Finite Group Databasebeta

← #29580511

Subgroups — #29580511

Order 15616.

744 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 103 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#21456594C211yesmaximal, characteristic
2C2×#16839918C211yesmaximal, characteristic
3#21449459C211yesmaximal, characteristic
4#21446803C211yesmaximal, characteristic
5#21444390C211yesmaximal, characteristic
6#21443666C211yesmaximal, characteristic
7C61×#1787C211yesmaximal, characteristic
8C22×#11758202—21no—
9C2×#14519775—21no—
10C2×#14519756C2211yescharacteristic
11C2×#14519756C2211yescharacteristic
12#16840098—21no—
13#16840098—21no—
14#16840003—22no—
15#16839920—21no—
16#16839920—21no—
17#16839918—22no—
18#16839918—22no—
19C2×#14519701—21no—
20C2×C22⋊#92418C2211yescharacteristic
21#16839681—22no—
22#16839669—21no—
23#16839635—22no—
24C4×C22⋊Dic61C2211yescharacteristic
25#16839528—22no—
26#16839498—22no—
27C23×Q8×C61C2211yescharacteristic
28C2×#103×C61—22no—
29C2×C22⋊C8×C61C2211yescharacteristic
30C61×#380C2211yescharacteristic
31C22×Dic122D411yescharacteristic
32C22×Dic122D411yescharacteristic
33C2×Dic244—22no—
34C2×#11758202—24no—
35C2×#11758202—44no—
36C22×#92418—21no—
37#14519775—44no—
38#14519756—21no—
39#14519756—21no—
40#14519756—21no—
41#14519756—21no—
42#14519756—21no—
43#14519756—21no—
44#14519756—21no—
45#14519756—21no—
46C2×C4⋊Dic61D411yescharacteristic
47C2×C4⋊Dic61D411yescharacteristic
48C2×#11758196—24no—
49C2×#11758196—21no—
50C2×#11758196—21no—
51C2×C22⋊Dic61—21no—
52C2×C22⋊Dic61—21no—
53C2×C22⋊Dic61D412yes—
54C2×C4×Dic61—21no—
55#14519711—22no—
56#14519702—22no—
57#14519702—22no—
58#14519701—24no—
59C22⋊#92418—21no—
60C22⋊#92418D412yes—
61#14519684—24no—
62#14519684—21no—
63C22×Q8×C61—21no—
64C22×Q8×C61—21no—
65C22×Q8×C61—21no—
66C22×Q8×C61—21no—
67C22×Q8×C61—21no—
68C22×Q8×C61—44no—
69C22×Q8×C61D412yes—
70C23×C4×C61—21no—
71C23×C4×C61C2311yescharacteristic
72C22×C8×C61D412yes—
73C2×C4⋊C4×C61—21no—
74C2×C22⋊C4×C61—21no—
75#103×C61—44no—
76C22⋊C8×C61—21no—
77C22⋊C8×C61—21no—
78#95×C61—21no—
79C24×C61C2×D411yescharacteristic
80C2×Q8×C61—24no—
81C2×Q8×C61—24no—
82C2×Q8×C61—24no—
83C2×Q8×C61—22no—
84C2×Q8×C61—22no—
85C2×Q8×C61—44no—
86C2×Q8×C61—42no—
87C2×Q8×C61—42no—
88C2×Q8×C61—42no—
89C2×Q8×C61—44no—
90C2×Q8×C61—44no—
91C2×Q8×C61—41no—
92C2×Q8×C61—41no—
93C22×C4×C61—22no—
94C22×C4×C61—42no—
95C22×C4×C61—44no—
96C22×C4×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61#4111yescharacteristic
105C22×Dic61—22no—
106C22×Dic61—21no—
107C22×Dic61—21no—
108C22×Dic61—21no—
109C22×Dic61—21no—
110C2×Dic122—21no—
111C2×Dic122—21no—
112C2×Dic122—21no—
113C2×Dic122—21no—
114C2×Dic122—42no—
115C2×Dic122—42no—
116C2×C8×C61—22no—
117C2×C8×C61—42no—
118C4⋊C4×C61—42no—
119C22⋊C4×C61—42no—
120C22⋊Dic61—22no—
121C22⋊Dic61—41no—
122C22⋊Dic61—41no—
123C22⋊Dic61—41no—
124C22⋊Dic61—41no—
125C22⋊Dic61Q1614yescomplemented
126#11758202—48no—
127C4⋊Dic61—21no—
128C4⋊Dic61—21no—
129C4⋊Dic61—21no—
