Finite Group Databasebeta

← #30598314

Subgroups — #30598314

Order 15616.

1068 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 108 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#21461662C212yesmaximal, complemented
2C4×#14519835C211yesmaximal, characteristic, complemented
3#21461219C212yesmaximal, complemented
4#21461074C211yesmaximal, characteristic, complemented
5C2×#16840423C211yesmaximal, characteristic
6C2×#14519842—24no—
7C2×#14519835—21no—
8#16840733C2212yes—
9C4×C22⋊#6923C2212yes—
10#16840640—21no—
11#16840616—24no—
12#16840608—21no—
13#16840605—21no—
14C22×#11758226C2211yescharacteristic
15#16840423—24no—
16#16840423—24no—
17C2×#14519766—21no—
18C2×#14519766C411yescharacteristic, complemented
19C2×#14519753—21no—
20C2×#14519753C2211yescharacteristic
21C2×#14519751—21no—
22C2×#14519751C411yescharacteristic, complemented
23C2×Q8×Dic61C411yescharacteristic, complemented
24C2×C4×C22⋊D61—21no—
25C2×C4×C22⋊D61C2211yescharacteristic
26C23×Q8×C61C411yescharacteristic, complemented
27C2×#11758213—22no—
28C2×C22⋊#6923—22no—
29C2×C4×#6923—22no—
30#14519842—48no—
31#14519835—24no—
32#14519834—22no—
33C22⋊#92420—22no—
34(C2×#6923)⋊C4—22no—
35C2×#11758226—22no—
36C2×#11758226—44no—
37C2×#11758226D414yes—
38C22×D244—21no—
39C22×D244C2×C411yescharacteristic
40C22×C22⋊D61—21no—
41C22×C22⋊D61D411yescharacteristic
42C22×C4×D61—21no—
43C22×C4×D61C2311yescharacteristic
44#14519766—28no—
45#14519766—22no—
46#14519766—22no—
47#14519753—28no—
48#14519753—22no—
49#14519753D414yes—
50#14519751—28no—
51#14519751—22no—
52#14519751—22no—
53Q8×Dic61—22no—
54Q8×Dic61—22no—
55C4×C22⋊D61—28no—
56C4×C22⋊D61—22no—
57C4×C22⋊D61—22no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61C2×C411yescharacteristic
60C2×#11758196—21no—
61C2×#11758196D411yescharacteristic
62C2×C22⋊Dic61D411yescharacteristic
63C2×#11758198—21no—
64C2×#11758198—21no—
65C2×#11758198—21no—
66C2×#11758198—21no—
67C2×#11758198D411yescharacteristic
68C2×C4×Dic61—21no—
69C2×C4×Dic61C2×C411yescharacteristic
70C22×Q8×C61—22no—
71C22×Q8×C61—22no—
72C22×Q8×C61—21no—
73C22×Q8×C61—44no—
74C22×Q8×C61C2×C411yescharacteristic
75C22×Q8×C61C2×C411yescharacteristic
76C23×C4×C61—21no—
77C23×C4×C61C2×C411yescharacteristic
78C24×C61C22⋊C411yescharacteristic
79C23×D61—21no—
80C23×D61C2×D411yescharacteristic
81C22×#6923—42no—
82C2×Q8×C61—24no—
83C2×Q8×C61—24no—
84C2×Q8×C61—21no—
85C2×Q8×C61—21no—
86C2×Q8×C61—44no—
87C2×Q8×C61—42no—
88C2×Q8×C61—48no—
89C2×Q8×C61—44no—
90C2×Q8×C61—44no—
91C2×Q8×C61C22⋊C412yes—
92C2×Q8×C61C22⋊C412yes—
93C22×C4×C61—21no—
94C22×C4×C61—24no—
95C22×C4×C61—21no—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—44no—
99C22×C4×C61—42no—
100C22×C4×C61C22⋊C411yescharacteristic
101C22×C4×C61C22×C411yescharacteristic
102C2×C22⋊D61—24no—
103C2×C22⋊D61—22no—
104C2×C22⋊D61—22no—
105C2×C22⋊D61—41no—
106C2×C22⋊D61—42no—
107C2×C22⋊D61—42no—
108C22×Dic61—21no—
109C22×Dic61C22⋊C411yescharacteristic
110C22×Dic61C22⋊C411yescharacteristic
111C22×Dic61C2×D411yescharacteristic
112#11758226—28no—
113#11758226—44no—
114#11758226—48no—
115C2×D244—24no—
116C2×D244—21no—
117C2×D244—42no—
118C2×D244—44no—
119C2×D244C22⋊C412yes—
120C2×C4×D61—24no—
121C2×C4×D61—42no—
122C2×C4×D61—44no—
123C2×C4×D61C2×D412yes—
124C2×C4×D61C2×D412yes—
125C22⋊#6923—24no—
126C22⋊#6923—42no—
127C2×#92420—42no—
128C4×#6923—44no—
129#11758213—44no—
130C22⋊Dic61—22no—
131#11758198—24no—
132#11758198—24no—
133#11758198—24no—
