Finite Group Databasebeta

← #30598228

Subgroups — #30598228

Order 15616.

876 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 128 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#21461711C211yesmaximal, characteristic
2C2×#16840616C212yesmaximal
3C2×#16840615C212yesmaximal
4#21460995C211yesmaximal, characteristic
5C2×#16840316C211yesmaximal, characteristic
6C2×#14519834C2212yes—
7C2×(C2×#6923)⋊C4—21no—
8C2×(C2×#6923)⋊C4—21no—
9C2×(C2×#6923)⋊C4C2212yes—
10#16840628—21no—
11#16840628—21no—
12#16840616—24no—
13#16840615—24no—
14#16840316—24no—
15#16840316—24no—
16C2×#14519776C411yescharacteristic, complemented
17C2×#14519775C411yescharacteristic, complemented
18C2×#14519767—21no—
19C2×#14519757C2211yescharacteristic
20C2×#14519755—21no—
21C2×#14519753C2211yescharacteristic
22C2×C4×Dic122—21no—
23C2×C4×Dic122C411yescharacteristic, complemented
24C2×#14519734—21no—
25C2×C42×D61C2211yescharacteristic
26C2×C4×Q8×C61C411yescharacteristic, complemented
27C23×#6923—22no—
28C22×#92420—22no—
29#14519834—22no—
30#14519834—22no—
31(C2×#6923)⋊C4—24no—
32(C2×#6923)⋊C4—22no—
33(C2×#6923)⋊C4—22no—
34(C2×#6923)⋊C4—42no—
35C22×Dic122—21no—
36C22×Dic122C2×C411yescharacteristic
37C22×C4×D61D411yescharacteristic
38C22×C4×D61D411yescharacteristic
39C22×C4×D61C2311yescharacteristic
40#14519776—22no—
41#14519776—22no—
42#14519775—22no—
43#14519775—22no—
44#14519767—28no—
45#14519757—22no—
46#14519757D414yes—
47#14519755—28no—
48#14519753—22no—
49#14519753D414yes—
50C4×Dic122—28no—
51C4×Dic122—22no—
52C4×Dic122—22no—
53#14519734—28no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61C2×C411yescharacteristic
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×#11758196—21no—
61C2×#11758198—21no—
62C2×#11758198—21no—
63C2×C4×Dic61—21no—
64C2×C4×Dic61C2×C411yescharacteristic
65C2×C4×Dic61D411yescharacteristic
66C2×C4×Dic61D411yescharacteristic
67C42×D61—22no—
68C42×D61—22no—
69C22×Q8×C61C2×C411yescharacteristic
70C4×Q8×C61—22no—
71C4×Q8×C61—22no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61C2×C411yescharacteristic
74C2×C42×C61—21no—
75C2×C42×C61C2×C411yescharacteristic
76C23×D61C2×D411yescharacteristic
77C22×#6923—24no—
78C22×#6923—42no—
79C22×#6923—42no—
80C22×#6923—42no—
81C22×#6923—42no—
82C22×#6923—42no—
83C22×#6923—42no—
84C2×Q8×C61—21no—
85C2×Q8×C61—41no—
86C2×Q8×C61—41no—
87C2×Q8×C61C22⋊C412yes—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61C22⋊C411yescharacteristic
91C22×C4×C61C22⋊C411yescharacteristic
92C22×C4×C61C22×C411yescharacteristic
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61C22⋊C411yescharacteristic
96C22×Dic61C22⋊C411yescharacteristic
97C22×Dic61C2×D411yescharacteristic
98C2×C4×D61—22no—
99C2×C4×D61—22no—
100C2×C4×D61—22no—
101C2×C4×D61—22no—
102C2×C4×D61—41no—
103C2×C4×D61—41no—
104C2×C4×D61—41no—
105C2×C4×D61SD1612yes—
106C2×C4×D61Q1612yes—
107C2×C4×D61C2×D412yes—
108C2×C4×D61C2×D412yes—
109C2×Dic122—24no—
110C2×Dic122—21no—
111C2×Dic122—42no—
112C2×Dic122—42no—
113C2×Dic122—41no—
114C2×Dic122—41no—
115C2×Dic122C22⋊C412yes—
116C2×#92420—42no—
117C2×#92420—42no—
118C2×#92420—42no—
119C4⋊C4×C61—24no—
120C4⋊C4×C61—21no—
121C4⋊C4×C61C22⋊C412yes—
122C42×C61—24no—
123C42×C61—21no—
124C42×C61—21no—
125#11758198—24no—
126#11758198—24no—
127C4⋊Dic61—21no—
128C4⋊Dic61—24no—
129C4⋊Dic61C22⋊C412yes—
130#11758196—24no—
131#11758196—24no—
132#11758196—24no—
133#11758196—24no—
134#11758196—24no—