130C4⋊Dic61—21no—
131#11758196—48no—
132#11758196—41no—
133#11758196—41no—
134#11758196—41no—
135#11758196—41no—
136C4×Dic61—42no—
137C2×#92418—22no—
138C2×#92418—42no—
139Dic244—82no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61#12011yescharacteristic
143C23×C61#12111yescharacteristic
144C23×C61#12711yescharacteristic
145C23×C61C2×Q1612yes—
146C23×C61#13712yes—
147Q8×C61—24no—
148Q8×C61—24no—
149Q8×C61—42no—
150Q8×C61—44no—
151Q8×C61—42no—
152Q8×C61—44no—
153C2×C4×C61—22no—
154C2×C4×C61—22no—
155C2×C4×C61—22no—
156C2×C4×C61—24no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—42no—
161C2×C4×C61—44no—
162C2×C4×C61—44no—
163C2×C4×C61—42no—
164C2×C4×C61—81no—
165C2×C4×C61—81no—
166C2×C4×C61—81no—
167C2×C4×C61#12011yescharacteristic
168C2×C4×C61#12011yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×Dic61—24no—
172C2×Dic61—42no—
173C2×Dic61—42no—
174C2×Dic61—42no—
175C2×Dic61—42no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—81no—
181C2×Dic61—81no—
182Dic122—42no—
183Dic122—42no—
184Dic122—42no—
185Dic122—81no—
186C8×C61—42no—
187#92418—42no—
188#9957—611nomaximal, Sylow
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—42no—
196C22×C61—42no—
197C22×C61#39211yescharacteristic
198C22×C61#45112yes—
199C22×C61#46212yes—
200C22×C61#49411yescharacteristic
201C22×C61#49711yescharacteristic
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—44no—
206C4×C61—161no—
207C4×C61B(4,𝔽2)12yes—
208Dic61—44no—
209Dic61—81no—
210Dic61—81no—
211Dic61—81no—
212Dic61—81no—
213#3189—611no—
214C2×#451—611no—
215#2434—611no—
216#1996—611no—
217#1787D6111yescharacteristic
218#1520—611no—
219#1282—611no—
220C2×C61—22no—
221C2×C61—21no—
222C2×C61—22no—
223C2×C61—21no—
224C2×C61#200911yescharacteristic
225C2×C61#201611yescharacteristic
226C2×C61#202111yescharacteristic
227C23×Q8C2×D6111yescharacteristic
228C22×Q16—1221no—
229C2×C4⋊Q8—1221no—
230C2×C22⋊Q8—611no—
231C2×C22⋊Q8—611no—
232#484—1221no—
233#484—1221no—
234#467—1222no—
235#451—1221no—
236#451—1221no—
237#451—1222no—
238#451—1222no—
239C2×#103—22no—
240C2×#103—1221no—
241C2×C22⋊C8—611no—
242C2×C22⋊C8C2×D6111yescharacteristic
243#395—1222no—
244#391—1221no—
245#385—1222no—
246#380C2×D6111yescharacteristic
247C4×C22⋊C4—611no—
248#342—1222no—
249#328—1222no—
250C61#995711yescharacteristic, complemented, Sylow
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8—21no—
254C22×Q8—21no—
255C22×Q8—21no—
256C22×Q8—44no—
257C22×Q8—611no—
258C22×Q8—611no—
259C22×Q8D24412yes—
260C23×C4—21no—
261C23×C4C22×D6111yescharacteristic
262C2×Q16—1222no—
263C2×Q16—1224no—
264C2×Q16—2444no—
265C22×C8—1221no—
266C22×C8C22⋊D6112yes—
267C4⋊Q8—2444no—
268C22⋊Q8—1221no—
269C22⋊Q8—1221no—
270C22⋊Q8—1221no—
271C22⋊Q8—1221no—
272C22⋊Q8—1221no—
273C22⋊Q8—1221no—
274C22⋊Q8—1221no—
275C22⋊Q8—1221no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—611no—
278C2×C4⋊C4—611no—
279C2×C4⋊C4—1224no—
280C2×C4⋊C4—1221no—