134#11758198—24no—
135#11758198—22no—
136C4⋊Dic61—24no—
137C4⋊Dic61—21no—
138C4⋊Dic61C22⋊C412yes—
139#11758196—24no—
140#11758196—22no—
141C4×Dic61—24no—
142C4×Dic61—21no—
143C4×Dic61—21no—
144C23×C61—21no—
145C23×C61—21no—
146C23×C61—41no—
147C23×C61—41no—
148C23×C61#10412yescomplemented
149C23×C61C2×C22⋊C411yescharacteristic
150C22×D61—21no—
151C22×D61—21no—
152C22×D61—22no—
153C22×D61—41no—
154C22×D61—41no—
155C22×D61—42no—
156C22×D61—42no—
157C22×D61—41no—
158C22×D61—41no—
159C22×D61#12012yes—
160C2×#6923—44no—
161C2×#6923—82no—
162C2×#6923—82no—
163Q8×C61—24no—
164Q8×C61—22no—
165Q8×C61—48no—
166Q8×C61—42no—
167Q8×C61—42no—
168Q8×C61#9914yescomplemented
169C2×C4×C61—24no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—21no—
174C2×C4×C61—48no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61—41no—
179C2×C4×C61C2×C22⋊C411yescharacteristic
180C2×C4×C61C2×C22⋊C411yescharacteristic
181C22⋊D61—44no—
182C22⋊D61—48no—
183C2×Dic61—21no—
184C2×Dic61—21no—
185C2×Dic61—22no—
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61#12012yes—
193D244—24no—
194D244—48no—
195D244—42no—
196C4×D61—24no—
197C4×D61—42no—
198C4×D61—48no—
199#92420—82no—
200#8366—611nomaximal, Sylow
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—22no—
204C22×C61—42no—
205C22×C61—42no—
206C22×C61—41no—
207C22×C61—41no—
208C22×C61#37911yescharacteristic
209C22×C61C2×#10411yescharacteristic
210C22×C61#42111yescharacteristic
211C2×D61—22no—
212C2×D61—22no—
213C2×D61—22no—
214C2×D61—42no—
215C2×D61—42no—
216C2×D61—42no—
217C2×D61—81no—
218C2×D61—81no—
219C2×D61—82no—
220C2×D61—82no—
221#6923—84no—
222C4×C61—24no—
223C4×C61—21no—
224C4×C61—42no—
225C4×C61—41no—
226C4×C61C2×#9912yes—
227Dic61—22no—
228Dic61—41no—
229Dic61—41no—
230Dic61—42no—
231C2×#548—611no—
232#1789—612no—
233C4×#99—611no—
234#1508—612no—
235#1465—611no—
236C2×C61—21no—
237C2×C61—21no—
238C2×C61—42no—
239C2×C61#178111yescharacteristic
240C2×C61#178911yescharacteristic
241C2×C61#179611yescharacteristic
242D61—42no—
243D61—82no—
244C22×#41—611no—
245C23×Q8#692311yescharacteristic, complemented
246#548—1224no—
247#548—1224no—
248C2×#124—611no—
249C2×#124—1221no—
250C2×C22⋊Q8—611no—
251C2×C22⋊Q8—1221no—
252C2×#121—611no—
253C2×#121—1221no—
254C2×C4×Q8—611no—
255C2×C4×D4—611no—
256C2×C4×D4—1221no—
257C2×#104—1224no—
258C2×#99—1221no—
259#407—612no—
260C4×C22⋊C4—612no—
261#342—1221no—
262#328—1224no—
263#324—1221no—
264#323—1221no—
265C61#836611yescharacteristic, complemented, Sylow
266C2×#41—614no—
267C2×#41—1222no—
268C2×#41—2444no—
269C22×Q8—22no—
270C22×Q8—22no—
271C22×Q8—21no—
272C22×Q8—44no—
273C22×Q8C2×#692311yescharacteristic
274C22×Q8C2×#692311yescharacteristic
275C22×D4—611no—
276C22×D4—611no—
277C22×D4—1221no—
278C22×D4—1221no—
279C23×C4—21no—
280C23×C4—611no—
281C23×C4—1221no—
282C23×C4C2×#692311yescharacteristic
283C2×M16—1222no—
284#124—1222no—
285#124—1228no—
286#124—1222no—
287C22⋊Q8—614no—
288C22⋊Q8—1222no—
289C22⋊Q8—1228no—
290#121—1222no—
291#121—1222no—
292#121—1228no—
293C4×Q8—1222no—
294C4×Q8—1222no—
295C4×D4—1222no—
296C4×D4—1222no—
297C4×D4—1228no—
298C2×C4⋊C4—611no—
299C2×C4⋊C4—611no—
300C2×C4⋊C4—1221no—
301C2×C4⋊C4—1221no—
302C2×C22⋊C4—611no—
303C2×C22⋊C4—611no—
304C2×C22⋊C4—1221no—
305C2×C22⋊C4—1221no—
306C2×C22⋊C4—1221no—
307C2×C22⋊C4—1221no—
308C2×C22⋊C4—1222no—
309C2×C42—611no—
310C2×C42—1221no—
311C2×C42—1222no—
312#104—2448no—
313#99—1224no—
314C22⋊C8—1222no—
315C22⋊C8—1222no—
316#95—1222no—
