Finite Group Databasebeta

← #29577203

Subgroups — #29577203

Order 15616.

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

normal subgroups only

This group has 802 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#21455073C211yesmaximal, characteristic
2#21448650C211yesmaximal, characteristic, complemented
3#21446655C211yesmaximal, characteristic, complemented
4#21445764C211yesmaximal, characteristic, complemented
5#21444334C211yesmaximal, characteristic, complemented
6#21443724C211yesmaximal, characteristic, complemented
7C61×#1297C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C2×#14519805C412yescomplemented
10C2×#14519751C2211yescharacteristic
11C2×C4×D244—21no—
12#16839909—22no—
13#16839903—21no—
14#16839898—21no—
15C2×#14519705—22no—
16#16839764C412yescomplemented
17C2×#14519691—22no—
18#16839726C2211yescharacteristic
19#16839668—21no—
20#16839638—22no—
21#16839624—21no—
22#16839622C2211yescharacteristic
23#95×D61—21no—
24#16839495—21no—
25#16839495—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×Dic122D411yescharacteristic
35C22×D244C2×C411yescharacteristic, complemented
36C22×D244D411yescharacteristic
37C22×C22⋊D61—21no—
38C22×C22⋊D61—21no—
39C22×C4×D61—21no—
40C22×C4×D61—22no—
41#14519805—44no—
42C2×#11758190—22no—
43C2×D488—22no—
44C2×#11758194—21no—
45#14519751—21no—
46#14519751—21no—
47#14519751—21no—
48#14519751—21no—
49C4×D244—44no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61C2×C411yescharacteristic, complemented
52C2×C22⋊Dic61D412yes—
53C2×#11758198—22no—
54C2×#11758198—21no—
55#14519705—44no—
56#14519702—22no—
57#14519699—22no—
58#14519691—44no—
59C22⋊#92418—21no—
60C22⋊#92418—21no—
61#14519684—22no—
62#14519684—22no—
63#14519684—21no—
64C23×C4×C61C2311yescharacteristic
65C2×M16×C61C2×C412yescomplemented
66C2×C22⋊C4×C61—21no—
67C2×C22⋊C4×C61—21no—
68C2×C42×C61—21no—
69C2×C42×C61—21no—
70C22⋊C8×C61—21no—
71C22⋊C8×C61—21no—
72#95×C61—21no—
73#95×C61D411yescharacteristic
74#95×C61D411yescharacteristic
75C24×C61C2×D411yescharacteristic
76C23×D61—21no—
77C23×D61—21no—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—42no—
81C22×C4×C61—41no—
82C22×C4×C61C22×C411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61#4111yescharacteristic
87C22×C4×C61#4111yescharacteristic
88C2×C22⋊D61—22no—
89C2×C22⋊D61—22no—
90C2×C22⋊D61—44no—
91C2×C22⋊D61—42no—
92C2×C22⋊D61—42no—
93C2×C22⋊D61—41no—
94C2×C22⋊D61—41no—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C22×Dic61—22no—
98#11758222—24no—
99#11758222—24no—
100#11758222—42no—
101#11758222—42no—
102#11758222—44no—
103#11758222—44no—
104C2×D244—21no—
105C2×D244—24no—
106C2×D244—21no—
107C2×D244—21no—
108C2×D244—42no—
109C2×D244C22⋊C412yes—
110C2×C4×D61—22no—
111C2×C4×D61—22no—
112C2×C4×D61—44no—
113C2×C4×D61—44no—
114C2×C4×D61—44no—
115C2×C4×D61—82no—
116C2×Dic122—21no—
117C2×Dic122—42no—
118C2×Dic122C22⋊C412yes—
119M16×C61—42no—
120C2×C8×C61—22no—
121C22⋊C4×C61—22no—
122C22⋊C4×C61—42no—
123C22⋊C4×C61—41no—
124C42×C61—42no—
125C42×C61—42no—
126C22⋊Dic61—42no—
127#11758198—44no—
128#11758198—41no—
129#11758198—41no—
130C4⋊Dic61—21no—
131C4⋊Dic61—21no—
132C4⋊Dic61—42no—
133#11758194—42no—
134C2×#92418—41no—
135D488—44no—
136#11758190—44no—
137C23×C61—21no—
