Finite Group Databasebeta

← #29577174

Subgroups — #29577174

Order 15616.

760 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 111 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#21455079C211yesmaximal, characteristic
2#21446803C211yesmaximal, characteristic, complemented
3#21445763C211yesmaximal, characteristic
4#21444338C211yesmaximal, characteristic, complemented
5#21443725C211yesmaximal, characteristic, complemented
6#21443705C211yesmaximal, characteristic, complemented
7C61×#2339C211yesmaximal, characteristic
8C2×#14519807C412yescomplemented
9C2×#14519756C2211yescharacteristic
10#16839924—22no—
11#16839918—21no—
12#16839913—21no—
13C2×#14519704—22no—
14#16839763C412yescomplemented
15#16839726C2211yescharacteristic
16#16839726C2211yescharacteristic
17C4×C22⋊Dic61C2211yescharacteristic
18#16839528—22no—
19#16839498—21no—
20#16839498—21no—
21C22×#41×C61C2211yescharacteristic
22C2×C4×D4×C61—21no—
23C61×#391—21no—
24C61×#386—22no—
25C61×#381—21no—
26C61×#380C2211yescharacteristic
27C4×C22⋊C4×C61C2211yescharacteristic
28C2×#95×C61—21no—
29C22×Dic122C2×C411yescharacteristic, complemented
30#14519807—44no—
31C2×#11758202—22no—
32C2×#11758200—22no—
33C2×#11758194—21no—
34C2×#11758194C2×C412yescomplemented
35#14519756—21no—
36#14519756—21no—
37#14519756—21no—
38#14519756—21no—
39C2×C4⋊Dic61C2×C411yescharacteristic, complemented
40C2×#11758196—21no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—21no—
43C2×C4×Dic61—21no—
44#14519704—44no—
45#14519701—22no—
46#14519698—22no—
47C22⋊#92418—21no—
48C22⋊#92418—21no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51#14519684—21no—
52C2×#41×C61—21no—
53C2×#41×C61—21no—
54C2×#41×C61—44no—
55C2×#41×C61C2×C412yes—
56C2×#41×C61D412yes—
57C22×Q8×C61D411yescharacteristic
58C22×D4×C61—21no—
59C22×D4×C61D411yescharacteristic
60C23×C4×C61—21no—
61C23×C4×C61—22no—
62C23×C4×C61C2311yescharacteristic
63C4×D4×C61—44no—
64C2×C4⋊C4×C61—21no—
65C2×C22⋊C4×C61—22no—
66C2×C22⋊C4×C61—21no—
67C2×C22⋊C4×C61D412yes—
68C2×C42×C61—21no—
69#95×C61—22no—
70#95×C61—22no—
71#95×C61—21no—
72#95×C61D411yescharacteristic
73#95×C61D411yescharacteristic
74C24×C61—21no—
75C24×C61C2×D411yescharacteristic
76#41×C61—24no—
77#41×C61—24no—
78#41×C61—42no—
79#41×C61—42no—
80#41×C61—44no—
81#41×C61—44no—
82C2×Q8×C61—21no—
83C2×Q8×C61—42no—
84C2×Q8×C61C22⋊C412yes—
85C2×D4×C61—24no—
86C2×D4×C61—22no—
87C2×D4×C61—22no—
88C2×D4×C61—21no—
89C2×D4×C61—44no—
90C2×D4×C61C22⋊C412yes—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61—22no—
96C22×C4×C61—21no—
97C22×C4×C61—42no—
98C22×C4×C61—44no—
99C22×C4×C61—44no—
100C22×C4×C61—44no—
101C22×C4×C61—82no—
102C22×C4×C61C22×C411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61#4111yescharacteristic
107C22×C4×C61#4111yescharacteristic
108C22×Dic61—21no—
109C22×Dic61—21no—
110C22×Dic61—41no—
111C2×Dic122—21no—
112C2×Dic122—21no—
113C2×Dic122—42no—
114C4⋊C4×C61—42no—
115C22⋊C4×C61—42no—
116C22⋊C4×C61—44no—
117C22⋊C4×C61—42no—
118C42×C61—42no—
119C22⋊Dic61—22no—
120C22⋊Dic61—41no—
121C22⋊Dic61—41no—
122C22⋊Dic61—41no—
123#11758202—44no—
124#11758200—44no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127#11758196—41no—
128#11758196—41no—
