Finite Group Databasebeta

← #29572526

Subgroups — #29572526

Order 15616.

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

normal subgroups only

This group has 690 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×#16840372C211yesmaximal, characteristic
2#21445900C211yesmaximal, characteristic
3#21445894C211yesmaximal, characteristic
4#21444353C211yesmaximal, characteristic
5#21444352C211yesmaximal, characteristic
6#21444351C211yesmaximal, characteristic
7C61×#1511C211yesmaximal, characteristic
8#16840372—22no—
9#16840372—22no—
10#16840372—22no—
11#16840372—22no—
12C2×#14519775C411yescharacteristic, complemented
13C2×#14519770—21no—
14C2×#14519770—21no—
15C2×#14519768—21no—
16C2×#14519768C411yescharacteristic, complemented
17C2×#14519767—21no—
18C2×#14519756C2211yescharacteristic
19C2×Q8×Dic61C411yescharacteristic, complemented
20C2×#14519742C411yescharacteristic, complemented
21C2×#14519736C2211yescharacteristic
22#16839726C2211yescharacteristic
23#16839625C2211yescharacteristic
24#16839530—21no—
25#16839530—21no—
26#16839528—21no—
27#16839528—21no—
28#16839500—22no—
29#16839499—22no—
30#16839498—22no—
31C2×C22⋊Q8×C61C2211yescharacteristic
32C61×#407C2211yescharacteristic
33C61×#381C2211yescharacteristic
34C61×#328—22no—
35C22×Dic122C2×C411yescharacteristic
36C2×#11758194—21no—
37#14519775—22no—
38#14519775—22no—
39#14519770—24no—
40#14519770—24no—
41#14519770—24no—
42#14519770—24no—
43#14519768—24no—
44#14519768—24no—
45#14519768—22no—
46#14519768—22no—
47#14519767—24no—
48#14519767—24no—
49#14519756—21no—
50#14519756—21no—
51#14519756D412yes—
52#14519756D412yes—
53Q8×Dic61—22no—
54Q8×Dic61—22no—
55#14519742—22no—
56#14519742—22no—
57#14519736—21no—
58#14519736—21no—
59#14519736—21no—
60#14519736—21no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61C2×C411yescharacteristic
64C2×C4⋊Dic61D411yescharacteristic
65C2×C4⋊Dic61D411yescharacteristic
66C2×#11758196—21no—
67C2×#11758196—21no—
68C2×#11758196—21no—
69C2×#11758196—21no—
70C2×#11758196—21no—
71C2×C22⋊Dic61—21no—
72C2×C22⋊Dic61—21no—
73C2×C22⋊Dic61D411yescharacteristic
74C2×C22⋊Dic61D411yescharacteristic
75C2×C4×Dic61—21no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61C2×C411yescharacteristic
78C2×C4×Dic61C2×C411yescharacteristic
79C22⋊#92418—21no—
80C22⋊#92418—21no—
81#14519684—21no—
82#14519684—21no—
83C22×Q8×C61C2×C411yescharacteristic
84C23×C4×C61C2311yescharacteristic
85C2×M16×C61—21no—
86C22⋊Q8×C61—21no—
87C22⋊Q8×C61—21no—
88C22⋊Q8×C61D412yes—
89C22⋊Q8×C61D412yes—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61C2×C411yescharacteristic
92C2×C22⋊C4×C61—21no—
93C2×C22⋊C4×C61—21no—
94C22⋊C8×C61—21no—
95C22⋊C8×C61—21no—
96#95×C61—21no—
97#95×C61—21no—
98C24×C61C2×D411yescharacteristic
99C2×Q8×C61—21no—
100C2×Q8×C61—42no—
101C2×Q8×C61C22⋊C412yes—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—42no—
105C22×C4×C61—41no—
106C22×C4×C61C22×C411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61—41no—
115C22×Dic61C22⋊C411yescharacteristic
116C22×Dic61C22⋊C411yescharacteristic
117C22×Dic61C22⋊C411yescharacteristic
118C22×Dic61C22⋊C411yescharacteristic
119C2×Dic122—21no—
120C2×Dic122—42no—
121C2×Dic122C22⋊C412yes—
122M16×C61—42no—
123C2×C8×C61—41no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61C22⋊C412yes—
