Finite Group Databasebeta

← #29580571

Subgroups — #29580571

Order 15616.

804 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 117 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×#16839918C211yesmaximal, characteristic
2#21451935C211yesmaximal, characteristic
3C2×#16839500C211yesmaximal, characteristic
4C2×#16839499C211yesmaximal, characteristic
5#21444090C211yesmaximal, characteristic
6#21443661C211yesmaximal, characteristic
7C2×C61×#451C211yesmaximal, characteristic
8C22×#11758202—21no—
9C2×#14519756C2211yescharacteristic
10#16839918—22no—
11#16839918—22no—
12C2×#14519701—21no—
13C2×C22⋊#92418C2211yescharacteristic
14#16839693—21no—
15#16839692—21no—
16#16839682—21no—
17#16839681—21no—
18#16839620—22no—
19C2×#14519684—22no—
20C2×#14519684C2211yescharacteristic
21C2×#14519684C2211yescharacteristic
22#16839500—22no—
23#16839499—22no—
24C23×Q8×C61C2211yescharacteristic
25C22×Q16×C61—21no—
26C2×C22⋊Q8×C61C2211yescharacteristic
27C61×#451—22no—
28C61×#451—22no—
29C2×#103×C61—21no—
30C2×C22⋊C8×C61C2211yescharacteristic
31C22×Dic122D411yescharacteristic
32C23×Dic61—21no—
33C23×Dic61D412yes—
34C2×#11758202—24no—
35C2×#11758202—44no—
36C22×#92418—21no—
37#14519756—21no—
38#14519756—21no—
39#14519756—21no—
40#14519756—21no—
41C2×C4⋊Dic61D411yescharacteristic
42C2×#11758196—21no—
43C2×#11758196—22no—
44C2×#11758196—22no—
45C2×C22⋊Dic61—21no—
46#14519701—24no—
47C22⋊#92418—21no—
48C22⋊#92418D412yes—
49#14519684—22no—
50#14519684—22no—
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—44no—
56C22×Q8×C61—21no—
57C22×Q8×C61—21no—
58C22×Q8×C61—21no—
59C22×Q8×C61—21no—
60C22×Q8×C61—21no—
61C22×Q8×C61—44no—
62C22×Q8×C61D411yescharacteristic
63C22×Q8×C61D412yes—
64C23×C4×C61—21no—
65C23×C4×C61C2311yescharacteristic
66C2×Q16×C61—24no—
67C2×Q16×C61—44no—
68C22×C8×C61—21no—
69C22⋊Q8×C61—21no—
70C22⋊Q8×C61—21no—
71C22⋊Q8×C61—21no—
72C22⋊Q8×C61—21no—
73C2×C4⋊C4×C61—21no—
74C2×C4⋊C4×C61D411yescharacteristic
75C2×C22⋊C4×C61—21no—
76#103×C61—24no—
77C22⋊C8×C61—21no—
78C22⋊C8×C61D412yes—
79C24×C61C2×D411yescharacteristic
80C2×Q8×C61—24no—
81C2×Q8×C61—24no—
82C2×Q8×C61—22no—
83C2×Q8×C61—22no—
84C2×Q8×C61—21no—
85C2×Q8×C61—21no—
86C2×Q8×C61—44no—
87C2×Q8×C61—42no—
88C2×Q8×C61—42no—
89C2×Q8×C61—44no—
90C2×Q8×C61—42no—
91C2×Q8×C61—44no—
92C2×Q8×C61—44no—
93C2×Q8×C61—41no—
94C2×Q8×C61—41no—
95C22×C4×C61—24no—
96C22×C4×C61—22no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—42no—
100C22×C4×C61—42no—
101C22×C4×C61—41no—
102C22×C4×C61—41no—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61#4111yescharacteristic
109C22×Dic61—22no—
110C22×Dic61—22no—
111C22×Dic61—22no—
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61—42no—
115C22×Dic61—42no—
116C22×Dic61—41no—
117C22×Dic61—42no—
118C22×Dic61—41no—
119C22×Dic61—42no—
120C22×Dic61Q1614yescomplemented
121C2×Dic122—21no—
122C2×Dic122—21no—
123C2×Dic122—42no—
124Q16×C61—48no—
125C2×C8×C61—22no—
126C2×C8×C61—42no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—41no—
130C4⋊C4×C61—41no—
131C22⋊C4×C61—41no—
132C22⋊C4×C61—41no—
133C22⋊Dic61—41no—
134C22⋊Dic61—41no—
135#11758202—48no—
136C4⋊Dic61—21no—
137C4⋊Dic61—21no—
138#11758196—41no—