135C4×Dic61—24no—
136C4×Dic61—21no—
137C4×Dic61—21no—
138C4×Dic61—22no—
139C4×Dic61—22no—
140C23×C61C2×C22⋊C411yescharacteristic
141C22×D61—21no—
142C22×D61—21no—
143C22×D61—21no—
144C22×D61—21no—
145C22×D61#12012yes—
146C22×D61C2×SD1611yescharacteristic
147C22×D61C2×Q1611yescharacteristic
148C22×D61#13712yes—
149C2×#6923—44no—
150C2×#6923—44no—
151C2×#6923—44no—
152C2×#6923—82no—
153C2×#6923—82no—
154C2×#6923—82no—
155C2×#6923—82no—
156Q8×C61—42no—
157Q8×C61—42no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—22no—
164C2×C4×C61—22no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61#10312yes—
172C2×C4×C61C2×C22⋊C411yescharacteristic
173C2×C4×C61C2×C22⋊C411yescharacteristic
174C2×Dic61—21no—
175C2×Dic61—21no—
176C2×Dic61—21no—
177C2×Dic61—21no—
178C2×Dic61—21no—
179C2×Dic61—21no—
180C2×Dic61—22no—
181C2×Dic61—22no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61#10312yes—
187C2×Dic61#10412yes—
188C2×Dic61#12012yes—
189C4×D61—24no—
190C4×D61—24no—
191C4×D61—42no—
192C4×D61—42no—
193C4×D61—44no—
194Dic122—48no—
195Dic122—42no—
196Dic122—42no—
197#92420—82no—
198#8333—611nomaximal, Sylow
199C22×C61#37911yescharacteristic
200C22×C61#41011yescharacteristic
201C22×C61#41011yescharacteristic
202C22×C61C2×#10311yescharacteristic
203C22×C61#41911yescharacteristic
204C22×C61C2×#10411yescharacteristic
205C22×C61#42111yescharacteristic
206C2×D61—21no—
207C2×D61—21no—
208C2×D61—22no—
209C2×D61—22no—
210C2×D61—21no—
211C2×D61—21no—
212C2×D61—22no—
213C2×D61—22no—
214C2×D61#44812yes—
215C2×D61#45112yes—
216#6923—84no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—22no—
220C4×C61—42no—
221C4×C61—42no—
222C4×C61—41no—
223Dic61—22no—
224Dic61—22no—
225Dic61—22no—
226Dic61—42no—
227Dic61—42no—
228C2×#533—611no—
229#1817—611no—
230C2×#328—612no—
231C2×#328—612no—
232#1434—611no—
233C2×C61#150311yescharacteristic
234C2×C61#150611yescharacteristic
235C2×C61#151511yescharacteristic
236C2×C61#151811yescharacteristic
237C2×C61#178211yescharacteristic
238C2×C61#178711yescharacteristic
239C2×C61#178911yescharacteristic
240D61—22no—
241D61—22no—
242#533—1224no—
243#533—1224no—
244C2×C4⋊Q8—611no—
245C2×C4⋊Q8—611no—
246C2×#125—1221no—
247C2×C22⋊Q8—611no—
248C2×C22⋊Q8—611no—
249C2×C22⋊Q8—1221no—
250C2×C4×Q8—611no—
251C2×C4×Q8—1221no—
252C2×C4×Q8#692311yescharacteristic, complemented
253C2×#117—1221no—
254C22×C42—611no—
255C2×C22⋊C8—612no—
256C2×#95—612no—
257C2×#95—1221no—
258C2×#95—1221no—
259#336—1221no—
260#336—1221no—
261#328—1224no—
262#328—1224no—
263C61#833311yescharacteristic, complemented, Sylow
264C22×Q8—611no—
265C22×Q8—1221no—
266C22×Q8C2×#692311yescharacteristic
267C23×C4—611no—
268C23×C4—611no—
269C23×C4—611no—
270C23×C4—1222no—
271C22×C8—1222no—
272C4⋊Q8—1222no—
273C4⋊Q8—1222no—
274C4⋊Q8—1222no—
275C4⋊Q8—1222no—
276#125—1228no—
277C22⋊Q8—614no—
278C22⋊Q8—614no—
279C22⋊Q8—1222no—
280C22⋊Q8—1228no—
281C22⋊Q8—1222no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—1222no—
285C4×Q8—1222no—
286C4×Q8—1228no—
287#117—1228no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—611no—
290C2×C4⋊C4—1221no—
291C2×C4⋊C4—1221no—
292C2×C4⋊C4—1221no—
293C2×C4⋊C4—1221no—
294C2×C4⋊C4—1221no—
295C2×C4⋊C4—1221no—
296C2×C4⋊C4C2×#692311yescharacteristic
297C2×C22⋊C4—1221no—
298C2×C22⋊C4—1221no—
299C2×C42—21no—
300C2×C42—611no—
301C2×C42—611no—
302C2×C42—611no—
303C2×C42—1222no—
304C2×C42—1222no—
305C2×C42—1221no—
306C2×C42C2×#692311yescharacteristic
