Finite Group Databasebeta

← #29577175

Subgroups — #29577175

Order 15616.

760 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 97 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#21455081C211yesmaximal, characteristic
2#21448650C211yesmaximal, characteristic
3#21446804C211yesmaximal, characteristic, complemented
4#21445764C211yesmaximal, characteristic
5#21444338C211yesmaximal, characteristic, complemented
6#21443724C211yesmaximal, characteristic, complemented
7C61×#1297C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C2×#14519810C412yescomplemented
10C2×#14519756C2211yescharacteristic
11C2×C4×D244—21no—
12#16839926—22no—
13#16839920—21no—
14#16839915—21no—
15C2×#14519705—22no—
16#16839764C412yescomplemented
17#16839726C2211yescharacteristic
18#16839668—21no—
19#16839638—22no—
20#16839624—21no—
21#16839622C2211yescharacteristic
22#95×D61—21no—
23#16839528—22no—
24#16839498—21no—
25#16839498—21no—
26C61×#407C2211yescharacteristic
27C4×C22⋊C4×C61C2211yescharacteristic
28C4×C22⋊C4×C61C2211yescharacteristic
29C2×#11758222—21no—
30C2×#11758222—21no—
31C2×#11758222—44no—
32C2×#11758222C2×C412yes—
33C2×#11758222D412yes—
34C22×Dic122C2×C411yescharacteristic, complemented
35C22×Dic122D411yescharacteristic
36C22×D244D411yescharacteristic
37C22×C22⋊D61—21no—
38C22×C4×D61—21no—
39C22×C4×D61—22no—
40#14519810—44no—
41C2×Dic244—22no—
42C2×#11758190—22no—
43C2×#11758194—21no—
44#14519756—21no—
45#14519756—21no—
46#14519756—21no—
47#14519756—21no—
48C4×D244—44no—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61C2×C411yescharacteristic, complemented
51C2×#11758196—21no—
52C2×C22⋊Dic61—21no—
53C2×C22⋊Dic61D412yes—
54C2×#11758198—22no—
55#14519705—44no—
56#14519702—22no—
57#14519699—22no—
58C22⋊#92418—21no—
59C22⋊#92418—21no—
60#14519684—22no—
61#14519684—22no—
62#14519684—21no—
63C23×C4×C61C2311yescharacteristic
64C2×M16×C61C2×C412yescomplemented
65C2×C22⋊C4×C61—21no—
66C2×C22⋊C4×C61—21no—
67C2×C42×C61—21no—
68C2×C42×C61—21no—
69C22⋊C8×C61—21no—
70C22⋊C8×C61—21no—
71#95×C61—21no—
72#95×C61D411yescharacteristic
73#95×C61D411yescharacteristic
74C24×C61C2×D411yescharacteristic
75C23×D61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—42no—
79C22×C4×C61—41no—
80C22×C4×C61C22×C411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61#4111yescharacteristic
85C22×C4×C61#4111yescharacteristic
86C2×C22⋊D61—22no—
87C2×C22⋊D61—22no—
88C2×C22⋊D61—44no—
89C22×Dic61—21no—
90C22×Dic61—22no—
91C22×Dic61—21no—
92C22×Dic61—21no—
93#11758222—24no—
94#11758222—24no—
95#11758222—42no—
96#11758222—42no—
97#11758222—44no—
98#11758222—44no—
99C2×D244—21no—
100C2×D244—24no—
101C2×D244C22⋊C412yes—
102C2×C4×D61—22no—
103C2×C4×D61—22no—
104C2×C4×D61—44no—
105C2×C4×D61—44no—
106C2×C4×D61—44no—
107C2×C4×D61—82no—
108C2×Dic122—21no—
109C2×Dic122—21no—
110C2×Dic122—21no—
111C2×Dic122—42no—
112C2×Dic122—42no—
113C2×Dic122C22⋊C412yes—
114M16×C61—42no—
115C2×C8×C61—22no—
116C22⋊C4×C61—22no—
117C22⋊C4×C61—42no—
118C22⋊C4×C61—41no—
119C42×C61—42no—
120C42×C61—42no—
121C22⋊Dic61—41no—
122C22⋊Dic61—41no—
123C22⋊Dic61—42no—
124#11758198—44no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127C4⋊Dic61—42no—