139#11758196—41no—
140#11758196—82no—
141#11758196—82no—
142C2×#92418—22no—
143C2×#92418—42no—
144C23×C61—21no—
145C23×C61—21no—
146C23×C61—22no—
147C23×C61#12011yescharacteristic
148C23×C61#12111yescharacteristic
149C23×C61#12711yescharacteristic
150C23×C61C2×Q1612yes—
151C23×C61#13712yes—
152Q8×C61—24no—
153Q8×C61—24no—
154Q8×C61—42no—
155Q8×C61—42no—
156Q8×C61—44no—
157Q8×C61—44no—
158Q8×C61—42no—
159Q8×C61—81no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—24no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—44no—
168C2×C4×C61—42no—
169C2×C4×C61—42no—
170C2×C4×C61—44no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—42no—
176C2×C4×C61#12011yescharacteristic
177C2×C4×C61#12011yescharacteristic
178C2×C4×C61#12111yescharacteristic
179C2×C4×C61#12111yescharacteristic
180C2×Dic61—24no—
181C2×Dic61—42no—
182C2×Dic61—42no—
183C2×Dic61—44no—
184C2×Dic61—44no—
185C2×Dic61—42no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—42no—
191C2×Dic61—82no—
192C2×Dic61—82no—
193C2×Dic61—81no—
194C2×Dic61—81no—
195Dic122—42no—
196Dic122—81no—
197C8×C61—42no—
198#92418—42no—
199#9967—611nomaximal, Sylow
200C22×C61—22no—
201C22×C61—22no—
202C22×C61—22no—
203C22×C61—22no—
204C22×C61—21no—
205C22×C61—21no—
206C22×C61—21no—
207C22×C61—21no—
208C22×C61#39211yescharacteristic
209C22×C61#45112yes—
210C22×C61#45511yescharacteristic
211C22×C61#45611yescharacteristic
212C22×C61B(4,𝔽2)12yes—
213C22×C61#46212yes—
214C22×C61#49411yescharacteristic
215C22×C61#49711yescharacteristic
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—44no—
221C4×C61—81no—
222C4×C61—81no—
223Dic61—44no—
224Dic61—82no—
225Dic61—81no—
226Dic61—81no—
227C2×#451—611no—
228C2×#451D6111yescharacteristic
229#2659—611no—
230C2×#328—611no—
231C2×#328—611no—
232#1446—611no—
233#1280—611no—
234C2×C61—21no—
235C2×C61—22no—
236C2×C61—21no—
237C2×C61#160212yes—
238C2×C61#160512yes—
239C2×C61#200711yescharacteristic
240C2×C61#201611yescharacteristic
241C2×C61#202111yescharacteristic
242C23×Q8C2×D6111yescharacteristic
243C22×Q16—21no—
244C22×Q16—1221no—
245C2×C22⋊Q8—611no—
246C2×C22⋊Q8C2×D6111yescharacteristic
247#451—22no—
248#451—22no—
249#451—1222no—
250#451—1222no—
251C2×#103—21no—
252C2×#103—1221no—
253C2×C22⋊C8—611no—
254C2×C22⋊C8C2×D6111yescharacteristic
255#398—1221no—
256#398—1221no—
257#395—1221no—
258#395—1221no—
259#380—1222no—
260C2×#95—611no—
261C2×#95—611no—
262C2×#95—1222no—
263#328—1222no—
264#328—1222no—
265C61#996711yescharacteristic, complemented, Sylow
266C22×Q8—21no—
267C22×Q8—21no—
268C22×Q8—21no—
269C22×Q8—21no—
270C22×Q8—21no—
271C22×Q8—44no—
272C22×Q8—611no—
273C22×Q8C22⋊D6111yescharacteristic
274C22×Q8C22⋊D6112yes—
275C23×C4—21no—
276C23×C4—612no—
277C23×C4—1221no—
278C23×C4C22×D6111yescharacteristic
279C2×Q16—24no—
280C2×Q16—44no—
281C2×Q16—1224no—
282C2×Q16—2444no—
283C22×C8—21no—
284C22×C8—1221no—
285C22⋊Q8—21no—
286C22⋊Q8—21no—
287C22⋊Q8—21no—
288C22⋊Q8—21no—
289C22⋊Q8—1221no—
290C22⋊Q8—1221no—
291C22⋊Q8—1221no—
292C22⋊Q8—1221no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—611no—
295C2×C4⋊C4—1222no—
296C2×C4⋊C4—1222no—
297C2×C4⋊C4—1221no—
298C2×C4⋊C4C22⋊D6111yescharacteristic
299C2×C22⋊C4—21no—
