Finite Group Databasebeta

← #30598352

Subgroups — #30598352

Order 15616.

1034 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 114 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, complemented
3#21461255C212yesmaximal, complemented
4#21461054C211yesmaximal, characteristic, complemented
5C2×#16840412C211yesmaximal, characteristic
6C2×#14519835—21no—
7D61⋊(C22⋊C8)C2212yes—
8#16840704C2212yes—
9#16840640—21no—
10#16840635—24no—
11#16840613—24no—
12#16840608—21no—
13#16840605—21no—
14C22×#11758224—21no—
15#16840412—24no—
16#16840412—24no—
17C2×#14519776C411yescharacteristic, complemented
18C2×#14519759—21no—
19C2×#14519758—21no—
20C2×#14519754—21no—
21C2×#14519748C2211yescharacteristic
22C2×D4×Dic61C411yescharacteristic, complemented
23C2×C4×C22⋊D61C2211yescharacteristic
24C22×C4⋊Dic61C411yescharacteristic, complemented
25C2×C4⋊C4×D61C2211yescharacteristic
26C2×#121×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—
34C2×#11758224—28no—
35C2×#11758224—44no—
36C2×#11758224—42no—
37C2×#11758224—42no—
38C22×Dic122—21no—
39C22×C22⋊D61—21no—
40C22×C22⋊D61D411yescharacteristic
41C23×Dic61—21no—
42C23×Dic61—21no—
43C22×C4×D61—21no—
44C22×C4×D61C2311yescharacteristic
45#14519776—22no—
46#14519776—22no—
47#14519759—28no—
48#14519758—28no—
49#14519754—28no—
50#14519748—22no—
51#14519748D414yes—
52D4×Dic61—22no—
53D4×Dic61—22no—
54C4×C22⋊D61—22no—
55C4×C22⋊D61—22no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61—22no—
59C2×C4⋊Dic61C2×C411yescharacteristic
60C2×C4⋊Dic61C2×C411yescharacteristic
61C2×#11758196—21no—
62C2×#11758196—21no—
63C2×#11758196D411yescharacteristic
64C4⋊C4×D61—22no—
65C4⋊C4×D61D414yes—
66C2×C22⋊Dic61—21no—
67C2×C22⋊Dic61—21no—
68C2×C22⋊Dic61D411yescharacteristic
69C2×#11758198D411yescharacteristic
70C2×C4×Dic61C2×C411yescharacteristic
71C22×D4×C61—21no—
72C22×D4×C61C2×C411yescharacteristic
73C23×C4×C61C2×C411yescharacteristic
74#121×C61—22no—
75#121×C61—22no—
76C2×C4⋊C4×C61C2×C411yescharacteristic
77C2×C22⋊C4×C61—21no—
78C24×C61—21no—
79C24×C61C22⋊C411yescharacteristic
80C23×D61C2×D411yescharacteristic
81C22×#6923—42no—
82C2×D4×C61—24no—
83C2×D4×C61—21no—
84C2×D4×C61—42no—
85C2×D4×C61—42no—
86C2×D4×C61—42no—
87C2×D4×C61C22⋊C412yes—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61—42no—
92C22×C4×C61C22⋊C411yescharacteristic
93C22×C4×C61C22×C411yescharacteristic
94C2×C22⋊D61—22no—
95C2×C22⋊D61—22no—
96C2×C22⋊D61—24no—
97C2×C22⋊D61—41no—
98C2×C22⋊D61—41no—
99C2×C22⋊D61—42no—
100C2×C22⋊D61D812yes—
101C2×C22⋊D61SD1612yes—
102C22×Dic61—24no—
103C22×Dic61—24no—
104C22×Dic61—21no—
105C22×Dic61—21no—
106C22×Dic61—42no—
107C22×Dic61—42no—
108C22×Dic61—42no—
109C22×Dic61—42no—
110C22×Dic61C22⋊C411yescharacteristic
111C22×Dic61C22⋊C411yescharacteristic
112C22×Dic61C2×D411yescharacteristic
113#11758224—416no—
114#11758224—416no—
115C2×C4×D61—24no—
116C2×C4×D61—42no—
117C2×C4×D61—41no—
118C2×C4×D61—41no—
119C2×C4×D61—42no—
120C2×C4×D61C2×D412yes—
121C2×C4×D61C2×D412yes—
122C2×Dic122—24no—
123C2×Dic122—42no—
124C2×Dic122—42no—
125C22⋊#6923—42no—
126C22⋊#6923—42no—
127C2×#92420—42no—
128C4⋊#6923—44no—
129D61⋊C8—44no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61C22⋊C412yes—
132C22⋊C4×C61—24no—
133C22⋊Dic61—24no—
134C22⋊Dic61—24no—
135C22⋊Dic61—22no—
136#11758198—22no—
137C4⋊Dic61—24no—
138C4⋊Dic61—21no—
139C4⋊Dic61—21no—
140C4⋊Dic61C22⋊C412yes—
141C4⋊Dic61C22⋊C412yes—
142#11758196—24no—
143#11758196—24no—
144#11758196—22no—