138C23×C61—21no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61C4×D411yescharacteristic
142C23×C61#12011yescharacteristic
143C23×C61#12111yescharacteristic
144C22×D61—24no—
145C22×D61—22no—
146C22×D61—42no—
147C22×D61—41no—
148C22×D61—41no—
149C22×D61—41no—
150C22×D61—42no—
151C22×D61—81no—
152C22×D61—81no—
153C2×C4×C61—22no—
154C2×C4×C61—22no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—81no—
167C2×C4×C61—81no—
168C2×C4×C61C2×C22⋊C411yescharacteristic
169C2×C4×C61C4×D411yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#12311yescharacteristic
172C22⋊D61—44no—
173C22⋊D61—44no—
174C22⋊D61—82no—
175C22⋊D61—81no—
176C22⋊D61—81no—
177C2×Dic61—22no—
178C2×Dic61—42no—
179C2×Dic61—42no—
180C2×Dic61—42no—
181C2×Dic61—42no—
182C2×Dic61—41no—
183C2×Dic61—42no—
184D244—24no—
185D244—22no—
186D244—42no—
187D244—81no—
188D244#9914yescomplemented
189C4×D61—44no—
190C4×D61—44no—
191C4×D61—88no—
192Dic122—22no—
193Dic122—22no—
194Dic122—42no—
195C8×C61—42no—
196#92418—81no—
197#9365—611nomaximal, Sylow
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—21no—
204C22×C61—21no—
205C22×C61—21no—
206C22×C61—81no—
207C22×C61—81no—
208C22×C61#38611yescharacteristic
209C22×C61#45311yescharacteristic
210C22×C61#45411yescharacteristic
211C2×D61—24no—
212C2×D61—44no—
213C2×D61—44no—
214C2×D61—42no—
215C2×D61—42no—
216C2×D61—81no—
217C2×D61—81no—
218C2×D61—81no—
219C2×D61—162no—
220C4×C61—21no—
221C4×C61—21no—
222C4×C61—21no—
223C4×C61—42no—
224C4×C61—81no—
225C4×C61—82no—
226C4×C61C2×#9912yes—
227Dic61—42no—
228Dic61—82no—
229Dic61—81no—
230#3006—611no—
231#2339—611no—
232#1964—611no—
233#1789—611no—
234#1507—611no—
235#1305—611no—
236#1297D6111yescharacteristic, complemented
237C2×C61—21no—
238C2×C61—21no—
239C2×C61—41no—
240C2×C61—41no—
241C2×C61#188311yescharacteristic
242C2×C61#189811yescharacteristic
243C2×C61#190611yescharacteristic
244D61—44no—
245D61—161no—
246C22×#41—611no—
247C2×AGL(1,ℤ/8)—612no—
248C2×#121—611no—
249C2×C4×D4—1221no—
250#449—1222no—
251#448—1221no—
252#447—1221no—
253C2×#104—1222no—
254#417—612no—
255C2×#99—1222no—
256#407—611no—
257#407C2×D6111yescharacteristic
258#391—1221no—
259#386—1222no—
260#381—1221no—
261#380—611no—
262C4×C22⋊C4C2×D6111yescharacteristic
263C4×C22⋊C4C2×D6111yescharacteristic
264C2×#95—1221no—
265#327—1221no—
266#327—1221no—
267C61#936511yescharacteristic, complemented, Sylow
268C2×#41—612no—
269C2×#41—612no—
270C2×#41—1221no—
271C2×#41—1221no—
272C2×#41—2444no—
273C22×Q8—611no—
274C22×D4—611no—
275C22×D4—611no—
276C22×D4—1221no—
277C22×D4—1221no—
278C23×C4—1222no—
279C23×C4—1221no—
280C23×C4C22×D6111yescharacteristic
281AGL(1,ℤ/8)—2444no—
282C2×SD16—1222no—
283C2×D8—1222no—
284C2×M16—1221no—
285C2×M16C4×D6112yescomplemented
286#121—1221no—
287#121—1221no—
288#121—1221no—
289#121—1221no—
290C4×D4—2444no—
291C2×C4⋊C4—611no—
292C2×C4⋊C4—1221no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—612no—
296C2×C22⋊C4—1222no—
297C2×C22⋊C4—1221no—
298C2×C42—21no—
299C2×C42—21no—
300#104—2444no—
301#103—1222no—
302#102—1222no—
303#99—2444no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—1221no—
307C22⋊C8—1221no—
308#95—21no—
309#95—1222no—
310#95—1222no—
311#95—1221no—
312#95D24411yescharacteristic
313#95D24411yescharacteristic
314C24—1221no—
315C24—1221no—
316C24D4×D6111yescharacteristic
