Finite Group Databasebeta

← #30603024

Subgroups — #30603024

Order 15616.

1258 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 178 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×#16840705C212yesmaximal
2#21462255C211yesmaximal, characteristic
3#21461516C211yesmaximal, characteristic
4C2×#16840639C212yesmaximal
5#381×D61C211yesmaximal, characteristic
6C22×C22⋊#6923C2212yes—
7C22×C22⋊#6923C2212yes—
8#16840705—24no—
9#16840705C418yescomplemented
10C22⋊C4×#6923—24no—
11C4×C22⋊#6923—22no—
12C2×(C2×#6923)⋊C4—24no—
13#16840640—22no—
14#16840639—24no—
15C23×C4×D61C2211yescharacteristic
16C2×C22⋊C4×D61C2211yescharacteristic
17C2×C22⋊C4×D61C2211yescharacteristic
18#16839625C411yescharacteristic, complemented
19#16839624—22no—
20#16839623C411yescharacteristic, complemented
21#16839623C411yescharacteristic, complemented
22#95×D61—22no—
23C61×#381C411yescharacteristic, complemented
24C23×#6923—22no—
25C23×#6923D414yes—
26C2×C22⋊#6923—22no—
27C2×C22⋊#6923—22no—
28C2×C22⋊#6923—22no—
29C2×C22⋊#6923—22no—
30C2×C22⋊#6923—22no—
31C2×C22⋊#6923—22no—
32C2×C22⋊#6923—22no—
33C2×C22⋊#6923—22no—
34C2×C22⋊#6923—22no—
35C2×C22⋊#6923C2×C414yes—
36C2×C22⋊#6923C2×C414yes—
37C2×C22⋊#6923D414yes—
38C2×C4×#6923—24no—
39(C2×#6923)⋊C4—24no—
40(C2×#6923)⋊C4—216no—
41C24×D61C2311yescharacteristic
42C23×Dic61C2×C411yescharacteristic
43C22×C4×D61—22no—
44C22×C4×D61—21no—
45C22×C4×D61—21no—
46C22×C4×D61—22no—
47C22×C4×D61—22no—
48C22×C4×D61—22no—
49C22×C4×D61—22no—
50C22×C4×D61—21no—
51C22×C4×D61C2×C414yes—
52C22×C4×D61D412yes—
53C2×C22⋊Dic61C2×C411yescharacteristic
54C2×C22⋊Dic61C2×C411yescharacteristic
55C2×#11758198—21no—
56C2×#11758198D412yes—
57C22⋊C4×D61—22no—
58C22⋊C4×D61—22no—
59C22⋊C4×D61—22no—
60C22⋊C4×D61—22no—
61#14519684—22no—
62#14519684—22no—
63#14519684—22no—
64C23×C4×C61C2×C411yescharacteristic
65C2×C22⋊C4×C61C2×C411yescharacteristic
66C2×C22⋊C4×C61C2×C411yescharacteristic
67#95×C61—22no—
68C24×C61C22×C411yescharacteristic
69C23×D61—44no—
70C23×D61C22×C412yes—
71C23×D61C2×D412yes—
72C23×D61C2×D411yescharacteristic
73C23×D61C2×D412yes—
74C23×D61C2×D411yescharacteristic
75C23×D61#4112yes—
76C23×D61#4112yes—
77C23×D61#4112yes—
78C22×#6923—24no—
79C22×#6923—24no—
80C22×#6923—24no—
81C22×#6923—48no—
82C22×#6923—48no—
83C22×#6923—42no—
84C22×#6923—42no—
85C22×#6923C22⋊C418yes—
86C22×C4×C61—21no—
87C22×C4×C61—21no—
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61—21no—
91C22×C4×C61C4212yescomplemented
92C22×C4×C61C22⋊C412yes—
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61—22no—
96C22×Dic61—22no—
97C22×Dic61—21no—
98C22×Dic61C4212yescomplemented
99C22×Dic61C22⋊C412yes—
