Finite Group Databasebeta

← #29589281

Subgroups — #29589281

Order 15616.

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

normal subgroups only

This group has 616 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
1C4×#14519773C211yesmaximal, characteristic
2#21444137C211yesmaximal, characteristic, complemented
3#21443764C212yesmaximal, complemented
4#21443750C212yesmaximal, complemented
5C61×#1459C211yesmaximal, characteristic, complemented
6C2×#14519773C411yescharacteristic
7C2×C4×D244C411yescharacteristic
8C2×C4×C22⋊D61—21no—
9C2×C4×C22⋊D61—21no—
10C2×C22⋊#92418—21no—
11#16839665C411yescharacteristic
12#16839663C2211yescharacteristic
13#16839662—21no—
14#16839662—21no—
15#16839631C2211yescharacteristic
16C42×Dic61C411yescharacteristic
17#16839488C2212yes—
18#16839486—21no—
19C2×C4×D4×C61C2211yescharacteristic
20C2×C22⋊C8×C61—21no—
21C61×#324C2212yes—
22C61×#323—21no—
23C22×D244C2×C411yescharacteristic
24C22×C22⋊D61—21no—
25C22×C22⋊D61—21no—
26C22×C4×D61—21no—
27C22×#92418—41no—
28#14519773—22no—
29#14519773C814yescomplemented
30C4×D244—41no—
31C4×D244—41no—
32C4×C22⋊D61—44no—
33C4×C22⋊D61—44no—
34C2×C4⋊Dic61C2×C411yescharacteristic
35C2×#11758196—21no—
36C2×#11758196—21no—
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×C4×Dic61—21no—
43C2×C4×Dic61—21no—
44C2×C4×Dic61C2×C411yescharacteristic
45C2×C4×Dic61C2×C411yescharacteristic
46C22⋊#92418—22no—
47C22⋊#92418—41no—
48C22×D4×C61C2×C411yescharacteristic
49C23×C4×C61D411yescharacteristic
50C23×C4×C61D411yescharacteristic
51C22×C8×C61—41no—
52C4×D4×C61—41no—
53C4×D4×C61—41no—
54C2×C4⋊C4×C61C2311yescharacteristic
55C2×C22⋊C4×C61D411yescharacteristic
56C2×C22⋊C4×C61D411yescharacteristic
57C2×C42×C61C2×C411yescharacteristic
58C22⋊C8×C61—22no—
59C22⋊C8×C61—41no—
60C24×C61C22⋊C411yescharacteristic
61C24×C61C22⋊C411yescharacteristic
62C23×D61—21no—
63C2×D4×C61—41no—
64C2×D4×C61—41no—
65C2×D4×C61C2×C812yes—
66C2×D4×C61M1612yes—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—42no—
72C22×C4×C61—42no—
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22×C411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C2×C22⋊D61—24no—
79C2×C22⋊D61—24no—
80C2×C22⋊D61—42no—
81C2×C22⋊D61—42no—
82C2×C22⋊D61—42no—
83C2×C22⋊D61—42no—
84C22×Dic61—21no—
85C22×Dic61—21no—
86C22×Dic61—21no—
87C2×D244—21no—
88C2×D244—41no—
89C2×D244—41no—
90C2×D244C2×C812yescomplemented
91C2×C4×D61—42no—
92C2×C4×D61—42no—
93C2×C4×D61—41no—
94C2×C4×D61—41no—
95C2×C8×C61—42no—
96C2×C8×C61—82no—
97C4⋊C4×C61—41no—
98C22⋊C4×C61—41no—
99C22⋊C4×C61—41no—
100C42×C61—21no—
101C42×C61—21no—
102C22⋊Dic61—42no—
103C22⋊Dic61—42no—
104#11758198—42no—
105#11758198—42no—
106#11758198—42no—
107C4⋊Dic61—41no—
108#11758196—42no—
109#11758196—42no—
110C4×Dic61—24no—
111C4×Dic61—21no—
112C4×Dic61—42no—
113C4×Dic61—41no—
114C4×Dic61C2×C812yescomplemented
115C2×#92418—42no—
116C2×#92418—82no—
117C23×C61—21no—
118C23×C61—21no—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61C22⋊C812yes—