317C24—611no—
318C24—1221no—
319C24C22⋊#692311yescharacteristic
320#41—1228no—
321#41—2444no—
322#41—2448no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—42no—
328C2×Q8—44no—
329C2×Q8—48no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8C22⋊#692312yes—
333C2×Q8C22⋊#692312yes—
334C2×D4—612no—
335C2×D4—1224no—
336C2×D4—1222no—
337C2×D4—1224no—
338C2×D4—1222no—
339C2×D4—1221no—
340C2×D4—2442no—
341C2×D4—2444no—
342C2×D4—2441no—
343C2×D4—2442no—
344C2×D4—2442no—
345C22×C4—21no—
346C22×C4—24no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—611no—
353C22×C4—612no—
354C22×C4—612no—
355C22×C4—611no—
356C22×C4—611no—
357C22×C4—1224no—
358C22×C4—1221no—
359C22×C4—2442no—
360C22×C4—2444no—
361C22×C4—2442no—
362C22×C4C22⋊#692311yescharacteristic
363C22×C4C22×#692311yescharacteristic
364M16—2444no—
365C2×C8—2442no—
366C4⋊C4—612no—
367C4⋊C4—1222no—
368C4⋊C4—1224no—
369C4⋊C4—1224no—
370C4⋊C4—1221no—
371C22⋊C4—1222no—
372C22⋊C4—1224no—
373C22⋊C4—1224no—
374C22⋊C4—1224no—
375C22⋊C4—1224no—
376C22⋊C4—1222no—
377C22⋊C4—1224no—
378C22⋊C4—2442no—
379C42—1224no—
380C42—1221no—
381C42—1221no—
382C42—2444no—
383C23—21no—
384C23—21no—
385C23—41no—
386C23—41no—
387C23—612no—
388C23—1221no—
389C23—1221no—
390C23—1222no—
391C23—2441no—
392C23—2441no—
393C23—2442no—
394C23—2442no—
395C23—2441no—
396C23—2441no—
397C23#1451984212yescomplemented
398C23C2×C22⋊#692311yescharacteristic
399Q8—24no—
400Q8—22no—
401Q8—42no—
402Q8—48no—
403Q8—42no—
404Q8#1451983514yescomplemented
405D4—1224no—
406D4—2448no—
407D4—2442no—
408D4—2444no—
409D4—2448no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—48no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—612no—
421C2×C4—1224no—
422C2×C4—1221no—
423C2×C4—1221no—
424C2×C4—1221no—
425C2×C4—1222no—
426C2×C4—1221no—
427C2×C4—4882no—
428C2×C4—4882no—
429C2×C4—2442no—
430C2×C4—2448no—
431C2×C4—2441no—
432C2×C4—2444no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4C2×C22⋊#692311yescharacteristic
437C2×C4C2×C22⋊#692311yescharacteristic
438C8—4882no—
439C22—21no—
440C22—21no—
441C22—22no—
442C22—42no—
443C22—42no—
444C22—41no—
445C22—41no—
446C22—1222no—
447C22—1222no—
448C22—1222no—
449C22—4882no—
450C22—4882no—
451C22—4881no—
452C22—4881no—
453C22—2442no—
454C22—2442no—
455C22—2442no—
456C22#1684070311yescharacteristic
457C22C2×#1451984211yescharacteristic
458C22#1684076111yescharacteristic
459C4—24no—
460C4—21no—
461C4—42no—
462C4—41no—
463C4—1222no—
464C4—4884no—
465C4—2441no—
466C4—2442no—
467C4—2441no—
468C4C2×#1451983512yes—
469C2—21no—
470C2—21no—
471C2—42no—
472C2—4882no—
473C2—2442no—
474C2#2146164511yescharacteristic
475C2#2146166211yescharacteristic
476C2#2146167111yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC23×Q8×C61C4
socleC22×C61#379
prime radicalC2×#16840423C2
cosocleC22×C4×D61C23
hypercenterC23×Q8#6923
hypocenterC61#8366
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. #21461662C2semidirect
2. C4×#14519835C2semidirect
3. #21461219C2semidirect
4. #21461074C2semidirect
18. C2×#14519766C4semidirect
22. C2×#14519751C4semidirect
23. C2×Q8×Dic61C4semidirect
26. C23×Q8×C61C4semidirect
148. C23×C61#104semidirect
168. Q8×C61#99semidirect
245. C23×Q8#6923semidirect
265. C61#8366semidirect
397. C23#14519842semidirect
404. Q8#14519835semidirect

Maximal subgroups

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

subgroupindex
#214616622
C4×#145198352
#214612192
#214610742
C2×#168404232
#836661