Finite Group Databasebeta

← #29568640

Subgroups — #29568640

Order 15616.

506 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 93 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#21454152C211yesmaximal, characteristic
2#21445886C211yesmaximal, characteristic, complemented
3#21445886C211yesmaximal, characteristic, complemented
4#21443642C211yesmaximal, characteristic, complemented
5#21443639C211yesmaximal, characteristic, complemented
6#21443639C211yesmaximal, characteristic, complemented
7C61×#1271C211yesmaximal, characteristic, complemented
8C2×#14519762—21no—
9C2×#14519752C411yescharacteristic
10C2×C42×D61C2211yescharacteristic
11C2×#14519689—21no—
12C2×#14519689—21no—
13#16839664C2211yescharacteristic
14#16839663C411yescharacteristic
15#16839658C411yescharacteristic
16#16839652—21no—
17#16839651—21no—
18#16839640—21no—
19#16839619C411yescharacteristic
20#16839517C2211yescharacteristic
21#16839517C2211yescharacteristic
22#16839516—21no—
23#16839516—21no—
24C61×#389C2211yescharacteristic
25C61×#336C2211yescharacteristic
26C61×#336C2211yescharacteristic
27C22×D244—21no—
28C22×D244C2×C411yescharacteristic
29C22×C4×D61C2×C411yescharacteristic
30C22×C4×D61D411yescharacteristic
31C22×C4×D61D411yescharacteristic
32C22×#92418—21no—
33C22×#92418—21no—
34#14519762—42no—
35#14519762—42no—
36#14519752—42no—
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×#11758196—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×C4×Dic61C2×C411yescharacteristic
46C2×C4×Dic61D411yescharacteristic
47C2×C4×Dic61D411yescharacteristic
48C42×D61—42no—
49#14519689—42no—
50#14519689—42no—
51#14519684—21no—
52#14519684—21no—
53C22×C8×C61—21no—
54C22×C8×C61—21no—
55C2×C4⋊C4×C61—21no—
56C2×C4⋊C4×C61C2×C411yescharacteristic
57C2×C4⋊C4×C61C2×C411yescharacteristic
58C2×C42×C61C2311yescharacteristic
59#95×C61—21no—
60C23×D61—21no—
61C23×D61C22⋊C411yescharacteristic
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61C22×C411yescharacteristic
65C22×C4×C61C2×D411yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C22×Dic61—21no—
68C22×Dic61C22⋊C411yescharacteristic
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61C22⋊C411yescharacteristic
71C2×D244—21no—
72C2×D244—21no—
73C2×D244—41no—
74C2×D244—41no—
75C2×D244—42no—
76C2×D244—42no—
77C2×D244—42no—
78C2×C4×D61—22no—
79C2×C4×D61—22no—
80C2×C4×D61—22no—
81C2×C4×D61—21no—
82C2×C4×D61—21no—
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—21no—
86C2×C4×D61—21no—
87C2×C4×D61—41no—
88C2×C4×D61—41no—
89C2×C4×D61—41no—
90C2×C8×C61—41no—
91C2×C8×C61—41no—
92C2×C8×C61—41no—
93C2×C8×C61—41no—
94C2×C8×C61—41no—
95C2×C8×C61—41no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—41no—
99C4⋊C4×C61—41no—
100C4⋊C4×C61—41no—
101C42×C61—41no—
102#11758198—42no—
103#11758198—42no—
104#11758198—42no—
105#11758198—42no—
106#11758198—41no—
107#11758198—41no—
108#11758198—41no—
109#11758198—41no—
110C4⋊Dic61—41no—
111#11758196—42no—
112C4×Dic61—41no—
113C4×Dic61—41no—
114C4×Dic61—41no—
115C2×#92418—41no—
116C2×#92418—41no—
117C2×#92418—41no—
118C2×#92418—41no—
119C2×#92418—41no—
120C2×#92418—41no—
121C23×C61C2×C22⋊C411yescharacteristic