281C2×C4⋊C4—1221no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—612no—
284C2×C22⋊C4—1221no—
285C2×C22⋊C4—1221no—
286C2×C42—1221no—
287#107—1222no—
288#103—44no—
289#103—1222no—
290#103—1222no—
291#103—1224no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—612no—
295C22⋊C8—1221no—
296#95—21no—
297#95—1224no—
298#95—1221no—
299C24D4×D6111yescharacteristic
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—44no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—1221no—
314C2×Q8—1221no—
315C2×Q8—1221no—
316C2×Q8—1221no—
317C2×Q8—2442no—
318C2×Q8—2442no—
319C22×C4—22no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—1222no—
326C22×C4—1221no—
327C22×C4—1221no—
328C22×C4—1221no—
329C22×C4—1221no—
330C22×C4C2×D24411yescharacteristic
331C22×C4#1175822211yescharacteristic
332C22×C4D4×D6111yescharacteristic
333C22×C4D4×D6111yescharacteristic
334C22×C4D4×D6111yescharacteristic
335C22×C4C2×C22⋊D6111yescharacteristic
336Q16—4882no—
337Q16—2448no—
338C2×C8—22no—
339C2×C8—42no—
340C2×C8—1222no—
341C2×C8—2442no—
342C4⋊C4—42no—
343C4⋊C4—1221no—
344C4⋊C4—1221no—
345C4⋊C4—1221no—
346C4⋊C4—1221no—
347C4⋊C4—2448no—
348C4⋊C4—2441no—
349C4⋊C4—2441no—
350C4⋊C4—2441no—
351C4⋊C4—2441no—
352C22⋊C4—42no—
353C22⋊C4—614no—
354C22⋊C4—1222no—
355C22⋊C4—2441no—
356C22⋊C4—2441no—
357C22⋊C4—2441no—
358C22⋊C4—2441no—
359C42—2442no—
360C23—41no—
361C23—41no—
362C23#1451974711yescharacteristic
363C23#1451975011yescharacteristic
364C23#1451977311yescharacteristic
365C23Q16×D6112yes—
366C23#1451980612yes—
367Q8—24no—
368Q8—24no—
369Q8—42no—
370Q8—44no—
371Q8—42no—
372Q8—44no—
373Q8—4881no—
374Q8—2442no—
375Q8—2442no—
376Q8—2442no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—1224no—
392C2×C4—4881no—
393C2×C4—4881no—
394C2×C4—2442no—
395C2×C4—2442no—
396C2×C4—2441no—
397C2×C4—2441no—
398C2×C4—2441no—
399C2×C4—2441no—
400C2×C4—2442no—
401C2×C4—2442no—
402C2×C4#1451974511yescharacteristic
403C2×C4#1451974711yescharacteristic
404C2×C4#1451975011yescharacteristic
405C2×C4#1451975111yescharacteristic
406C8—42no—
407C8—2442no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—42no—
415C22—42no—
416C22#1683967111yescharacteristic
417C22#1683991912yes—
418C22#1683997112yes—
419C22#1684014011yescharacteristic
420C22#1684015311yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—44no—
425C4—161no—
426C4—4881no—
427C4—4881no—
428C4—4881no—
429C4—4881no—
430C4—2444no—
431C4#1683994812yes—
432C2—21no—
433C2—22no—
434C2—22no—
435C2—21no—
436C2#2144688111yescharacteristic
437C2#2144692011yescharacteristic
438C2#2144694511yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1787C2
socleC22×C61#392
prime radicalC24×C61C2×D4
hypercenter#1787D61
hypocenterC61#9957
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
125. C22⋊Dic61Q16semidirect
250. C61#9957semidirect

Maximal subgroups

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

subgroupindex
#214565942
C2×#168399182
#214494592
#214468032
#214443902
#214436662
C61×#17872
#995761