126C23×C61C22⋊C812yes—
127C23×C61C2×C22⋊C411yescharacteristic
128C22×D61—22no—
129C22×D61—41no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—41no—
135D4×C61—42no—
136D4×C61—42no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—22no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—42no—
144C2×C4×C61—42no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—81no—
149C2×C4×C61—81no—
150C2×C4×C61#9911yescharacteristic
151C2×C4×C61#9911yescharacteristic
152C2×C4×C61C22×C811yescharacteristic
153C2×C4×C61C2×M1611yescharacteristic
154C22⋊D61—48no—
155C22⋊D61—48no—
156C2×Dic61—22no—
157C2×Dic61—22no—
158C2×Dic61—22no—
159C2×Dic61—22no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165D244—42no—
166D244—42no—
167C4×D61—82no—
168C4×D61—82no—
169C8×C61—82no—
170#92418—82no—
171#11475—611nomaximal, Sylow
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—22no—
175C22×C61—22no—
176C22×C61—42no—
177C22×C61—42no—
178C22×C61—41no—
179C22×C61—41no—
180C22×C61—41no—
181C22×C61—41no—
182C22×C61—81no—
183C22×C61—81no—
184C22×C61C2×C22⋊C811yescharacteristic
185C22×C61C2×#9911yescharacteristic
186C22×C61#41311yescharacteristic
187C2×D61—42no—
188C2×D61—42no—
189C2×D61—42no—
190C2×D61—81no—
191C2×D61—81no—
192C2×D61—81no—
193C2×D61—81no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—41no—
198C4×C61—41no—
199C4×C61—41no—
200C4×C61#35312yes—
201Dic61—24no—
202Dic61—42no—
203Dic61—81no—
204C4×#127—611no—
205#1459—611no—
206#1459D6111yescharacteristic, complemented
207#1325—612no—
208#1318—612no—
209C2×C61—21no—
210C2×C61—21no—
211C2×C61—42no—
212C2×C61—42no—
213C2×C61#145911yescharacteristic
214C2×C61C2×#35311yescharacteristic
215C2×C61#213211yescharacteristic
216D61—82no—
217C2×#127—611no—
218C2×C4×D4—611no—
219C2×C4×D4—1221no—
220C2×C4×D4—1221no—
221C2×C4×D4C2×D6111yescharacteristic
222C2×C22⋊C8—21no—
223C2×C22⋊C8—1221no—
224#390—611no—
225#390—611no—
226#390—1221no—
227#390—1221no—
228#384—611no—
229C43—611no—
230#324—612no—
231#324C2×D6112yes—
232#323—21no—
233#323—1221no—
234C61#1147511yescharacteristic, complemented, Sylow
235C22×D4—611no—
236C22×D4—1221no—
237C22×D4—1221no—
238C22×D4C4×D6111yescharacteristic
239C23×C4—1221no—
240C23×C4D24411yescharacteristic
241C23×C4C22⋊D6111yescharacteristic
242C22×C8—41no—
243C22×C8—2441no—
244#127—614no—
245#127—1222no—
246C4×D4—41no—
247C4×D4—41no—
248C4×D4—2441no—
249C4×D4—2441no—
250C4×D4—2444no—
251C4×D4—2444no—
252C2×C4⋊C4—611no—
253C2×C4⋊C4—1221no—
254C2×C4⋊C4—1221no—
255C2×C4⋊C4C22×D6111yescharacteristic
256C2×C22⋊C4—1221no—
257C2×C22⋊C4—1221no—
258C2×C22⋊C4—1221no—
259C2×C22⋊C4—1221no—
260C2×C22⋊C4—1221no—
261C2×C22⋊C4D24411yescharacteristic
262C2×C22⋊C4C22⋊D6111yescharacteristic
263C2×C42—611no—
264C2×C42—611no—
265C2×C42—1221no—
266C2×C42—1221no—
267C2×C42C4×D6111yescharacteristic
268C22⋊C8—22no—
269C22⋊C8—41no—
270C22⋊C8—1222no—
271C22⋊C8—2441no—
272C24—1221no—
273C24#1175819811yescharacteristic
274C24#1175819811yescharacteristic