300C2×C22⋊C4—1221no—
301#103—24no—
302#103—1224no—
303C22⋊C8—21no—
304C22⋊C8—612no—
305C22⋊C8—1221no—
306C22⋊C8C22⋊D6112yes—
307#95—1222no—
308#95—1222no—
309#95—1221no—
310#95—1221no—
311#95—1221no—
312#95—1221no—
313#95—2444no—
314C24D4×D6111yescharacteristic
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—44no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—44no—
325C2×Q8—42no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—1221no—
331C2×Q8—1221no—
332C2×Q8—2442no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—614no—
342C22×C4—1222no—
343C22×C4—1222no—
344C22×C4—1222no—
345C22×C4—1221no—
346C22×C4—1221no—
347C22×C4—2442no—
348C22×C4—2441no—
349C22×C4—2442no—
350C22×C4—2441no—
351C22×C4—2442no—
352C22×C4—2442no—
353C22×C4D4×D6111yescharacteristic
354C22×C4D4×D6111yescharacteristic
355C22×C4#1175822411yescharacteristic
356C22×C4C2×C22⋊D6111yescharacteristic
357C22×C4C2×C22⋊D6111yescharacteristic
358C22×C4C2×C22⋊D6111yescharacteristic
359Q16—48no—
360Q16—2448no—
361C2×C8—22no—
362C2×C8—42no—
363C2×C8—1222no—
364C2×C8—2442no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—1221no—
370C4⋊C4—1221no—
371C4⋊C4—4882no—
372C4⋊C4—4882no—
373C4⋊C4—2441no—
374C4⋊C4—2441no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—2441no—
378C22⋊C4—2441no—
379C23—21no—
380C23—21no—
381C23—22no—
382C23#1451974511yescharacteristic
383C23#1451974911yescharacteristic
384C23#1451977311yescharacteristic
385C23Q16×D6112yes—
386C23#1451980912yes—
387Q8—24no—
388Q8—24no—
389Q8—42no—
390Q8—42no—
391Q8—44no—
392Q8—44no—
393Q8—42no—
394Q8—81no—
395Q8—4881no—
396Q8—2442no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—1224no—
414C2×C4—4881no—
415C2×C4—4882no—
416C2×C4—4881no—
417C2×C4—4882no—
418C2×C4—2444no—
419C2×C4—2442no—
420C2×C4—2441no—
421C2×C4—2441no—
422C2×C4—2442no—
423C2×C4—2441no—
424C2×C4—2441no—
425C2×C4—2442no—
426C2×C4—2442no—
427C2×C4—2444no—
428C2×C4#1451974511yescharacteristic
429C2×C4#1451974611yescharacteristic
430C2×C4#1451974811yescharacteristic
431C2×C4#1451974911yescharacteristic
432C8—42no—
433C8—2442no—
434C22—21no—
435C22—21no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—21no—
441C22—21no—
442C22#1683967011yescharacteristic
443C22#1683991612yes—
444C22#1683993811yescharacteristic
445C22#1683994311yescharacteristic
446C22#1683994812yes—
447C22#1683997012yes—
448C22#1684014011yescharacteristic
449C22#1684015511yescharacteristic
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—44no—
455C4—81no—
456C4—81no—
457C4—4882no—
458C4—4881no—
459C4—4881no—
460C4—2444no—
461C2—21no—
462C2—22no—
463C2—21no—
464C2#2144477412yes—
465C2#2144479112yes—
466C2#2144686411yescharacteristic
467C2#2144691711yescharacteristic
468C2#2144694211yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#451C2
socleC23×C61#120
prime radicalC24×C61C2×D4
hypercenterC2×#451D61
hypocenterC61#9967
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
120. C22×Dic61Q16semidirect
265. C61#9967semidirect

Maximal subgroups

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

subgroupindex
C2×#168399182
#214519352
C2×#168395002
C2×#168394992
#214440902
#214436612
C2×C61×#4512
#996761