Finite Group Databasebeta

← #29578074

Subgroups — #29578074

Order 15616.

538 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 99 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#21455755C211yesmaximal, characteristic
2#21445919C211yesmaximal, characteristic, complemented
3#21445894C211yesmaximal, characteristic, complemented
4#21444348C211yesmaximal, characteristic
5#21443727C211yesmaximal, characteristic, complemented
6#21443672C211yesmaximal, characteristic, complemented
7C61×#2333C211yesmaximal, characteristic
8C2×#14519736C2211yescharacteristic
9#16840083—22no—
10#16840078C412yescomplemented
11#16840073C412yescomplemented
12#16839802—21no—
13#16839767—21no—
14#16839726C2211yescharacteristic
15#16839726C2211yescharacteristic
16#16839625C2211yescharacteristic
17#16839528—21no—
18#16839528—21no—
19#16839522—21no—
20#16839522—21no—
21#16839496—22no—
22C2×#121×C61C2211yescharacteristic
23C61×#390—21no—
24C61×#388—21no—
25C61×#387—21no—
26C61×#386—21no—
27C61×#386—21no—
28C61×#381C2211yescharacteristic
29C4×C22⋊C4×C61—21no—
30C4×C22⋊C4×C61C2211yescharacteristic
31C2×#11758194—21no—
32C2×#11758194—21no—
33#14519736—21no—
34#14519736—21no—
35#14519736—21no—
36#14519736—21no—
37C2×C4⋊Dic61C2×C411yescharacteristic, complemented
38C2×C4⋊Dic61C2×C411yescharacteristic, complemented
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61—21no—
41C2×C4×Dic61—21no—
42#14519710—22no—
43#14519708—22no—
44#14519698—22no—
45#14519698—22no—
46C22⋊#92418—21no—
47C22⋊#92418—21no—
48C22⋊#92418—21no—
49C22⋊#92418C2×C412yescomplemented
50#14519684—21no—
51#14519684—21no—
52C22×D4×C61—21no—
53C22×D4×C61D411yescharacteristic
54C23×C4×C61—21no—
55C23×C4×C61C2311yescharacteristic
56#121×C61—21no—
57#121×C61—21no—
58#121×C61C2×C412yes—
59#121×C61D412yes—
60C2×C4⋊C4×C61—21no—
61C2×C4⋊C4×C61D411yescharacteristic
62C2×C22⋊C4×C61—21no—
63C2×C22⋊C4×C61—21no—
64C2×C22⋊C4×C61—21no—
65C2×C22⋊C4×C61—21no—
66C2×C22⋊C4×C61—21no—
67C2×C22⋊C4×C61D411yescharacteristic
68C2×C22⋊C4×C61D411yescharacteristic
69C2×C42×C61—21no—
70C2×C42×C61D411yescharacteristic
71C2×C42×C61D411yescharacteristic
72#95×C61—21no—
73#95×C61—21no—
74#95×C61—21no—
75#95×C61—21no—
76#95×C61—21no—
77C24×C61—21no—
78C24×C61C2×D411yescharacteristic
79C2×D4×C61—22no—
80C2×D4×C61—22no—
81C2×D4×C61—21no—
82C2×D4×C61—41no—
83C2×D4×C61—41no—
84C2×D4×C61—81no—
85C2×D4×C61—81no—
86C2×D4×C61C22⋊C412yes—
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—41no—
93C22×C4×C61—41no—
94C22×C4×C61—42no—
95C22×C4×C61—81no—
96C22×C4×C61—81no—
97C22×C4×C61C22×C411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61#4111yescharacteristic
102C22×C4×C61#4111yescharacteristic
103C22×Dic61—21no—
104C22×Dic61—21no—
105C22×Dic61—41no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—81no—
108C4⋊C4×C61C22⋊C412yes—
109C22⋊C4×C61—22no—
110C22⋊C4×C61—22no—
111C22⋊C4×C61—42no—
112C22⋊C4×C61—42no—
113C22⋊C4×C61—41no—
114C22⋊C4×C61—41no—
115C22⋊C4×C61—81no—
116C22⋊C4×C61—81no—
117C42×C61—41no—
118C42×C61—41no—
119C42×C61—81no—
120C22⋊Dic61—41no—
121C22⋊Dic61—41no—
122C22⋊Dic61—41no—
123C22⋊Dic61—41no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128C4×Dic61—41no—
129C4×Dic61—41no—
130#11758194—41no—
131#11758194—41no—
132#11758194—41no—
133#11758194—41no—
134C2×#92418—22no—
135C2×#92418—41no—
136C23×C61—21no—
137C23×C61—21no—
138C23×C61—22no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61—41no—
143C23×C61—41no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61—41no—
147C23×C61C4×D411yescharacteristic
148C23×C61#12011yescharacteristic
149C23×C61#12111yescharacteristic
150D4×C61—42no—
151D4×C61—81no—
152D4×C61—82no—
153D4×C61—82no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—22no—
159C2×C4×C61—22no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—84no—
178C2×C4×C61C2×C22⋊C411yescharacteristic
179C2×C4×C61C4×D411yescharacteristic
180C2×C4×C61#12111yescharacteristic
181C2×C4×C61#12311yescharacteristic
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61—81no—
185C2×Dic61—81no—
186C2×Dic61—81no—
187C2×Dic61—81no—
188C2×Dic61—81no—
189#92418—42no—
190#92418—81no—
191#9525—611nomaximal, Sylow
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—42no—
201C22×C61—42no—
202C22×C61—42no—
203C22×C61—81no—
204C22×C61—81no—
205C22×C61—81no—
206C22×C61—81no—
207C22×C61—81no—
208C22×C61—81no—
209C22×C61#38611yescharacteristic
210C22×C61#45811yescharacteristic
211C22×C61#45811yescharacteristic
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—42no—
217C4×C61—42no—