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—612no—
278C2×D4—1224no—
279C2×D4—1224no—
280C2×D4—1221no—
281C2×D4—2441no—
282C2×D4—2441no—
283C2×D4—2442no—
284C2×D4—2442no—
285C2×D4—2442no—
286C2×D4—2442no—
287C2×D4C8×D6112yes—
288C2×D4#1175818912yes—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—1221no—
296C22×C4—1221no—
297C22×C4—1221no—
298C22×C4—2442no—
299C22×C4—2442no—
300C22×C4—2441no—
301C22×C4—2441no—
302C22×C4#1175819811yescharacteristic
303C22×C4#1175819811yescharacteristic
304C22×C4C2×C4×D6111yescharacteristic
305C22×C4C2×D24411yescharacteristic
306C22×C4C2×C22⋊D6111yescharacteristic
307C2×C8—42no—
308C2×C8—82no—
309C2×C8—4882no—
310C2×C8—2442no—
311C4⋊C4—41no—
312C4⋊C4—2441no—
313C4⋊C4—2442no—
314C4⋊C4—2442no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—2442no—
318C22⋊C4—2442no—
319C22⋊C4—2442no—
320C22⋊C4—2442no—
321C22⋊C4—2442no—
322C42—21no—
323C42—21no—
324C42—612no—
325C42—1224no—
326C42—1221no—
327C42—2441no—
328C42—2442no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—1222no—
338C23—2441no—
339C23—2441no—
340C23—2441no—
341C23—2441no—
342C23—2441no—
343C23—2441no—
344C23#1451968912yes—
345C23#1451968912yes—
346C23C2×#1175819811yescharacteristic
347D4—42no—
348D4—42no—
349D4—2442no—
350D4—2442no—
351D4—2448no—
352D4—2448no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—1222no—
367C2×C4—1222no—
368C2×C4—1222no—
369C2×C4—1222no—
370C2×C4—4882no—
371C2×C4—4882no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4#1451969111yescharacteristic
378C2×C4#1451969111yescharacteristic
379C2×C4C2×C8×D6111yescharacteristic
380C2×C4C2×#1175818911yescharacteristic
381C8—82no—
382C8—4882no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—42no—
388C22—42no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—81no—
394C22—81no—
395C22—4881no—
396C22—4881no—
397C22—4881no—
398C22—4881no—
399C22—2442no—
400C22—2442no—
401C22—2442no—
402C22C2×#1451968911yescharacteristic
403C22C2×#1451969111yescharacteristic
404C22#1683974411yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—1224no—
412C4—4881no—
413C4—2442no—
414C4#1683955412yes—
415C2—21no—
416C2—21no—
417C2—42no—
418C2—42no—
419C2—4882no—
420C2#2144413511yescharacteristic
421C2C2×#1683955411yescharacteristic
422C2#2144750211yescharacteristic
subgroupG/H
centerC2×C4#14519691
commutatorC2×C4×C61C22×C8
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1459C2
socleC22×C61C2×#99
prime radicalC2×#14519773C4
cosocleC2×C4⋊C4×C61C23
hypercenter#1459D61
hypocenterC61#11475
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
2. #21444137C2semidirect
3. #21443764C2semidirect
4. #21443750C2semidirect
5. C61×#1459C2semidirect
29. #14519773C8semidirect
90. C2×D244C2×C8semidirect
114. C4×Dic61C2×C8semidirect
206. #1459D61semidirect
234. C61#11475semidirect

Maximal subgroups

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

subgroupindex
C4×#145197732
#214441372
#214437642
#214437502
C61×#14592
#1147561