Finite Group Databasebeta

← #29572534

Subgroups — #29572534

Order 15616.

720 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 107 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×#16840381C211yesmaximal, characteristic
2#21444537C211yesmaximal, characteristic, complemented
3#21444535C211yesmaximal, characteristic, complemented
4#21444443C211yesmaximal, characteristic, complemented
5#21444151C211yesmaximal, characteristic, complemented
6#21444151C211yesmaximal, characteristic, complemented
7C61×#1533C211yesmaximal, characteristic, complemented
8#16840381—22no—
9#16840381—22no—
10#16840381—22no—
11#16840381—22no—
12C2×#14519777C2211yescharacteristic
13C2×#14519770—21no—
14C2×#14519763—21no—
15C2×#14519761—21no—
16C2×#14519754—21no—
17C2×#14519753C411yescharacteristic
18C2×#14519748C411yescharacteristic
19C2×Q8×Dic61C411yescharacteristic
20C2×D4×Dic61C411yescharacteristic
21C2×#14519734C2211yescharacteristic
22#16839819C2211yescharacteristic
23#16839819C2211yescharacteristic
24#16839509—22no—
25#16839507—22no—
26#16839502—22no—
27#16839487—21no—
28#16839487—21no—
29#16839485—21no—
30#16839485—21no—
31C2×#124×C61C2211yescharacteristic
32C61×#432C2211yescharacteristic
33C61×#432C2211yescharacteristic
34C61×#329—22no—
35C22×Dic122—21no—
36C22×C22⋊D61—21no—
37C23×Dic61—21no—
38C22×C4×D61C2×C411yescharacteristic
39#14519777—21no—
40#14519777—21no—
41#14519777D412yes—
42#14519777D412yes—
43#14519770—24no—
44#14519770—24no—
45#14519763—24no—
46#14519763—24no—
47#14519761—24no—
48#14519761—24no—
49#14519754—24no—
50#14519754—24no—
51#14519753—22no—
52#14519753—22no—
53#14519748—22no—
54#14519748—22no—
55Q8×Dic61—22no—
56Q8×Dic61—22no—
57D4×Dic61—22no—
58D4×Dic61—22no—
59#14519734—21no—
60#14519734—21no—
61#14519734—21no—
62#14519734—21no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61C2×C411yescharacteristic
66C2×C4⋊Dic61C2×C411yescharacteristic
67C2×#11758196—21no—
68C2×#11758196D411yescharacteristic
69C2×#11758196D411yescharacteristic
70C2×C22⋊Dic61—21no—
71C2×C22⋊Dic61—21no—
72C2×#11758198—21no—
73C2×#11758198D411yescharacteristic
74C2×#11758198D411yescharacteristic
75C2×C4×Dic61—21no—
76C2×C4×Dic61C2×C411yescharacteristic
77C4⋊#92418—21no—
78C4⋊#92418—21no—
79C22⋊#92418—21no—
80C22⋊#92418—21no—
81#14519688—21no—
82#14519688—21no—
83C22×Q8×C61C2×C411yescharacteristic
84C22×D4×C61C2×C411yescharacteristic
85#124×C61—21no—
86#124×C61—21no—
87#124×C61D412yes—
88#124×C61D412yes—
89C2×C22⋊C4×C61—21no—
90C2×C22⋊C4×C61—21no—
91C2×C42×C61C2311yescharacteristic
92C4⋊C8×C61—21no—
93C4⋊C8×C61—21no—
94C22⋊C8×C61—21no—
95C22⋊C8×C61—21no—
96#97×C61—21no—
97#97×C61—21no—
98C24×C61—21no—
99C23×D61C22⋊C411yescharacteristic
100C2×Q8×C61—21no—
101C2×Q8×C61—42no—
102C2×Q8×C61C22⋊C412yes—
103C2×D4×C61—21no—
104C2×D4×C61—42no—
105C2×D4×C61C22⋊C412yes—
106C22×C4×C61—21no—
107C22×C4×C61C22×C411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C2×C22⋊D61—24no—
111C2×C22⋊D61—82no—
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61—24no—
115C22×Dic61—82no—
116C22×Dic61C22⋊C411yescharacteristic
117C22×Dic61C22⋊C411yescharacteristic
118C22×Dic61C22⋊C411yescharacteristic
119C2×C4×D61—21no—
120C2×C4×D61—21no—
121C2×C4×D61—42no—
122C2×Dic122—22no—
123C2×Dic122—22no—
124C2×Dic122—82no—
125C2×C8×C61—41no—
126C2×C8×C61—41no—
127C22⋊C4×C61—22no—
128C22⋊C4×C61—22no—
129C22⋊C4×C61—22no—
130C22⋊C4×C61—22no—
131C42×C61C2×D411yescharacteristic
132C42×C61C2×D411yescharacteristic