128#11758196—41no—
129#11758196—41no—
130#11758194—42no—
131C2×#92418—41no—
132Dic244—44no—
133#11758190—44no—
134C23×C61—21no—
135C23×C61—21no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61C4×D411yescharacteristic
139C23×C61#12011yescharacteristic
140C23×C61#12111yescharacteristic
141C22×D61—24no—
142C22×D61—22no—
143C22×D61—42no—
144C22×D61—41no—
145C22×D61—41no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—42no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—81no—
160C2×C4×C61—81no—
161C2×C4×C61C2×C22⋊C411yescharacteristic
162C2×C4×C61C4×D411yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12311yescharacteristic
165C22⋊D61—44no—
166C22⋊D61—44no—
167C2×Dic61—22no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170C2×Dic61—42no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—42no—
174C2×Dic61—81no—
175C2×Dic61—81no—
176D244—24no—
177D244—22no—
178D244#9914yes—
179C4×D61—44no—
180C4×D61—44no—
181C4×D61—88no—
182Dic122—22no—
183Dic122—22no—
184Dic122—42no—
185Dic122—42no—
186Dic122—81no—
187C8×C61—42no—
188#92418—81no—
189#9361—611nomaximal, Sylow
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—21no—
198C22×C61—81no—
199C22×C61—81no—
200C22×C61#38611yescharacteristic
201C22×C61#45511yescharacteristic
202C22×C61#45611yescharacteristic
203C2×D61—24no—
204C2×D61—44no—
205C2×D61—44no—
206C2×D61—42no—
207C2×D61—42no—
208C4×C61—21no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—42no—
212C4×C61—81no—
213C4×C61—82no—
214C4×C61C2×#9912yes—
215Dic61—42no—
216Dic61—82no—
217Dic61—81no—
218Dic61—81no—
219#3007—611no—
220#2339—611no—
221#1996—611no—
222#1789—611no—
223#1508—611no—
224#1305—611no—
225#1297D6111yescharacteristic, complemented
226C2×C61—21no—
227C2×C61—21no—
228C2×C61—41no—
229C2×C61—41no—
230C2×C61#188211yescharacteristic
231C2×C61#189811yescharacteristic
232C2×C61#190611yescharacteristic
233D61—44no—
234C22×#41—611no—
235C2×#137—612no—
236C2×C22⋊Q8—611no—
237C2×C4×D4—1221no—
238#452—1222no—
239#451—1221no—
240#450—1221no—
241C2×#104—1222no—
242#417—612no—
243#407—611no—
244#407C2×D6111yescharacteristic
245#391—1221no—
246#386—1222no—
247#381—1221no—
248#380—611no—
249C4×C22⋊C4C2×D6111yescharacteristic
250C4×C22⋊C4C2×D6111yescharacteristic
251C2×#95—1221no—
252#342—1222no—
253#328—1221no—
254#328—1221no—
255C61#936111yescharacteristic, complemented, Sylow
256C2×#41—612no—
257C2×#41—612no—
258C2×#41—1221no—
259C2×#41—1221no—
260C2×#41—2444no—
261C22×Q8—611no—
262C22×Q8—611no—
263C22×D4—611no—
264C22×D4—1221no—
265C23×C4—1222no—
266C23×C4—1221no—
267C23×C4C22×D6111yescharacteristic
268#137—2444no—
269C2×Q16—1222no—
270C2×SD16—1222no—
271C2×M16—1221no—
272C2×M16C4×D6112yescomplemented
273C22⋊Q8—1221no—
274C22⋊Q8—1221no—
275C22⋊Q8—1221no—
276C22⋊Q8—1221no—
277C4×D4—2444no—
278C2×C4⋊C4—611no—
279C2×C4⋊C4—1221no—
280C2×C4⋊C4—1221no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—612no—
284C2×C22⋊C4—1222no—
285C2×C22⋊C4—1221no—
286C2×C42—21no—
287C2×C42—21no—
288#104—2444no—
289#103—1222no—
290#102—1222no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—1221no—
294C22⋊C8—1221no—
295#95—21no—
296#95—1222no—
297#95—1222no—
298#95—1221no—