100C2×C4×D61—24no—
101C2×C4×D61—24no—
102C2×C4×D61—24no—
103C2×C4×D61—24no—
104C2×C4×D61—24no—
105C2×C4×D61—24no—
106C2×C4×D61—22no—
107C2×C4×D61—22no—
108C2×C4×D61—41no—
109C2×C4×D61—42no—
110C2×C4×D61—42no—
111C2×C4×D61—42no—
112C2×C4×D61—42no—
113C2×C4×D61—41no—
114C2×C4×D61—41no—
115C2×C4×D61—42no—
116C2×C4×D61—42no—
117C2×C4×D61—41no—
118C2×C4×D61—44no—
119C2×C4×D61—44no—
120C2×C4×D61—42no—
121C22⋊#6923—24no—
122C22⋊#6923—24no—
123C22⋊#6923—24no—
124C22⋊#6923—24no—
125C22⋊#6923—24no—
126C22⋊#6923—24no—
127C22⋊#6923—42no—
128C22⋊#6923—42no—
129C4×#6923—84no—
130C22⋊C4×C61—22no—
131C22⋊C4×C61—22no—
132C22⋊Dic61—22no—
133C22⋊Dic61—22no—
134#11758198—24no—
135#11758198—41no—
136#11758198—41no—
137C23×C61—42no—
138C23×C61C2×C4211yescharacteristic
139C23×C61C2×C22⋊C411yescharacteristic
140C23×C61#11711yescharacteristic
141C23×C61C4×D412yes—
142C23×C61C4×D412yes—
143C22×D61—24no—
144C22×D61—28no—
145C22×D61—44no—
146C22×D61—44no—
147C22×D61—44no—
148C22×D61—44no—
149C22×D61—42no—
150C22×D61—44no—
151C22×D61C2×C22⋊C412yes—
152C22×D61#11712yes—
153C22×D61C4×D412yes—
154C22×D61C4×D412yes—
155C22×D61C4×D412yes—
156C22×D61C4×D412yes—
157C22×D61#12011yescharacteristic
158C22×D61#12112yes—
159C22×D61#12112yes—
160C22×D61#12112yes—
161C22×D61#12312yes—
162C22×D61#12311yescharacteristic
163C22×D61#12411yescharacteristic
164C22×D61#12412yes—
165C22×D61#12612yes—
166C22×D61#12711yescharacteristic
167C2×#6923—28no—
168C2×#6923—48no—
169C2×#6923—48no—
170C2×#6923—48no—
171C2×#6923—44no—
172C2×#6923—84no—
173C2×#6923—84no—
174C2×#6923—84no—
175C2×C4×C61—22no—
176C2×C4×C61—24no—
177C2×C4×C61—22no—
178C2×C4×C61—41no—
179C2×C4×C61—42no—
180C2×C4×C61—41no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—41no—
184C2×Dic61—22no—
185C2×Dic61—24no—
186C2×Dic61—22no—
187C2×Dic61—41no—
188C2×Dic61—42no—
189C2×Dic61—41no—
190C2×Dic61—42no—
191C2×Dic61—42no—
192C2×Dic61—41no—
193C4×D61—44no—
194C4×D61—44no—
195C4×D61—44no—
196C4×D61—44no—
197C4×D61—44no—
198C4×D61—82no—
199C4×D61—84no—
200#10936—611nomaximal, Sylow
201C22×C61—22no—
202C22×C61—24no—
203C22×C61—41no—
204C22×C61—41no—
205C22×C61—42no—
206C22×C61C4×C22⋊C411yescharacteristic
207C22×C61C4×C22⋊C412yes—
208C22×C61#38611yescharacteristic
209C22×C61#38812yes—
210C22×C61#39011yescharacteristic
211C2×D61—22no—
212C2×D61—24no—
213C2×D61—24no—
214C2×D61—24no—
215C2×D61—24no—
216C2×D61—24no—
217C2×D61—24no—
218C2×D61—24no—
219C2×D61—42no—
220C2×D61—42no—
221C2×D61—42no—
222C2×D61—42no—
223C2×D61—41no—
224C2×D61—42no—
225C2×D61—41no—
226C2×D61#38612yes—
227C2×D61#38812yes—
228C2×D61#38812yes—
229C2×D61#39012yes—
230C2×D61#45814yes—