128C22⋊C4×C61—22no—
129C22⋊C4×C61—22no—
130C22⋊C4×C61—42no—
131C22⋊Dic61—22no—
132C22⋊Dic61—22no—
133C22⋊Dic61—21no—
134C22⋊Dic61—21no—
135C22⋊Dic61—21no—
136C22⋊Dic61—21no—
137C22⋊Dic61—42no—
138C4⋊Dic61—24no—
139C4⋊Dic61—24no—
140C4⋊Dic61—21no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61C22⋊C412yes—
146#11758196—24no—
147#11758196—24no—
148#11758196—24no—
149#11758196—24no—
150#11758196—22no—
151#11758196—22no—
152C4×Dic61—24no—
153C4×Dic61—24no—
154C4×Dic61—21no—
155C4×Dic61—21no—
156C4×Dic61—21no—
157C4×Dic61—21no—
158#11758194—42no—
159C2×#92418—41no—
160C23×C61—22no—
161C23×C61—41no—
162C23×C61—41no—
163C23×C61C2×C22⋊C411yescharacteristic
164C23×C61#12011yescharacteristic
165C23×C61#12011yescharacteristic
166Q8×C61—42no—
167Q8×C61—42no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61—81no—
179C2×C4×C61—81no—
180C2×C4×C61—81no—
181C2×C4×C61C2×C22⋊C411yescharacteristic
182C2×C4×C61C2×C22⋊C411yescharacteristic
183C2×C4×C61#12011yescharacteristic
184C2×C4×C61#12011yescharacteristic
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61—22no—
189C2×Dic61—21no—
190C2×Dic61—22no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—81no—
198C2×Dic61—81no—
199C2×Dic61—81no—
200Dic122—42no—
201Dic122—42no—
202C8×C61—81no—
203#92418—81no—
204#8521—611nomaximal, Sylow
205C22×C61—21no—
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—82no—
214C22×C61#37911yescharacteristic
215C22×C61#41012yes—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—42no—
221C4×C61—42no—
222C4×C61—161no—
223Dic61—41no—
224Dic61—41no—
225Dic61—41no—
226Dic61—41no—
227Dic61—42no—
228Dic61—42no—
229Dic61—161no—
230C2×#541—611no—
231#1822—611no—
232#1820—611no—
233#1511—611no—
234#1511—611no—
235#1511—611no—
236#1511D6111yescharacteristic
237C2×C61—22no—
238C2×C61—41no—
239C2×C61—41no—
240C2×C61#178211yescharacteristic
241C2×C61#179611yescharacteristic
242C2×C61#179611yescharacteristic
243#541—1222no—
244#541—1222no—
245#541—1222no—
246#541—1222no—
247C2×C4⋊Q8—611no—
248C2×#126—1221no—
249C2×#126—1221no—
250C2×#125—611no—
251C2×#125—1221no—
252C2×#125—1221no—
253C2×C22⋊Q8—611no—
254C2×C22⋊Q8C2×D6111yescharacteristic
255C2×C4×Q8—611no—
256C2×C4×Q8—611no—
257C2×#117—611no—
258#407—611no—
259#407C2×D6111yescharacteristic
260#381—611no—
261#381C2×D6111yescharacteristic
262#343—1221no—
263#343—1221no—
264#342—1221no—
265#342—1221no—
266#328—22no—
267#328—1222no—
268#328—1222no—
269#328—1222no—
270C61#852111yescharacteristic, complemented, Sylow
271C22×Q8—611no—
272C22×Q8C4×D6111yescharacteristic
273C23×C4C22×D6111yescharacteristic
274C2×M16—21no—
275C2×M16—1221no—
276C4⋊Q8—1222no—
277C4⋊Q8—1222no—
278#126—1224no—
279#126—1224no—
280#126—1224no—
281#126—1224no—
282#125—1222no—
283#125—1224no—
284#125—1224no—
285#125—1224no—
286#125—1224no—
287#125—1222no—
288C22⋊Q8—21no—
289C22⋊Q8—21no—
290C22⋊Q8—612no—
291C22⋊Q8—612no—
292C22⋊Q8—1221no—
293C22⋊Q8—1221no—
294C22⋊Q8D24412yes—
295C22⋊Q8C22⋊D6112yes—
296C4×Q8—1222no—
297C4×Q8—1222no—
298C4×Q8—1222no—
299C4×Q8—1222no—
300#117—1221no—
301#117—1221no—
302#117—1221no—
303#117—1221no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—611no—
306C2×C4⋊C4—611no—
307C2×C4⋊C4—611no—
308C2×C4⋊C4—1221no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4—1221no—