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—21no—
125C22×D61—21no—
126C22×D61—21no—
127C22×D61—21no—
128C22×D61—41no—
129C22×D61—41no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—41no—
135C22×D61#9912yes—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61#13111yescharacteristic
149C2×C4×C61#13111yescharacteristic
150C2×C4×C61#13511yescharacteristic
151C2×C4×C61#13511yescharacteristic
152C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
154C2×Dic61—21no—
155C2×Dic61—21no—
156C2×Dic61—21no—
157C2×Dic61—21no—
158C2×Dic61—21no—
159C2×Dic61—21no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166D244—81no—
167D244—81no—
168D244—84no—
169C4×D61—42no—
170C4×D61—42no—
171C4×D61—42no—
172C4×D61—42no—
173C4×D61—42no—
174C4×D61—42no—
175C8×C61—81no—
176C8×C61—81no—
177#92418—81no—
178#92418—81no—
179#7718—611nomaximal, Sylow
180C22×C61#40811yescharacteristic
181C22×C61C2×#9911yescharacteristic
182C22×C61#41311yescharacteristic
183C22×C61#41611yescharacteristic
184C22×C61#41811yescharacteristic
185C22×C61#42111yescharacteristic
186C22×C61#42111yescharacteristic
187C2×D61—22no—
188C2×D61—22no—
189C2×D61—22no—
190C2×D61—41no—
191C2×D61—41no—
192C2×D61—41no—
193C2×D61—41no—
194C2×D61—81no—
195C2×D61—81no—
196C2×D61—81no—
197C2×D61—81no—
198C2×D61—81no—
199C2×D61—81no—
200C2×D61—81no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—81no—
208C4×C61—81no—
209Dic61—41no—
210Dic61—41no—
211Dic61—41no—
212Dic61—81no—
213#2854—611no—
214#1817—611no—
215#1817—611no—
216#1271—611no—
217#1271D6111yescharacteristic, complemented
218#1270—611no—
219#1270—611no—
220C2×C61#146811yescharacteristic
221C2×C61#148211yescharacteristic
222C2×C61#149611yescharacteristic
223C2×C61#151011yescharacteristic
224C2×C61#151611yescharacteristic
225C2×C61#153511yescharacteristic
226C2×C61#155311yescharacteristic
227D61—42no—
228D61—161no—
229C2×#123—1221no—
230C2×#121—611no—
231C22×C42—611no—
232C2×C22⋊C8—1221no—
233C2×C22⋊C8—1221no—
234#390—611no—
235#390—611no—
236#389—611no—
237#389C2×D6111yescharacteristic
238#388—1221no—
239#388—1221no—
240#386—1221no—
241#380—611no—
242#336—611no—
243#336—611no—
244#336—1221no—
245#336—1221no—
246#336C2×D6111yescharacteristic
247#336C2×D6111yescharacteristic
248C61#771811yescharacteristic, complemented, Sylow
249C22×D4—611no—
250C22×D4—1221no—
251C23×C4—611no—
252C23×C4—611no—
253C23×C4—611no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—1221no—
257C22×C8—1221no—
258#123—2442no—
259#123—2442no—
260#121—2442no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—611no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4C4×D6111yescharacteristic
265C2×C4⋊C4C4×D6111yescharacteristic
266C2×C22⋊C4—1221no—
267C2×C22⋊C4—1221no—
268C2×C22⋊C4—1221no—
269C2×C22⋊C4—1221no—
270C2×C22⋊C4—1221no—
271C2×C22⋊C4—1221no—
272C2×C42—611no—
273C2×C42—611no—
274C2×C42—611no—
275C2×C42—2442no—
276C2×C42C22×D6111yescharacteristic
277C22⋊C8—2442no—
278C22⋊C8—2442no—
279#95—21no—
280#95—1221no—
281#95—1221no—
282C24—611no—
283C24—1221no—
284C2×D4—1221no—
285C2×D4—1221no—
286C2×D4—2441no—
287C2×D4—2441no—
288C2×D4—2442no—
289C2×D4—2442no—
290C2×D4—2442no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—611no—
