Finite Group Databasebeta

← #29578076

Subgroups — #29578076

Order 15616.

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

normal subgroups only

This group has 538 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#21448586C211yesmaximal, characteristic
2#21445918C211yesmaximal, characteristic, complemented
3#21445896C211yesmaximal, characteristic, complemented
4#21444348C211yesmaximal, characteristic
5#21443728C211yesmaximal, characteristic, complemented
6#21443673C211yesmaximal, characteristic, complemented
7C61×#3094C211yesmaximal, characteristic
8#16839726C2211yescharacteristic
9#16839665—21no—
10#16839655—21no—
11#16839648—21no—
12#16839642—21no—
13#16839642—21no—
14#16839625C2211yescharacteristic
15#16839625C2211yescharacteristic
16C22⋊C4×Dic61—21no—
17C4×C22⋊Dic61C2211yescharacteristic
18#16839529—21no—
19#16839529—21no—
20#16839521—21no—
21#16839521—21no—
22#16839496—22no—
23C2×#121×C61C2211yescharacteristic
24C2×#117×C61C2211yescharacteristic
25C61×#482—22no—
26C61×#481C412yescomplemented
27C61×#480C412yescomplemented
28C61×#428—21no—
29C61×#418—21no—
30C61×#407C2211yescharacteristic
31C23×Dic61—21no—
32C2×#11758194—21no—
33C2×C4⋊Dic61—21no—
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61—21no—
39C2×C22⋊Dic61D411yescharacteristic
40C2×C22⋊Dic61D411yescharacteristic
41C2×C4×Dic61—21no—
42C2×C4×Dic61D411yescharacteristic
43C2×C4×Dic61D411yescharacteristic
44C22⋊#92418—21no—
45C22⋊#92418—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53C22×D4×C61—21no—
54C22×D4×C61D411yescharacteristic
55C23×C4×C61C2311yescharacteristic
56C2×M16×C61—21no—
57#121×C61—21no—
58#121×C61—21no—
59#121×C61C2×C412yes—
60#121×C61D412yes—
61#117×C61—21no—
62#117×C61—21no—
63#117×C61—21no—
64#117×C61—21no—
65C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
66C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
67C2×C4⋊C4×C61D411yescharacteristic
68C2×C22⋊C4×C61—21no—
69C2×C22⋊C4×C61—21no—
70C2×C42×C61—21no—
71#107×C61—22no—
72#106×C61—22no—
73#102×C61—22no—
74#102×C61—22no—
75C22⋊C8×C61—21no—
76C22⋊C8×C61C2×C412yescomplemented
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—41no—
91C22×C4×C61—41no—
92C22×C4×C61C22×C411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61#4111yescharacteristic
97C22×C4×C61#4111yescharacteristic
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61—42no—
103C22×Dic61—41no—
104C22×Dic61—81no—
105C22×Dic61—81no—
106M16×C61—41no—
107M16×C61—41no—
108C2×C8×C61—22no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61C22⋊C412yes—
115C22⋊C4×C61—22no—
116C22⋊C4×C61—22no—
117C22⋊C4×C61—41no—
118C22⋊C4×C61—41no—
119C42×C61—41no—
120C42×C61—41no—
121C22⋊Dic61—42no—
122C22⋊Dic61—42no—
123C22⋊Dic61—41no—
124C22⋊Dic61—41no—
125C22⋊Dic61—41no—
126C22⋊Dic61—41no—
127C22⋊Dic61—81no—
128C22⋊Dic61—81no—
129C4⋊Dic61—81no—
130C4×Dic61—41no—
131C4×Dic61—41no—
132C4×Dic61—81no—
133#11758194—41no—
134#11758194—41no—
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—41no—
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—81no—
168C2×C4×C61—81no—
169C2×C4×C61C2×C22⋊C411yescharacteristic
170C2×C4×C61C4×D411yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61#12311yescharacteristic
173C2×Dic61—22no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61—41no—
185C2×Dic61—84no—
186C2×Dic61—81no—
187C2×Dic61—81no—
188C2×Dic61—81no—
189C8×C61—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—81no—
218C4×C61—81no—
219Dic61—42no—
220Dic61—81no—
221Dic61—81no—
222Dic61—81no—
223Dic61—161no—
224#3094D6111yescharacteristic
225#2333—611no—
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×#117C2×D6111yescharacteristic
241#482—22no—
242#481Dic6112yescomplemented
243#480Dic6112yescomplemented
244#428—21no—
245#418—21no—
246#407—611no—
247#407C2×D6111yescharacteristic
248#390—1221no—
249#388—1221no—
250#387—1221no—
251#386—1221no—
252#386—1221no—
253#381—611no—
254#381—611no—
255C4×C22⋊C4—611no—