145C4×Dic61—21no—
146C4×Dic61—21no—
147C23×C61—21no—
148C23×C61—21no—
149C23×C61—22no—
150C23×C61—41no—
151C23×C61—41no—
152C23×C61—42no—
153C23×C61—42no—
154C23×C61—41no—
155C23×C61—41no—
156C23×C61#10212yes—
157C23×C61C2×C22⋊C411yescharacteristic
158C22×D61—41no—
159C22×D61—41no—
160C22×D61#12012yes—
161C22×D61C2×D811yescharacteristic
162C22×D61C2×SD1611yescharacteristic
163C22×D61AGL(1,ℤ/8)12yes—
164C2×#6923—82no—
165C2×#6923—82no—
166C2×#6923—82no—
167D4×C61—42no—
168D4×C61—48no—
169D4×C61—42no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—21no—
173C2×C4×C61—42no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—42no—
177C2×C4×C61—41no—
178C2×C4×C61C2×C22⋊C411yescharacteristic
179C2×C4×C61C2×C22⋊C411yescharacteristic
180C22⋊D61—24no—
181C22⋊D61—24no—
182C22⋊D61—44no—
183C22⋊D61—44no—
184C2×Dic61—21no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—22no—
188C2×Dic61—22no—
189C2×Dic61—41no—
190C2×Dic61—48no—
191C2×Dic61—48no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61#10212yes—
198C2×Dic61#12012yes—
199C4×D61—24no—
200C4×D61—42no—
201C4×D61—44no—
202C4×D61—44no—
203Dic122—48no—
204#92420—82no—
205#8376—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—22no—
209C22×C61—42no—
210C22×C61—42no—
211C22×C61—41no—
212C22×C61—41no—
213C22×C61—42no—
214C22×C61—42no—
215C22×C61—42no—
216C22×C61—82no—
217C22×C61—82no—
218C22×C61#37911yescharacteristic
219C22×C61C2×#10211yescharacteristic
220C22×C61#41811yescharacteristic
221C2×D61—22no—
222C2×D61—22no—
223C2×D61—41no—
224C2×D61—41no—
225C2×D61—42no—
226C2×D61#44712yes—
227C2×D61#45012yes—
228#6923—162no—
229C4×C61—21no—
230C4×C61—41no—
231C4×C61—42no—
232C4×C61#41012yes—
233Dic61—22no—
234Dic61—22no—
235Dic61—41no—
236Dic61—42no—
237Dic61—42no—
238C2×#547—611no—
239#1823—612no—
240#1799—611no—
241#1519—612no—
242#1459—611no—
243C2×C61—21no—
244C2×C61—21no—
245C2×C61—42no—
246C2×C61—82no—
247C2×C61#178211yescharacteristic
248C2×C61#178611yescharacteristic
249C2×C61#179611yescharacteristic
250D61—22no—
251D61—22no—
252C22×#41—1221no—
253#547—1224no—
254#547—1224no—
255C2×C4⋊Q8—611no—
256C2×#123—1221no—
257C2×#123—1221no—
258C2×C22⋊Q8—1221no—
259C2×#121—611no—
260C2×#121#692311yescharacteristic, complemented
261C2×C4×D4—611no—
262C2×C4×D4—611no—
263C22×C4⋊C4—611no—
264C22×C4⋊C4—611no—
265C2×#99—1221no—
266C2×C22⋊C8—612no—
267#380—612no—
268#342—1221no—
269#340—1224no—
270#327—1224no—
271#324—1221no—
272#323—1221no—
273C61#837611yescharacteristic, complemented, Sylow
274C2×#41—1228no—
275C2×#41—2442no—
276C2×#41—2442no—
277C2×#41—2444no—
278C22×Q8—1221no—
279C22×D4—21no—
280C22×D4—611no—
281C22×D4—1221no—
282C22×D4C2×#692311yescharacteristic
283C23×C4—611no—
284C23×C4—1221no—
285C23×C4—1221no—
286C23×C4—1221no—
287C23×C4C2×#692311yescharacteristic
288C22×C8—1222no—
289C4⋊Q8—1222no—
290C4⋊Q8—1222no—
291#123—1228no—
292#123—1228no—
293C22⋊Q8—1228no—
294#121—22no—
295#121—22no—
296#121—614no—
297#121—1222no—
298C4×D4—1222no—
299C4×D4—1222no—
300C4×D4—1222no—
301C4×D4—1222no—
302C2×C4⋊C4—611no—
303C2×C4⋊C4—611no—
304C2×C4⋊C4—614no—
305C2×C4⋊C4—611no—
306C2×C4⋊C4—1222no—
307C2×C4⋊C4—1222no—
308C2×C4⋊C4—1222no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4—1221no—
312C2×C4⋊C4—1222no—
313C2×C4⋊C4C2×#692311yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—611no—
316C2×C22⋊C4—611no—
317C2×C22⋊C4—1221no—
318C2×C22⋊C4—1221no—
319C2×C22⋊C4—1222no—
320C2×C42—611no—
321#99—2442no—
322C22⋊C8—1222no—