294C22×C4—611no—
295C22×C4—611no—
296C22×C4—1222no—
297C22×C4—1222no—
298C22×C4—1222no—
299C22×C4—1221no—
300C22×C4—1221no—
301C22×C4—1221no—
302C22×C4—1221no—
303C22×C4—1221no—
304C22×C4—1221no—
305C22×C4—1221no—
306C22×C4—2441no—
307C22×C4—2441no—
308C22×C4—2441no—
309C22×C4C2×C4×D6111yescharacteristic
310C22×C4D4×D6111yescharacteristic
311C22×C4D4×D6111yescharacteristic
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—2441no—
319C2×C8—2441no—
320C2×C8—2441no—
321C2×C8—2441no—
322C2×C8—2441no—
323C2×C8—2441no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—2441no—
330C4⋊C4—2442no—
331C22⋊C4—2442no—
332C22⋊C4—2442no—
333C22⋊C4—2442no—
334C22⋊C4—2442no—
335C22⋊C4—2441no—
336C22⋊C4—2441no—
337C22⋊C4—2441no—
338C22⋊C4—2441no—
339C42—41no—
340C42—2441no—
341C42—2441no—
342C42—2441no—
343C23—612no—
344C23—1221no—
345C23—1221no—
346C23—1221no—
347C23—1221no—
348C23—1221no—
349C23—1221no—
350C23—2441no—
351C23—2441no—
352C23—2441no—
353C23—2441no—
354C23—2441no—
355C23—2441no—
356C23—2441no—
357C23C22⋊C4×D6111yescharacteristic
358D4—4881no—
359D4—4881no—
360D4—4884no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—1221no—
374C2×C4—1221no—
375C2×C4—1221no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1221no—
379C2×C4—1221no—
380C2×C4—1221no—
381C2×C4—1221no—
382C2×C4—2442no—
383C2×C4—2442no—
384C2×C4—2442no—
385C2×C4—2442no—
386C2×C4—2442no—
387C2×C4—2442no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4#1451978311yescharacteristic
392C2×C4#1451978511yescharacteristic
393C2×C4#1451979811yescharacteristic
394C2×C4#1451980011yescharacteristic
395C2×C4#1451980111yescharacteristic
396C2×C4#1451980411yescharacteristic
397C8—81no—
398C8—81no—
399C8—4881no—
400C8—4881no—
401C22—1222no—
402C22—1222no—
403C22—1222no—
404C22—4881no—
405C22—4881no—
406C22—4881no—
407C22—4881no—
408C22—4881no—
409C22—4881no—
410C22—4881no—
411C22—2441no—
412C22—2441no—
413C22—2441no—
414C22—2441no—
415C22#1683972711yescharacteristic
416C22#99×D6111yescharacteristic
417C22#1683974511yescharacteristic
418C22#1683975711yescharacteristic
419C22#1683976511yescharacteristic
420C22#1683977711yescharacteristic
421C22#1683977711yescharacteristic
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—81no—
429C4—81no—
430C4—4881no—
431C4—2441no—
432C4—2441no—
433C4—2441no—
434C2—9761no—
435C2—2442no—
436C2#2144416611yescharacteristic
437C2#2144422611yescharacteristic
438C2#2144428611yescharacteristic
439C2#2144434611yescharacteristic
440C2#2144437411yescharacteristic
441C2#2144444911yescharacteristic
442C2#2144452311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1271C2
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519752C4
cosocleC2×C42×C61C23
hypercenter#1271D61
hypocenterC61#7718
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
2. #21445886C2semidirect
3. #21445886C2semidirect
4. #21443642C2semidirect
5. #21443639C2semidirect
6. #21443639C2semidirect
7. C61×#1271C2semidirect
217. #1271D61semidirect
248. C61#7718semidirect

Maximal subgroups

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

subgroupindex
#214541522
#214458862
#214436422
#214436392
C61×#12712
#771861