312C2×C4⋊C4—1221no—
313C2×C4⋊C4—1221no—
314C2×C4⋊C4—1221no—
315C2×C4⋊C4C4×D6111yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—611no—
319C2×C22⋊C4—611no—
320C2×C22⋊C4—1221no—
321C2×C22⋊C4—1221no—
322C2×C42—611no—
323C2×C42—611no—
324C2×C42—1221no—
325C2×C42—1221no—
326C22⋊C8—21no—
327C22⋊C8—21no—
328C22⋊C8—1221no—
329C22⋊C8—1221no—
330#95—21no—
331#95—21no—
332#95—1221no—
333#95—1221no—
334C24D4×D6111yescharacteristic
335C2×Q8—21no—
336C2×Q8—42no—
337C2×Q8—612no—
338C2×Q8—1221no—
339C2×Q8—2442no—
340C2×Q8#1175819812yes—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—611no—
346C22×C4—611no—
347C22×C4—611no—
348C22×C4—611no—
349C22×C4—1221no—
350C22×C4—1221no—
351C22×C4—2441no—
352C22×C4C2×C4×D6111yescharacteristic
353C22×C4C2×D24411yescharacteristic
354C22×C4D4×D6111yescharacteristic
355C22×C4D4×D6111yescharacteristic
356C22×C4D4×D6111yescharacteristic
357C22×C4C2×C22⋊D6111yescharacteristic
358M16—42no—
359M16—2442no—
360C2×C8—41no—
361C2×C8—2441no—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—612no—
366C4⋊C4—1224no—
367C4⋊C4—1224no—
368C4⋊C4—1224no—
369C4⋊C4—1224no—
370C4⋊C4—1224no—
371C4⋊C4—1224no—
372C4⋊C4—1221no—
373C4⋊C4—1221no—
374C4⋊C4—1221no—
375C4⋊C4—1222no—
376C4⋊C4—1221no—
377C4⋊C4—1221no—
378C4⋊C4—1222no—
379C4⋊C4#1175819812yes—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—42no—
383C22⋊C4—1221no—
384C22⋊C4—1221no—
385C22⋊C4—1222no—
386C22⋊C4—1221no—
387C22⋊C4—1221no—
388C22⋊C4—1222no—
389C22⋊C4—2442no—
390C42—1224no—
391C42—1224no—
392C42—1221no—
393C42—1221no—
394C42—1221no—
395C42—1221no—
396C23—22no—
397C23—41no—
398C23—41no—
399C23C22⋊C4×D6111yescharacteristic
400C23#1451974511yescharacteristic
401C23#1451974711yescharacteristic
402Q8—42no—
403Q8—42no—
404Q8—2442no—
405Q8—2442no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—1221no—
420C2×C4—1221no—
421C2×C4—1222no—
422C2×C4—1221no—
423C2×C4—1221no—
424C2×C4—1222no—
425C2×C4—4881no—
426C2×C4—4881no—
427C2×C4—4881no—
428C2×C4—2442no—
429C2×C4—2442no—
430C2×C4—2441no—
431C2×C4—2441no—
432C2×C4—2441no—
433C2×C4—2441no—
434C2×C4C22⋊C4×D6111yescharacteristic
435C2×C4C2×#1175819811yescharacteristic
436C2×C4#1451974511yescharacteristic
437C2×C4#1451974711yescharacteristic
438C8—81no—
439C8—4881no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—82no—
449C22#1683961711yescharacteristic
450C22#1683973412yes—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—42no—
456C4—42no—
457C4—161no—
458C4—9761no—
459C4—2442no—
460C4—2441no—
461C4—2441no—
462C4—2442no—
463C4—2441no—
464C4—2441no—
465C2—22no—
466C2—41no—
467C2—41no—
468C2#2144573511yescharacteristic
469C2#2144579111yescharacteristic
470C2#2144579311yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1511C2
socleC22×C61#379
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#1511D61
hypocenterC61#8521
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1511

Complements

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

NH ≅ G/Ntype
12. C2×#14519775C4semidirect
16. C2×#14519768C4semidirect
19. C2×Q8×Dic61C4semidirect
20. C2×#14519742C4semidirect
270. C61#8521semidirect

Maximal subgroups

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

subgroupindex
C2×#168403722
#214459002
#214458942
#214443532
#214443522
#214443512
C61×#15112
#852161