129C4×Dic61—42no—
130#11758194—42no—
131#11758194—42no—
132C2×#92418—22no—
133C2×#92418—41no—
134C23×C61—24no—
135C23×C61—22no—
136C23×C61—21no—
137C23×C61—21no—
138C23×C61—41no—
139C23×C61—41no—
140C23×C61—42no—
141C23×C61—41no—
142C23×C61—41no—
143C23×C61C4×D411yescharacteristic
144C23×C61#12011yescharacteristic
145C23×C61#12111yescharacteristic
146Q8×C61—22no—
147Q8×C61—22no—
148Q8×C61—42no—
149D4×C61—24no—
150D4×C61—22no—
151D4×C61—44no—
152D4×C61—44no—
153D4×C61#9914yes—
154C2×C4×C61—22no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—22no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—44no—
163C2×C4×C61—44no—
164C2×C4×C61—41no—
165C2×C4×C61—42no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—42no—
169C2×C4×C61—41no—
170C2×C4×C61—42no—
171C2×C4×C61—42no—
172C2×C4×C61—41no—
173C2×C4×C61—88no—
174C2×C4×C61C2×C22⋊C411yescharacteristic
175C2×C4×C61C4×D411yescharacteristic
176C2×C4×C61#12111yescharacteristic
177C2×C4×C61#12311yescharacteristic
178C2×Dic61—42no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—81no—
182C2×Dic61—81no—
183C2×Dic61—81no—
184C2×Dic61—81no—
185Dic122—42no—
186Dic122—81no—
187#92418—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—24no—
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—44no—
200C22×C61—44no—
201C22×C61—42no—
202C22×C61—42no—
203C22×C61—81no—
204C22×C61—81no—
205C22×C61#38611yescharacteristic
206C22×C61#45511yescharacteristic
207C22×C61#45611yescharacteristic
208C4×C61—21no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—42no—
212C4×C61—42no—
213C4×C61—81no—
214C4×C61—82no—
215C4×C61C2×#9912yes—
216Dic61—82no—
217Dic61—81no—
218Dic61—81no—
219#3007—611no—
220#2339D6111yescharacteristic
221#1996—611no—
222#1789—611no—
223#1508—611no—
224#1305—611no—
225#1297—611no—
226C2×C61—21no—
227C2×C61—21no—
228C2×C61—41no—
229C2×C61—41no—
230C2×C61—44no—
231C2×C61#188211yescharacteristic
232C2×C61#189811yescharacteristic
233C2×C61#190611yescharacteristic
234C22×#41C2×D6111yescharacteristic
235C2×#137—612no—
236C2×C22⋊Q8—611no—
237C2×C4×D4—21no—
238#452—1222no—
239#451—1221no—
240#450—1221no—
241C2×#104—1222no—
242#417—612no—
243#407—611no—
244#407—611no—
245#391—21no—
246#386—22no—
247#381—21no—
248#380C2×D6111yescharacteristic
249C4×C22⋊C4—611no—
250C4×C22⋊C4C2×D6111yescharacteristic
251C2×#95—21no—
252#342—1222no—
253#328—1221no—
254#328—1221no—
255C61#936111yescharacteristic, complemented, Sylow
256C2×#41—21no—
257C2×#41—21no—
258C2×#41—44no—
259C2×#41C4×D6112yes—
260C2×#41D24412yes—
261C22×Q8—611no—
262C22×Q8C22⋊D6111yescharacteristic
263C22×D4—21no—
264C22×D4C22⋊D6111yescharacteristic
265C23×C4—21no—
266C23×C4—22no—
267C23×C4C22×D6111yescharacteristic
268#137—2444no—
269C2×Q16—1222no—
270C2×SD16—1222no—
271C2×M16—612no—
272C2×M16—1221no—
273C22⋊Q8—1221no—
274C22⋊Q8—1221no—
275C22⋊Q8—1221no—
276C22⋊Q8—1221no—
277C4×D4—44no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—611no—
280C2×C4⋊C4—1221no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—1221no—
284C2×C22⋊C4—1221no—
285C2×C22⋊C4C22⋊D6112yes—
286C2×C42—21no—
287C2×C42—1221no—
288#104—2444no—
289#103—1222no—
290#102—1222no—
291C22⋊C8—1221no—
292C22⋊C8—1221no—
293C22⋊C8—1221no—