231#6923—48no—
232#6923—84no—
233C4×C61—42no—
234C4×C61—42no—
235C4×C61—81no—
236Dic61—42no—
237Dic61—42no—
238Dic61—81no—
239C2×#381—612no—
240C2×#381—611no—
241#2273—611no—
242C2×#342—612no—
243C2×#342—611no—
244C2×C61—21no—
245C2×C61—21no—
246C2×C61—22no—
247C2×C61—22no—
248C2×C61#143711yescharacteristic
249C2×C61#189012yes—
250D61—22no—
251D61—22no—
252D61—22no—
253D61#189014yes—
254C24×C4—611no—
255C22×C22⋊C4—612no—
256C22×C22⋊C4—611no—
257C22×C22⋊C4—612no—
258C22×C22⋊C4—611no—
259#381—618no—
260#381—611no—
261#381—611no—
262#381—611no—
263#381—1222no—
264#381—1224no—
265#381#692311yescharacteristic, complemented
266C4×C22⋊C4—1222no—
267C4×C22⋊C4—1224no—
268C2×#95—1222no—
269C2×#95—1224no—
270#342—1224no—
271#342—1222no—
272C61#1093611yescharacteristic, complemented, Sylow
273C25—611no—
274C23×C4—611no—
275C23×C4—614no—
276C23×C4—614no—
277C23×C4—612no—
278C23×C4—1222no—
279C23×C4—1222no—
280C23×C4—1222no—
281C23×C4—1221no—
282C23×C4—1222no—
283C23×C4—1221no—
284C23×C4—1222no—
285C23×C4—1221no—
286C23×C4—1222no—
287C23×C4C2×#692311yescharacteristic
288C2×C22⋊C4—614no—
289C2×C22⋊C4—614no—
290C2×C22⋊C4—611no—
291C2×C22⋊C4—612no—
292C2×C22⋊C4—614no—
293C2×C22⋊C4—611no—
294C2×C22⋊C4—1222no—
295C2×C22⋊C4—1222no—
296C2×C22⋊C4—1222no—
297C2×C22⋊C4—1222no—
298C2×C22⋊C4—1222no—
299C2×C22⋊C4—1222no—
300C2×C22⋊C4—1222no—
301C2×C22⋊C4—1222no—
302C2×C22⋊C4—1222no—
303C2×C22⋊C4—1222no—
304C2×C22⋊C4—1222no—
305C2×C22⋊C4—1222no—
306C2×C22⋊C4—1221no—
307C2×C22⋊C4—1222no—
308C2×C22⋊C4C2×#692311yescharacteristic
309C2×C22⋊C4C2×#692311yescharacteristic
310C2×C42—1224no—
311#95—22no—
312#95—1222no—
313#95—1222no—
314#95—12216no—
315#95—1224no—
316#95—1222no—
317C24—611no—
318C24—612no—
319C24—612no—
320C24—611no—
321C24—612no—
322C24—612no—
323C24—612no—
324C24—612no—
325C24—2444no—
326C24C22×#692311yescharacteristic
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—612no—
333C22×C4—618no—
334C22×C4—612no—
335C22×C4—1222no—
336C22×C4—1224no—
337C22×C4—1222no—
338C22×C4—1224no—
339C22×C4—1224no—
340C22×C4—1224no—
341C22×C4—1224no—
342C22×C4—1224no—
343C22×C4—1224no—
344C22×C4—1224no—
345C22×C4—1221no—
346C22×C4—1221no—
347C22×C4—1222no—
348C22×C4—1222no—
349C22×C4—1224no—
350C22×C4—1221no—
351C22×C4—2442no—
352C22×C4—2444no—
353C22×C4—2444no—
354C22×C4—2441no—
355C22×C4—2442no—
356C22×C4—2441no—
357C22×C4—2442no—
358C22×C4—2442no—
359C22×C4—2442no—
360C22×C4—2442no—
361C22×C4—2442no—
362C22×C4—2441no—
363C22×C4—2441no—
364C22×C4—2442no—
365C22×C4—2442no—
366C22×C4—2448no—
367C22×C4—2448no—
368C22×C4C4×#692312yescomplemented
369C22×C4C22⋊#692312yes—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—1224no—