256C4×C22⋊C4—1221no—
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—1221no—
266C23×C4C22×D6111yescharacteristic
267C2×M16—21no—
268C2×M16—1221no—
269#121—21no—
270#121—21no—
271#121C2×Dic6112yes—
272#121C22⋊D6112yes—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277C2×C4⋊C4—1221no—
278C2×C4⋊C4C2×Dic6111yescharacteristic, complemented
279C2×C4⋊C4C2×Dic6111yescharacteristic, complemented
280C2×C4⋊C4C22⋊D6111yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—611no—
284C2×C22⋊C4—611no—
285C2×C22⋊C4—1221no—
286C2×C22⋊C4—1221no—
287C2×C22⋊C4—1221no—
288C2×C22⋊C4—1221no—
289C2×C22⋊C4—1221no—
290C2×C42—21no—
291C2×C42—611no—
292C2×C42—611no—
293C2×C42—1221no—
294#107—22no—
295#106—22no—
296#102—22no—
297#102—22no—
298C22⋊C8—21no—
299C22⋊C8—1221no—
300C22⋊C8—1221no—
301C22⋊C8C2×Dic6112yescomplemented
302#95—1221no—
303#95—1221no—
304#95—1221no—
305#95—1221no—
306#95—1221no—
307#95—1221no—
308#95—1221no—
309C24—21no—
310C24D4×D6111yescharacteristic
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—21no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—81no—
317C2×D4—81no—
318C2×D4C22⋊Dic6112yes—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—1221no—
325C22×C4—1221no—
326C22×C4—1221no—
327C22×C4—1221no—
328C22×C4—4881no—
329C22×C4—4881no—
330C22×C4—2442no—
331C22×C4—2441no—
332C22×C4D4×D6111yescharacteristic
333C22×C4#1175822411yescharacteristic
334C22×C4#1175822411yescharacteristic
335C22×C4C22×Dic6111yescharacteristic
336C22×C4C2×C22⋊D6111yescharacteristic
337C22×C4C2×C22⋊D6111yescharacteristic
338M16—41no—
339M16—41no—
340M16—2441no—
341M16—2441no—
342C2×C8—22no—
343C2×C8—2441no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—4881no—
350C4⋊C4C22⋊Dic6112yes—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—4881no—
356C22⋊C4—4881no—
357C22⋊C4—2442no—
358C22⋊C4—2442no—
359C22⋊C4—2441no—
360C22⋊C4—2441no—
361C22⋊C4—2441no—
362C22⋊C4—2441no—
363C42—41no—
364C42—41no—
365C42—4881no—
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—
379C23D4×Dic6111yescharacteristic
380C23#1451974511yescharacteristic
381C23#1451974911yescharacteristic
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—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—1222no—
402C2×C4—4881no—
403C2×C4—4881no—
404C2×C4—4881no—
405C2×C4—4884no—
406C2×C4—2441no—
407C2×C4—2441no—
408C2×C4—2441no—
409C2×C4—2441no—
410C2×C4—2441no—
411C2×C4—2441no—
412C2×C4—2441no—
413C2×C4—2441no—
414C2×C4—2441no—
415C2×C4—2441no—
416C2×C4—2441no—
417C2×C4C2×C22⋊Dic6111yescharacteristic
418C2×C4D4×Dic6111yescharacteristic
419C2×C4#1451974911yescharacteristic
420C2×C4#1451975811yescharacteristic
421C8—42no—
422C8—4881no—
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#1683964211yescharacteristic
441C22#1683995111yescharacteristic
442C22#1683995111yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—42no—
448C4—81no—
449C4—81no—
450C4—9761no—
451C4—4881no—
452C4—4881no—
453C4—4881no—
454C4—2442no—
455C2—21no—
456C2—21no—
457C2—41no—
458C2—41no—
459C2—82no—
460C2#2144623311yescharacteristic
461C2#2144630811yescharacteristic
462C2#2144632211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#3094C2
socleC22×C61#386
prime radicalC2×#121×C61C22
cosocleC23×C4×C61C23
hypercenter#3094D61
hypocenterC61#9525
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3094

Complements

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

NH ≅ G/Ntype
2. #21445918C2semidirect
3. #21445896C2semidirect
5. #21443728C2semidirect
6. #21443673C2semidirect
26. C61×#481C4semidirect
27. C61×#480C4semidirect
65. C2×C4⋊C4×C61C2×C4semidirect
66. C2×C4⋊C4×C61C2×C4semidirect
76. C22⋊C8×C61C2×C4semidirect
242. #481Dic61semidirect
243. #480Dic61semidirect
262. C61#9525semidirect
278. C2×C4⋊C4C2×Dic61semidirect
279. C2×C4⋊C4C2×Dic61semidirect
301. C22⋊C8C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214485862
#214459182
#214458962
#214443482
#214437282
#214436732
C61×#30942
#952561