294C22⋊C8—1221no—
295#95—22no—
296#95—22no—
297#95—21no—
298#95—1221no—
299#95C22⋊D6111yescharacteristic
300#95C22⋊D6111yescharacteristic
301C24—21no—
302C24C2×C22⋊D6111yescharacteristic
303#41—24no—
304#41—24no—
305#41—42no—
306#41—42no—
307#41—44no—
308#41—44no—
309C2×Q8—21no—
310C2×Q8—42no—
311C2×Q8—1221no—
312C2×Q8—1221no—
313C2×Q8—2442no—
314C2×Q8#1175819812yes—
315C2×D4—24no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—21no—
319C2×D4—44no—
320C2×D4#1175819812yes—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—82no—
332C22×C4—1221no—
333C22×C4—1221no—
334C22×C4—2441no—
335C22×C4C2×C4×D6111yescharacteristic
336C22×C4C2×D24411yescharacteristic
337C22×C4#1175822211yescharacteristic
338C22×C4#1175822211yescharacteristic
339C22×C4C2×C22⋊D6111yescharacteristic
340C22×C4C2×C22⋊D6111yescharacteristic
341Q16—2444no—
342SD16—2444no—
343M16—2442no—
344M16—2442no—
345C2×C8—1222no—
346C2×C8—2441no—
347C4⋊C4—42no—
348C4⋊C4—1221no—
349C4⋊C4—1221no—
350C4⋊C4—2441no—
351C4⋊C4—2441no—
352C22⋊C4—42no—
353C22⋊C4—44no—
354C22⋊C4—42no—
355C22⋊C4—1222no—
356C22⋊C4—2441no—
357C22⋊C4—2441no—
358C22⋊C4—2441no—
359C42—42no—
360C42—2442no—
361C23—24no—
362C23—22no—
363C23—21no—
364C23—21no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—41no—
369C23—41no—
370C23C4×C22⋊D6111yescharacteristic
371C23#1451974611yescharacteristic
372C23#1451975111yescharacteristic
373Q8—22no—
374Q8—22no—
375Q8—42no—
376Q8—4881no—
377Q8—2442no—
378D4—24no—
379D4—22no—
380D4—44no—
381D4—44no—
382D4#1451969114yes—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—88no—
403C2×C4—4881no—
404C2×C4—4881no—
405C2×C4—4881no—
406C2×C4—4881no—
407C2×C4—2442no—
408C2×C4—2441no—
409C2×C4—2441no—
410C2×C4C2×#1175819811yescharacteristic
411C2×C4C4×C22⋊D6111yescharacteristic
412C2×C4#1451975111yescharacteristic
413C2×C4#1451976011yescharacteristic
414C8—4881no—
415C8—2442no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—24no—
423C22—21no—
424C22—21no—
425C22—44no—
426C22—44no—
427C22—42no—
428C22—42no—
429C22—81no—
430C22—81no—
431C22#1683964111yescharacteristic
432C22#1683994011yescharacteristic
433C22#1683994511yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—42no—
438C4—42no—
439C4—81no—
440C4—82no—
441C4—4882no—
442C4—4881no—
443C4—4881no—
444C4C2×#1451969112yes—
445C2—21no—
446C2—21no—
447C2—41no—
448C2—41no—
449C2—44no—
450C2#2144618811yescharacteristic
451C2#2144627211yescharacteristic
452C2#2144632011yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2339C2
socleC22×C61#386
prime radicalC22×#41×C61C22
cosocleC23×C4×C61C23
hypercenter#2339D61
hypocenterC61#9361
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2339

Complements

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

NH ≅ G/Ntype
2. #21446803C2semidirect
4. #21444338C2semidirect
5. #21443725C2semidirect
6. #21443705C2semidirect
8. C2×#14519807C4semidirect
14. #16839763C4semidirect
29. C22×Dic122C2×C4semidirect
34. C2×#11758194C2×C4semidirect
39. C2×C4⋊Dic61C2×C4semidirect
255. C61#9361semidirect

Maximal subgroups

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

subgroupindex
#214550792
#214468032
#214457632
#214443382
#214437252
#214437052
C61×#23392
#936161