323C22⋊C8—1222no—
324#95—1222no—
325C24—21no—
326C24—611no—
327C24C22⋊#692311yescharacteristic
328#41—24416no—
329#41—24416no—
330C2×Q8—1224no—
331C2×Q8—2442no—
332C2×Q8—2442no—
333C2×D4—24no—
334C2×D4—21no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—612no—
339C2×D4—612no—
340C2×D4—1222no—
341C2×D4—1222no—
342C2×D4—1224no—
343C2×D4—2441no—
344C2×D4—2441no—
345C2×D4—2442no—
346C2×D4C22⋊#692312yes—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—42no—
351C22×C4—611no—
352C22×C4—611no—
353C22×C4—612no—
354C22×C4—612no—
355C22×C4—611no—
356C22×C4—1224no—
357C22×C4—1224no—
358C22×C4—1224no—
359C22×C4—1221no—
360C22×C4—1221no—
361C22×C4—2442no—
362C22×C4—2441no—
363C22×C4—2441no—
364C22×C4—2442no—
365C22×C4—2442no—
366C22×C4—2442no—
367C22×C4—2442no—
368C22×C4—2442no—
369C22×C4—2442no—
370C22×C4C22⋊#692311yescharacteristic
371C22×C4C22×#692311yescharacteristic
372C2×C8—2442no—
373C2×C8—2444no—
374C4⋊C4—21no—
375C4⋊C4—612no—
376C4⋊C4—612no—
377C4⋊C4—1224no—
378C4⋊C4—1222no—
379C4⋊C4—1224no—
380C4⋊C4—1224no—
381C4⋊C4—1221no—
382C4⋊C4—1221no—
383C4⋊C4—2444no—
384C4⋊C4C22⋊#692312yes—
385C22⋊C4—24no—
386C22⋊C4—1222no—
387C22⋊C4—1222no—
388C22⋊C4—1224no—
389C22⋊C4—1224no—
390C22⋊C4—2442no—
391C22⋊C4—2442no—
392C42—1221no—
393C42—1221no—
394C23—21no—
395C23—21no—
396C23—22no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23—41no—
402C23—41no—
403C23—612no—
404C23—611no—
405C23—611no—
406C23—612no—
407C23—2441no—
408C23—2441no—
409C23#1451983912yes—
410C23C2×C22⋊#692311yescharacteristic
411Q8—2448no—
412D4—42no—
413D4—48no—
414D4—42no—
415D4—1224no—
416D4—1224no—
417D4—2444no—
418D4—2444no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—612no—
428C2×C4—612no—
429C2×C4—1221no—
430C2×C4—1221no—
431C2×C4—1224no—
432C2×C4—1221no—
433C2×C4—1222no—
434C2×C4—1222no—
435C2×C4—4882no—
436C2×C4—4882no—
437C2×C4—4882no—
438C2×C4—2442no—
439C2×C4—2444no—
440C2×C4—2444no—
441C2×C4—2441no—
442C2×C4—2448no—
443C2×C4—2448no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4C2×C22⋊#692311yescharacteristic
450C2×C4C2×C22⋊#692311yescharacteristic
451C8—4882no—
452C22—21no—
453C22—21no—
454C22—22no—
455C22—42no—
456C22—42no—
457C22—41no—
458C22—41no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—82no—
463C22—82no—
464C22—612no—
465C22—612no—
466C22—1222no—
467C22—1222no—
468C22—2441no—
469C22—2441no—
470C22—2442no—
471C22#1684070311yescharacteristic
472C22C2×#1451983911yescharacteristic
473C22#1684075411yescharacteristic
474C4—21no—
475C4—41no—
476C4—42no—
477C4—1222no—
478C4—1222no—
479C4—9762no—
480C4—2442no—
481C4—2442no—
482C4—2441no—
483C4#1684074212yes—
484C2—21no—
485C2—21no—
486C2—42no—
487C2—82no—
488C2—1222no—
489C2—1222no—
490C2#2146164911yescharacteristic
491C2#2146165711yescharacteristic
492C2#2146167211yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#121×C61C4
socleC22×C61#379
prime radicalC2×#16840412C2
cosocleC22×C4×D61C23
hypercenterC2×#121#6923
hypocenterC61#8376
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #21461722C2semidirect
2. #21461679C2semidirect
3. #21461255C2semidirect
4. #21461054C2semidirect
17. C2×#14519776C4semidirect
22. C2×D4×Dic61C4semidirect
24. C22×C4⋊Dic61C4semidirect
26. C2×#121×C61C4semidirect
260. C2×#121#6923semidirect
273. C61#8376semidirect

Maximal subgroups

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

subgroupindex
#214617222
#214616792
#214612552
#214610542
C2×#168404122
#837661