299#95D24411yescharacteristic
300#95D24411yescharacteristic
301C24—1221no—
302C24D4×D6111yescharacteristic
303#41—1224no—
304#41—1224no—
305#41—2442no—
306#41—2442no—
307#41—2444no—
308#41—2444no—
309C2×Q8—612no—
310C2×Q8—1221no—
311C2×Q8—1221no—
312C2×Q8—1221no—
313C2×Q8—2442no—
314C2×Q8—2442no—
315C2×D4—612no—
316C2×D4—1224no—
317C2×D4—1222no—
318C2×D4—1222no—
319C2×D4—1221no—
320C2×D4—2444no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—1221no—
326C22×C4—1222no—
327C22×C4—1222no—
328C22×C4—1221no—
329C22×C4—1221no—
330C22×C4—1222no—
331C22×C4—4882no—
332C22×C4—2444no—
333C22×C4—2444no—
334C22×C4—2444no—
335C22×C4C2×C4×D6111yescharacteristic
336C22×C4C2×D24411yescharacteristic
337C22×C4#1175822211yescharacteristic
338C22×C4D4×D6111yescharacteristic
339C22×C4D4×D6111yescharacteristic
340C22×C4#1175822611yescharacteristic
341Q16—2444no—
342SD16—2444no—
343M16—42no—
344M16—2442no—
345C2×C8—22no—
346C2×C8—2441no—
347C4⋊C4—1221no—
348C4⋊C4—1221no—
349C4⋊C4—2442no—
350C4⋊C4—2441no—
351C4⋊C4—2441no—
352C22⋊C4—22no—
353C22⋊C4—42no—
354C22⋊C4—41no—
355C22⋊C4—2444no—
356C22⋊C4—2441no—
357C22⋊C4—2441no—
358C22⋊C4—2442no—
359C42—42no—
360C42—42no—
361C23—21no—
362C23—21no—
363C23—41no—
364C23—41no—
365C23—1224no—
366C23—1222no—
367C23—2442no—
368C23—2441no—
369C23—2441no—
370C23#1451973911yescharacteristic
371C23#1451974711yescharacteristic
372C23#1451975011yescharacteristic
373Q8—1222no—
374Q8—1222no—
375Q8—4881no—
376Q8—2442no—
377Q8—2442no—
378D4—614no—
379D4—1224no—
380D4—1222no—
381D4—2444no—
382D4—2444no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—1222no—
399C2×C4—4888no—
400C2×C4—4881no—
401C2×C4—4881no—
402C2×C4—2444no—
403C2×C4—2444no—
404C2×C4—2442no—
405C2×C4—2442no—
406C2×C4—2442no—
407C2×C4—2441no—
408C2×C4—2441no—
409C2×C4—2442no—
410C2×C4C22⋊C4×D6111yescharacteristic
411C2×C4C4×D24411yescharacteristic
412C2×C4#1451975211yescharacteristic
413C2×C4#1451976211yescharacteristic
414C8—42no—
415C8—4881no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—81no—
425C22—81no—
426C22—1224no—
427C22—2444no—
428C22—2444no—
429C22—2442no—
430C22—2442no—
431C22#1683964011yescharacteristic
432C22#1683994111yescharacteristic
433C22#1683994611yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—42no—
438C4—81no—
439C4—82no—
440C4—4882no—
441C4—4881no—
442C4—4881no—
443C4—2442no—
444C4#99×D6112yes—
445C2—21no—
446C2—21no—
447C2—41no—
448C2—41no—
449C2—2444no—
450C2#2144618911yescharacteristic
451C2#2144627111yescharacteristic
452C2#2144631911yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1297C2
socleC22×C61#386
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenter#1297D61
hypocenterC61#9361
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
3. #21446804C2semidirect
5. #21444338C2semidirect
6. #21443724C2semidirect
7. C61×#1297C2semidirect
9. C2×#14519810C4semidirect
16. #16839764C4semidirect
34. C22×Dic122C2×C4semidirect
50. C2×C4⋊Dic61C2×C4semidirect
64. C2×M16×C61C2×C4semidirect
225. #1297D61semidirect
255. C61#9361semidirect
272. C2×M16C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214550812
#214486502
#214468042
#214457642
#214443382
#214437242
C61×#12972
#936161