133C42×C61C2×D411yescharacteristic
134C42×C61C2×D411yescharacteristic
135C22⋊Dic61—24no—
136C22⋊Dic61—24no—
137#11758198—24no—
138#11758198—21no—
139#11758198—21no—
140#11758198—21no—
141#11758198—21no—
142C4⋊Dic61—24no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—22no—
146C4⋊Dic61—22no—
147C4⋊Dic61C22⋊C412yes—
148C4⋊Dic61C22⋊C412yes—
149#11758196—24no—
150#11758196—21no—
151#11758196—21no—
152#11758196—21no—
153#11758196—21no—
154C4×Dic61—24no—
155C4×Dic61—21no—
156C4×Dic61—21no—
157C2×#92418—41no—
158C2×#92418—41no—
159C23×C61—22no—
160C23×C61—42no—
161C23×C61—42no—
162C23×C61—42no—
163C23×C61C2×C22⋊C411yescharacteristic
164C22×D61—21no—
165C22×D61—22no—
166C22×D61—41no—
167C22×D61—41no—
168Q8×C61—42no—
169Q8×C61—42no—
170D4×C61—42no—
171D4×C61—42no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—42no—
177C2×C4×C61C2×C22⋊C411yescharacteristic
178C2×C4×C61C2×C22⋊C411yescharacteristic
179C2×C4×C61#12011yescharacteristic
180C2×C4×C61#12011yescharacteristic
181C2×C4×C61#12011yescharacteristic
182C2×C4×C61#12011yescharacteristic
183C22⋊D61—84no—
184C2×Dic61—21no—
185C2×Dic61—21no—
186C2×Dic61—22no—
187C2×Dic61—21no—
188C2×Dic61—22no—
189C2×Dic61—22no—
190C2×Dic61—42no—
191C2×Dic61—42no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—84no—
195C4×D61—42no—
196C4×D61—42no—
197Dic122—82no—
198Dic122—82no—
199C8×C61—81no—
200C8×C61—81no—
201#92418—81no—
202#92418—81no—
203#8523—611nomaximal, Sylow
204C22×C61—21no—
205C22×C61—21no—
206C22×C61—42no—
207C22×C61—42no—
208C22×C61—42no—
209C22×C61—84no—
210C22×C61#37911yescharacteristic
211C22×C61#41312yes—
212C2×D61—41no—
213C2×D61—41no—
214C2×D61—41no—
215C2×D61—82no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—21no—
221C4×C61—21no—
222C4×C61—42no—
223Dic61—42no—
224Dic61—42no—
225Dic61—41no—
226Dic61—41no—
227Dic61—41no—
228C2×#542—611no—
229#1556—611no—
230#1556—611no—
231#1533—611no—
232#1533D6111yescharacteristic, complemented
233#1464—611no—
234#1464—611no—
235C2×C61—22no—
236C2×C61—82no—
237C2×C61#178511yescharacteristic
238C2×C61#179511yescharacteristic
239C2×C61#179611yescharacteristic
240D61—81no—
241#542—1222no—
242#542—1222no—
243#542—1222no—
244#542—1222no—
245C2×C4⋊Q8—611no—
246C2×#126—1221no—
247C2×#124—1221no—
248C2×#124C2×D6111yescharacteristic
249C2×#123—1221no—
250C2×C22⋊Q8—611no—
251C2×C22⋊Q8—1221no—
252C2×#121—611no—
253C2×C4×Q8—611no—
254C2×C4×D4—611no—
255C2×#117—611no—
256#432—611no—
257#432—611no—
258#432C2×D6111yescharacteristic
259#432C2×D6111yescharacteristic
260#332—1222no—
261#332—1222no—
262#329—22no—
263#329—1222no—
264#324—1221no—
265#324—1221no—
266#323—1221no—
267#323—1221no—
268C61#852311yescharacteristic, complemented, Sylow
269C22×Q8—1221no—
270C22×Q8C4×D6111yescharacteristic
271C22×D4—1221no—
272C22×D4C4×D6111yescharacteristic
273C23×C4—611no—
274C23×C4—1221no—
275C4⋊Q8—612no—
276C4⋊Q8—612no—
277C4⋊Q8—1221no—
278C4⋊Q8—1221no—
279#126—1224no—
280#126—1224no—
281#124—21no—
282#124—21no—
283#124—1224no—
284#124—1224no—
285#124D24412yes—
286#124C22⋊D6112yes—
287#123—1224no—
288#123—1224no—
289C22⋊Q8—1224no—
290C22⋊Q8—1224no—
291C22⋊Q8—1222no—
292C22⋊Q8—1222no—
293#121—1222no—
294#121—1222no—
295C4×Q8—1222no—
296C4×Q8—1222no—
297C4×D4—1222no—
298C4×D4—1222no—
299#117—1221no—
300#117—1221no—
301#117—1221no—
302#117—1221no—
303C2×C4⋊C4—611no—
304C2×C4⋊C4—611no—
305C2×C4⋊C4—611no—
306C2×C4⋊C4—611no—
307C2×C4⋊C4—1221no—
308C2×C4⋊C4—1221no—
309C2×C4⋊C4—1221no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—611no—