317#41—1224no—
318#41—1224no—
319#41—2442no—
320#41—2442no—
321#41—2444no—
322#41—2444no—
323C2×Q8—612no—
324C2×Q8—1221no—
325C2×Q8—2442no—
326C2×D4—612no—
327C2×D4—1224no—
328C2×D4—1222no—
329C2×D4—1222no—
330C2×D4—1221no—
331C2×D4—1221no—
332C2×D4—1221no—
333C2×D4—2442no—
334C2×D4—2442no—
335C2×D4—2442no—
336C2×D4—2441no—
337C2×D4—2441no—
338C2×D4—2444no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—1221no—
344C22×C4—1222no—
345C22×C4—1222no—
346C22×C4—1221no—
347C22×C4—1222no—
348C22×C4—4882no—
349C22×C4—2444no—
350C22×C4—2444no—
351C22×C4—2444no—
352C22×C4C2×C4×D6111yescharacteristic
353C22×C4C2×D24411yescharacteristic
354C22×C4#1175822211yescharacteristic
355C22×C4D4×D6111yescharacteristic
356C22×C4D4×D6111yescharacteristic
357C22×C4#1175822611yescharacteristic
358SD16—2444no—
359D8—2444no—
360M16—42no—
361M16—2442no—
362C2×C8—22no—
363C2×C8—2441no—
364C4⋊C4—1221no—
365C4⋊C4—1221no—
366C4⋊C4—2442no—
367C22⋊C4—22no—
368C22⋊C4—42no—
369C22⋊C4—41no—
370C22⋊C4—2444no—
371C22⋊C4—2441no—
372C22⋊C4—2441no—
373C22⋊C4—2442no—
374C42—42no—
375C42—42no—
376C23—21no—
377C23—21no—
378C23—41no—
379C23—41no—
380C23—1224no—
381C23—1222no—
382C23—4881no—
383C23—4881no—
384C23—2442no—
385C23—2442no—
386C23—2441no—
387C23—2441no—
388C23—2441no—
389C23#1451973911yescharacteristic
390C23#1451974711yescharacteristic
391C23#1451975011yescharacteristic
392Q8—1222no—
393Q8—1222no—
394Q8—2442no—
395D4—614no—
396D4—1224no—
397D4—1222no—
398D4—4881no—
399D4—4882no—
400D4—4881no—
401D4—4881no—
402D4—2444no—
403D4—2444no—
404D4—2442no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—1222no—
421C2×C4—4888no—
422C2×C4—2444no—
423C2×C4—2444no—
424C2×C4—2442no—
425C2×C4—2442no—
426C2×C4—2442no—
427C2×C4—2441no—
428C2×C4—2442no—
429C2×C4—2442no—
430C2×C4C22⋊C4×D6111yescharacteristic
431C2×C4C4×D24411yescharacteristic
432C2×C4#1451975211yescharacteristic
433C2×C4#1451976211yescharacteristic
434C8—42no—
435C8—4881no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—81no—
445C22—81no—
446C22—1224no—
447C22—9762no—
448C22—4881no—
449C22—4881no—
450C22—4881no—
451C22—2444no—
452C22—2444no—
453C22—2442no—
454C22—2442no—
455C22#1683964011yescharacteristic
456C22#1683993111yescharacteristic
457C22#1683993611yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—42no—
462C4—81no—
463C4—82no—
464C4—4882no—
465C4—4881no—
466C4—2442no—
467C4#99×D6112yes—
468C2—21no—
469C2—21no—
470C2—41no—
471C2—41no—
472C2—9761no—
473C2—2444no—
474C2#2144619611yescharacteristic
475C2#2144627111yescharacteristic
476C2#2144631911yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1297C2
socleC22×C61#386
prime radical#21455073C2
cosocleC23×C4×C61C23
hypercenter#1297D61
hypocenterC61#9365
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
2. #21448650C2semidirect
3. #21446655C2semidirect
4. #21445764C2semidirect
5. #21444334C2semidirect
6. #21443724C2semidirect
7. C61×#1297C2semidirect
9. C2×#14519805C4semidirect
16. #16839764C4semidirect
35. C22×D244C2×C4semidirect
51. C2×C4⋊Dic61C2×C4semidirect
65. C2×M16×C61C2×C4semidirect
188. D244#99semidirect
236. #1297D61semidirect
267. C61#9365semidirect
285. C2×M16C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214550732
#214486502
#214466552
#214457642
#214443342
#214437242
C61×#12972
#936561