4⋊C4—611no—
268C2×C4⋊C4—1222no—
269C2×C4⋊C4—1221no—
270C2×C4⋊C4—1221no—
271C2×C4⋊C4—1221no—
272C2×C4⋊C4—1221no—
273C2×C4⋊C4—1221no—
274C2×C4⋊C4—1221no—
275C2×C4⋊C4—1222no—
276C2×C4⋊C4C2×#692311yescharacteristic
277C2×C4⋊C4C2×#692311yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—611no—
280C2×C22⋊C4—611no—
281C2×C22⋊C4—1221no—
282C2×C22⋊C4—1221no—
283C2×C22⋊C4—1222no—
284C2×C42—21no—
285C2×C42—611no—
286C2×C42—1221no—
287#99—2442no—
288C22⋊C8—1222no—
289C22⋊C8—1222no—
290#95—1222no—
291C24—611no—
292C24C22⋊#692311yescharacteristic
293C2×Q8—612no—
294C2×Q8—1221no—
295C2×Q8—2442no—
296C2×D4—1221no—
297C2×D4—1222no—
298C2×D4—1222no—
299C2×D4—1221no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—42no—
305C22×C4—611no—
306C22×C4—612no—
307C22×C4—612no—
308C22×C4—611no—
309C22×C4—611no—
310C22×C4—1224no—
311C22×C4—1221no—
312C22×C4—1221no—
313C22×C4—2442no—
314C22×C4—2442no—
315C22×C4—2441no—
316C22×C4—2441no—
317C22×C4—2442no—
318C22×C4C22⋊#692311yescharacteristic
319C22×C4C22×#692311yescharacteristic
320C2×C8—2442no—
321C2×C8—2444no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—612no—
325C4⋊C4—1224no—
326C4⋊C4—1222no—
327C4⋊C4—1224no—
328C4⋊C4—1224no—
329C4⋊C4—1224no—
330C4⋊C4—1224no—
331C4⋊C4—1224no—
332C4⋊C4—1221no—
333C4⋊C4—2444no—
334C4⋊C4C22⋊#692312yes—
335C4⋊C4C22⋊#692312yes—
336C22⋊C4—24no—
337C22⋊C4—1222no—
338C22⋊C4—1224no—
339C22⋊C4—1222no—
340C22⋊C4—1224no—
341C22⋊C4—2442no—
342C22⋊C4—2442no—
343C42—24no—
344C42—1224no—
345C42—1221no—
346C42—1221no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23—612no—
353C23—612no—
354C23—611no—
355C23—611no—
356C23—2441no—
357C23—2441no—
358C23C2×C22⋊#692311yescharacteristic
359Q8—2442no—
360Q8—2442no—
361D4—2442no—
362D4—2442no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—612no—
374C2×C4—1224no—
375C2×C4—1221no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1222no—
379C2×C4—1221no—
380C2×C4—1222no—
381C2×C4—4882no—
382C2×C4—4882no—
383C2×C4—4882no—
384C2×C4—2442no—
385C2×C4—2444no—
386C2×C4—2444no—
387C2×C4—2441no—
388C2×C4—2442no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4—2441no—
392C2×C4C2×C22⋊#692311yescharacteristic
393C2×C4C2×C22⋊#692311yescharacteristic
394C8—4882no—
395C22—21no—
396C22—21no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—81no—
401C22—81no—
402C22—612no—
403C22—612no—
404C22—1222no—
405C22—1222no—
406C22—2442no—
407C22—2441no—
408C22—2441no—
409C22#1684070311yescharacteristic
410C22#1684075211yescharacteristic
411C22#1684075311yescharacteristic
412C4—21no—
413C4—41no—
414C4—42no—
415C4—42no—
416C4—1222no—
417C4—9762no—
418C4—2442no—
419C4—2441no—
420C4—2442no—
421C4—2441no—
422C4#1684074112yes—
423C2—21no—
424C2—21no—
425C2—81no—
426C2—1222no—
427C2—1222no—
428C2#2146164811yescharacteristic
429C2#2146165911yescharacteristic
430C2#2146167211yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#117×C61C4
socleC22×C61#379
prime radicalC22×C22⋊D61D4
cosocleC22×C4×D61C23
hypercenterC2×#117#6923
hypocenterC61#8382
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #21461722C2semidirect
3. #21461258C2semidirect
18. C2×#14519768C4semidirect
21. C2×#14519756C4semidirect
22. C2×#14519742C4semidirect
26. C2×#117×C61C4semidirect
232. C2×#117#6923semidirect
242. C61#8382semidirect

Maximal subgroups

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

subgroupindex
#214617222
#214616792
#214612582
#214610542
C2×#168404512
#838261