218C4×C61—81no—
219C4×C61—81no—
220C4×C61—81no—
221Dic61—81no—
222Dic61—81no—
223Dic61—161no—
224#3094—611no—
225#2333D6111yescharacteristic
226#1826—611no—
227#1820—611no—
228#1510—611no—
229#1306—611no—
230#1284—611no—
231C2×C61—21no—
232C2×C61—21no—
233C2×C61—41no—
234C2×C61—41no—
235C2×C61—82no—
236C2×C61#189011yescharacteristic
237C2×C61#190411yescharacteristic
238C2×C61#190611yescharacteristic
239C2×#121C2×D6111yescharacteristic
240C2×#117—611no—
241#482—1222no—
242#481—612no—
243#480—612no—
244#428—1221no—
245#418—1221no—
246#407—611no—
247#407—611no—
248#390—21no—
249#388—21no—
250#387—21no—
251#386—21no—
252#386—21no—
253#381—611no—
254#381C2×D6111yescharacteristic
255C4×C22⋊C4—21no—
256C4×C22⋊C4C2×D6111yescharacteristic
257#342—1221no—
258#342—1221no—
259#339—1221no—
260#339—1221no—
261#327—1222no—
262C61#952511yescharacteristic, complemented, Sylow
263C22×D4—21no—
264C22×D4C22⋊D6111yescharacteristic
265C23×C4—21no—
266C23×C4C22×D6111yescharacteristic
267C2×M16—1221no—
268C2×M16—1221no—
269#121—21no—
270#121—21no—
271#121C4×D6112yes—
272#121D24412yes—
273#117—1221no—
274#117—1221no—
275#117—1221no—
276#117—1221no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—611no—
279C2×C4⋊C4—611no—
280C2×C4⋊C4C22⋊D6111yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—1221no—
287C2×C22⋊C4—1221no—
288C2×C22⋊C4C22⋊D6111yescharacteristic
289C2×C22⋊C4C22⋊D6111yescharacteristic
290C2×C42—21no—
291C2×C42—1221no—
292C2×C42C22⋊D6111yescharacteristic
293C2×C42C22⋊D6111yescharacteristic
294#107—1222no—
295#106—1222no—
296#102—1222no—
297#102—1222no—
298C22⋊C8—612no—
299C22⋊C8—1221no—
300C22⋊C8—1221no—
301C22⋊C8—1221no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—1221no—
308#95—1221no—
309C24—21no—
310C24C2×C22⋊D6111yescharacteristic
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—21no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—81no—
317C2×D4—81no—
318C2×D4#1175819812yes—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—81no—
328C22×C4—81no—
329C22×C4—1221no—
330C22×C4—1221no—
331C22×C4—2441no—
332C22×C4C2×C4×D6111yescharacteristic
333C22×C4C2×D24411yescharacteristic
334C22×C4#1175822211yescharacteristic
335C22×C4#1175822211yescharacteristic
336C22×C4C2×C22⋊D6111yescharacteristic
337C22×C4C2×C22⋊D6111yescharacteristic
338M16—2441no—
339M16—2441no—
340M16—2441no—
341M16—2441no—
342C2×C8—1222no—
343C2×C8—2441no—
344C4⋊C4—21no—
345C4⋊C4—81no—
346C4⋊C4—1221no—
347C4⋊C4—1221no—
348C4⋊C4—1221no—
349C4⋊C4—1221no—
350C4⋊C4#1175819812yes—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—81no—
358C22⋊C4—81no—
359C22⋊C4—2441no—
360C22⋊C4—2441no—
361C22⋊C4—2441no—
362C22⋊C4—2441no—
363C42—41no—
364C42—41no—
365C42—81no—
366C42—2441no—
367C42—2441no—
368C23—21no—
369C23—21no—
370C23—22no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23C4×C22⋊D6111yescharacteristic
380C23#1451974611yescharacteristic
381C23#1451975111yescharacteristic
382D4—42no—
383D4—81no—
384D4—82no—
385D4—82no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—84no—
410C2×C4—4881no—
411C2×C4—4881no—
412C2×C4—4881no—
413C2×C4—4881no—
414C2×C4—4881no—
415C2×C4—2441no—
416C2×C4—2441no—
417C2×C4C2×#1175819811yescharacteristic
418C2×C4C4×C22⋊D6111yescharacteristic
419C2×C4#1451975111yescharacteristic
420C2×C4#1451976011yescharacteristic
421C8—4881no—
422C8—2442no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22#1683964111yescharacteristic
441C22#1683995211yescharacteristic
442C22#1683995211yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—42no—
448C4—42no—
449C4—81no—
450C4—81no—
451C4—81no—
452C4—9761no—
453C4—4881no—
454C4—4881no—
455C2—21no—
456C2—21no—
457C2—41no—
458C2—41no—
459C2—82no—
460C2#2144623211yescharacteristic
461C2#2144630611yescharacteristic
462C2#2144632011yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2333C2
socleC22×C61#386
prime radicalC2×#121×C61C22
cosocleC23×C4×C61C23
hypercenter#2333D61
hypocenterC61#9525
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2333

Complements

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

NH ≅ G/Ntype
2. #21445919C2semidirect
3. #21445894C2semidirect
5. #21443727C2semidirect
6. #21443672C2semidirect
10. #16840078C4semidirect
11. #16840073C4semidirect
37. C2×C4⋊Dic61C2×C4semidirect
38. C2×C4⋊Dic61C2×C4semidirect
49. C22⋊#92418C2×C4semidirect
262. C61#9525semidirect

Maximal subgroups

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

subgroupindex
#214557552
#214459192
#214458942
#214443482
#214437272
#214436722
C61×#23332
#952561