307C22⋊C8—1222no—
308C22⋊C8—1222no—
309#95—1222no—
310#95—1222no—
311#95—1224no—
312#95—2442no—
313C24—611no—
314C2×Q8—21no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—612no—
318C2×Q8—1224no—
319C2×Q8—1221no—
320C2×Q8—2441no—
321C2×Q8—2441no—
322C2×Q8—2442no—
323C2×Q8—2442no—
324C2×Q8C22⋊#692312yes—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—611no—
328C22×C4—611no—
329C22×C4—611no—
330C22×C4—612no—
331C22×C4—612no—
332C22×C4—612no—
333C22×C4—612no—
334C22×C4—1222no—
335C22×C4—1222no—
336C22×C4—1222no—
337C22×C4—1222no—
338C22×C4—1221no—
339C22×C4—1221no—
340C22×C4—1224no—
341C22×C4—2441no—
342C22×C4—2441no—
343C22×C4—2441no—
344C22×C4—2442no—
345C22×C4—2442no—
346C22×C4—2442no—
347C22×C4—2442no—
348C22×C4—2442no—
349C22×C4—2442no—
350C22×C4C22⋊#692311yescharacteristic
351C22×C4C22⋊#692311yescharacteristic
352C22×C4C22×#692311yescharacteristic
353C2×C8—2442no—
354C2×C8—2442no—
355C2×C8—2442no—
356C4⋊C4—24no—
357C4⋊C4—21no—
358C4⋊C4—612no—
359C4⋊C4—1224no—
360C4⋊C4—1224no—
361C4⋊C4—1224no—
362C4⋊C4—1224no—
363C4⋊C4—1224no—
364C4⋊C4—1224no—
365C4⋊C4—1221no—
366C4⋊C4C22⋊#692312yes—
367C22⋊C4—1224no—
368C22⋊C4—1224no—
369C42—24no—
370C42—21no—
371C42—21no—
372C42—1222no—
373C42—1222no—
374C42—1224no—
375C42—1221no—
376C42—1221no—
377C23—611no—
378C23—611no—
379C23—612no—
380C23—612no—
381C23—1221no—
382C23—1221no—
383C23—1221no—
384C23—1221no—
385C23C2×C22⋊#692311yescharacteristic
386Q8—42no—
387Q8—42no—
388Q8—2442no—
389Q8—2442no—
390Q8—2448no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—612no—
405C2×C4—612no—
406C2×C4—612no—
407C2×C4—1224no—
408C2×C4—1224no—
409C2×C4—1221no—
410C2×C4—1221no—
411C2×C4—1221no—
412C2×C4—1221no—
413C2×C4—1221no—
414C2×C4—1221no—
415C2×C4—1222no—
416C2×C4—1222no—
417C2×C4—4882no—
418C2×C4—4882no—
419C2×C4—4882no—
420C2×C4—4882no—
421C2×C4—2442no—
422C2×C4—2442no—
423C2×C4—2444no—
424C2×C4—2444no—
425C2×C4—2444no—
426C2×C4—2444no—
427C2×C4—2441no—
428C2×C4—2441no—
429C2×C4—2441no—
430C2×C4—2441no—
431C2×C4#1451984012yes—
432C2×C4C2×C22⋊#692311yescharacteristic
433C2×C4C2×C22⋊#692311yescharacteristic
434C8—4882no—
435C22—612no—
436C22—612no—
437C22—1221no—
438C22—1221no—
439C22—1222no—
440C22—1222no—
441C22—1221no—
442C22—1221no—
443C22—1222no—
444C22—1222no—
445C22#1684070311yescharacteristic
446C22#1684074111yescharacteristic
447C22#1684074111yescharacteristic
448C22C2×#1451984011yescharacteristic
449C22#1684075511yescharacteristic
450C22D61⋊#10411yescharacteristic
451C22#1684075911yescharacteristic
452C4—21no—
453C4—21no—
454C4—22no—
455C4—42no—
456C4—41no—
457C4—42no—
458C4—1222no—
459C4—1222no—
460C4—1222no—
461C4—4884no—
462C4—2442no—
463C4—2442no—
464C2—1222no—
465C2—1222no—
466C2#2146120111yescharacteristic
467C2#2146121211yescharacteristic
468C2#2146124111yescharacteristic
469C2#2146125211yescharacteristic
470C2#2146164811yescharacteristic
471C2#2146165811yescharacteristic
472C2#2146166111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenterC2×C4×Q8#6923
hypocenterC61#8333
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
16. C2×#14519776C4semidirect
17. C2×#14519775C4semidirect
23. C2×C4×Dic122C4semidirect
26. C2×C4×Q8×C61C4semidirect
252. C2×C4×Q8#6923semidirect
263. C61#8333semidirect

Maximal subgroups

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

subgroupindex
#214617112
C2×#168406162
C2×#168406152
#214609952
C2×#168403162
#833361