373C22⋊C4—1224no—
374C22⋊C4—1224no—
375C22⋊C4—1224no—
376C22⋊C4—1224no—
377C22⋊C4—1222no—
378C22⋊C4—1224no—
379C22⋊C4—1224no—
380C22⋊C4—1222no—
381C22⋊C4—2441no—
382C22⋊C4—2441no—
383C22⋊C4—2442no—
384C22⋊C4—2442no—
385C42—4884no—
386C23—42no—
387C23—611no—
388C23—612no—
389C23—611no—
390C23—612no—
391C23—612no—
392C23—612no—
393C23—612no—
394C23—612no—
395C23—612no—
396C23—612no—
397C23—612no—
398C23—611no—
399C23—611no—
400C23—612no—
401C23—612no—
402C23—612no—
403C23—1224no—
404C23—1228no—
405C23—2442no—
406C23—2444no—
407C23—2444no—
408C23—2444no—
409C23—2444no—
410C23—2444no—
411C23C2×C4×#692311yescharacteristic
412C23C2×C22⋊#692311yescharacteristic
413C23#1451985611yescharacteristic
414C23D4×#692312yes—
415C23D4×#692312yes—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—1228no—
426C2×C4—1222no—
427C2×C4—1222no—
428C2×C4—1224no—
429C2×C4—4882no—
430C2×C4—4884no—
431C2×C4—4884no—
432C2×C4—4884no—
433C2×C4—4884no—
434C2×C4—2444no—
435C2×C4—2444no—
436C2×C4—2444no—
437C2×C4—2444no—
438C2×C4—2444no—
439C2×C4—2442no—
440C2×C4—2444no—
441C2×C4—2448no—
442C2×C4—2441no—
443C2×C4—2448no—
444C2×C4—2441no—
445C2×C4—2442no—
446C2×C4—2441no—
447C2×C4—2442no—
448C2×C4—2448no—
449C22—22no—
450C22—24no—
451C22—41no—
452C22—41no—
453C22—42no—
454C22—612no—
455C22—614no—
456C22—612no—
457C22—612no—
458C22—612no—
459C22—1224no—
460C22—1224no—
461C22—1224no—
462C22—1224no—
463C22—1224no—
464C22—1224no—
465C22—1224no—
466C22—1222no—
467C22—2441no—
468C22—2441no—
469C22—2442no—
470C22—2442no—
471C22—2442no—
472C22—2442no—
473C22—2442no—
474C22C4×C22⋊#692311yescharacteristic
475C22C22⋊C4×#692312yes—
476C22#1684071011yescharacteristic
477C22#1684071212yes—
478C22#1684071411yescharacteristic
479C4—42no—
480C4—42no—
481C4—81no—
482C4—4881no—
483C4—4884no—
484C4—2448no—
485C4—2442no—
486C4—2442no—
487C2—21no—
488C2—22no—
489C2—21no—
490C2—22no—
491C2—614no—
492C2—1222no—
493C2—1222no—
494C2—1222no—
495C2#2146100011yescharacteristic
496C2#2146182912yes—
subgroupG/H
centerC22#16840710
commutatorC23×C61C2×C42
FrattiniC24C22×#6923
FittingC61×#381C4
socleC22×C61#386
prime radical, cosocleC24×D61C23
hypercenter#381#6923
hypocenterC61#10936
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
9. #16840705C4semidirect
18. #16839625C4semidirect
20. #16839623C4semidirect
21. #16839623C4semidirect
23. C61×#381C4semidirect
91. C22×C4×C61C42semidirect
98. C22×Dic61C42semidirect
265. #381#6923semidirect
272. C61#10936semidirect
368. C22×C4C4×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407052
#214622552
#214615162
C2×#168406392
#381×D612
#1093661