313C2×C22⋊C4—611no—
314C2×C22⋊C4—1221no—
315C2×C22⋊C4—1221no—
316C2×C22⋊C4—1221no—
317C2×C42—611no—
318C2×C42—1221no—
319C2×C42C22×D6111yescharacteristic
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—1221no—
323C4⋊C8—1221no—
324C22⋊C8—21no—
325C22⋊C8—21no—
326C22⋊C8—1221no—
327C22⋊C8—1221no—
328#97—21no—
329#97—21no—
330#97—1221no—
331#97—1221no—
332C24—21no—
333C24—611no—
334C2×Q8—21no—
335C2×Q8—42no—
336C2×Q8—1222no—
337C2×Q8—1222no—
338C2×Q8—4882no—
339C2×Q8#1175819812yes—
340C2×D4—21no—
341C2×D4—42no—
342C2×D4—1224no—
343C2×D4—4882no—
344C2×D4#1175819812yes—
345C22×C4—21no—
346C22×C4—611no—
347C22×C4—611no—
348C22×C4—611no—
349C22×C4—1224no—
350C22×C4—1221no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—1221no—
354C22×C4—4882no—
355C22×C4—2442no—
356C22×C4C2×C4×D6111yescharacteristic
357C22×C4D4×D6111yescharacteristic
358C22×C4D4×D6111yescharacteristic
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—2441no—
362C2×C8—2441no—
363C4⋊C4—612no—
364C4⋊C4—612no—
365C4⋊C4—1224no—
366C4⋊C4—1224no—
367C4⋊C4—1221no—
368C4⋊C4—1221no—
369C4⋊C4—1221no—
370C4⋊C4—1221no—
371C4⋊C4—1222no—
372C4⋊C4—1221no—
373C4⋊C4—1221no—
374C4⋊C4—1222no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—1224no—
380C22⋊C4—1224no—
381C22⋊C4—1224no—
382C22⋊C4—1221no—
383C22⋊C4—1221no—
384C22⋊C4—1221no—
385C22⋊C4—1221no—
386C42—1224no—
387C42—1221no—
388C42—1221no—
389C42C2×D24411yescharacteristic
390C42D4×D6111yescharacteristic
391C42D4×D6111yescharacteristic
392C42C2×C22⋊D6111yescharacteristic
393C23—22no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—1222no—
398C23—1221no—
399C23—2441no—
400C23—2441no—
401C23C22⋊C4×D6111yescharacteristic
402Q8—42no—
403Q8—42no—
404Q8—4882no—
405Q8—4882no—
406D4—42no—
407D4—42no—
408D4—4884no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—1222no—
415C2×C4—1221no—
416C2×C4—1222no—
417C2×C4—1221no—
418C2×C4—1222no—
419C2×C4—1221no—
420C2×C4—4884no—
421C2×C4—2442no—
422C2×C4—2442no—
423C2×C4—2442no—
424C2×C4—2442no—
425C2×C4—2441no—
426C2×C4—2441no—
427C2×C4C22⋊C4×D6111yescharacteristic
428C2×C4C2×#1175819811yescharacteristic
429C2×C4#1451974511yescharacteristic
430C2×C4#1451974511yescharacteristic
431C2×C4#1451974711yescharacteristic
432C2×C4#1451974711yescharacteristic
433C8—81no—
434C8—81no—
435C8—4881no—
436C8—4881no—
437C22—21no—
438C22—21no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—84no—
443C22—4882no—
444C22—2441no—
445C22—2441no—
446C22—2441no—
447C22#1683961711yescharacteristic
448C22#1683974312yes—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—42no—
456C4—2441no—
457C4—2441no—
458C4—2442no—
459C4—2442no—
460C4—2441no—
461C2—22no—
462C2—82no—
463C2—4881no—
464C2#2144574411yescharacteristic
465C2#2144578811yescharacteristic
466C2#2144579311yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1533C2
socleC22×C61#379
prime radicalC2×#14519748C4
cosocleC2×C42×C61C23
hypercenter#1533D61
hypocenterC61#8523
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1533

Complements

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

NH ≅ G/Ntype
2. #21444537C2semidirect
3. #21444535C2semidirect
4. #21444443C2semidirect
5. #21444151C2semidirect
6. #21444151C2semidirect
7. C61×#1533C2semidirect
232. #1533D61semidirect
268. C61#8523semidirect

Maximal subgroups

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

subgroupindex
C2×#168403812
#214445372
#214445352
#214444432
#214441512
C61×#15332
#852361