Finite Group Databasebeta

← #29577036

Subgroups — #29577036

Order 15616.

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

normal subgroups only

This group has 534 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#21448976C211yesmaximal, characteristic
2#21446805C211yesmaximal, characteristic, complemented
3#21444352C211yesmaximal, characteristic, complemented
4#21444335C211yesmaximal, characteristic
5#21443726C211yesmaximal, characteristic, complemented
6#21443673C211yesmaximal, characteristic, complemented
7C61×#3193C211yesmaximal, characteristic
8#16839726C2211yescharacteristic
9#16839665—21no—
10#16839655—21no—
11#16839648—21no—
12#16839642—21no—
13#16839642—21no—
14#16839637—21no—
15#16839625C2211yescharacteristic
16#16839622C2211yescharacteristic
17C4×C22⋊Dic61C2211yescharacteristic
18#16839529—21no—
19#16839529—21no—
20#16839499—21no—
21#16839499—21no—
22#16839496—22no—
23C2×C22⋊Q8×C61C2211yescharacteristic
24C2×#121×C61C2211yescharacteristic
25C61×#484—21no—
26C61×#480C412yescomplemented
27C61×#468—22no—
28C61×#450—21no—
29C61×#448C412yescomplemented
30C61×#407C2211yescharacteristic
31C23×Dic61—21no—
32C2×#11758194—21no—
33C2×C4⋊Dic61—21no—
34C2×C4⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61—21no—
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61D411yescharacteristic
41C2×C22⋊Dic61D411yescharacteristic
42C2×C4×Dic61—21no—
43C2×C4×Dic61—21no—
44C22⋊#92418—21no—
45C22⋊#92418—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684D411yescharacteristic
52#14519684D411yescharacteristic
53C22×Q8×C61C2×C411yescharacteristic, complemented
54C22×D4×C61—21no—
55C22×D4×C61D411yescharacteristic
56C23×C4×C61C2311yescharacteristic
57C2×SD16×C61—22no—
58C2×M16×C61—21no—
59C22⋊Q8×C61—21no—
60C22⋊Q8×C61—21no—
61C22⋊Q8×C61—21no—
62C22⋊Q8×C61—21no—
63#121×C61—21no—
64#121×C61—21no—
65#121×C61C2×C412yes—
66#121×C61D412yes—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
69C2×C4⋊C4×C61D411yescharacteristic
70C2×C22⋊C4×C61—21no—
71C2×C22⋊C4×C61—21no—
72#107×C61—22no—
73#103×C61—22no—
74#102×C61—22no—
75C22⋊C8×C61—21no—
76C22⋊C8×C61C2×C412yescomplemented
77C24×C61—21no—
78C24×C61C2×D411yescharacteristic
79C2×Q8×C61—21no—
80C2×Q8×C61—21no—
81C2×Q8×C61—81no—
82C2×D4×C61—22no—
83C2×D4×C61—22no—
84C2×D4×C61—21no—
85C2×D4×C61—81no—
86C2×D4×C61—81no—
87C2×D4×C61—81no—
88C2×D4×C61C22⋊C412yes—
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—81no—
93C22×C4×C61C22×C411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61#4111yescharacteristic
98C22×C4×C61#4111yescharacteristic
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61—21no—
103C22×Dic61—41no—
104C22×Dic61—81no—
105C22×Dic61—81no—
106C22×Dic61—81no—
107SD16×C61—82no—
108M16×C61—81no—
109C2×C8×C61—22no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C4⋊C4×C61C22⋊C412yes—
116C22⋊C4×C61—22no—
117C22⋊C4×C61—22no—
118C22⋊C4×C61—41no—
119C22⋊C4×C61—41no—
120C22⋊Dic61—22no—
121C22⋊Dic61—41no—
122C22⋊Dic61—41no—
123C22⋊Dic61—41no—
124C22⋊Dic61—81no—
125C22⋊Dic61—81no—
126C22⋊Dic61—81no—
127C22⋊Dic61—81no—
128C4⋊Dic61—81no—
129C4⋊Dic61—81no—
130C4×Dic61—81no—
131C4×Dic61—81no—
132#11758194—81no—
133C2×#92418—41no—
134C23×C61—21no—
135C23×C61—21no—
136C23×C61—22no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61—41no—
143C23×C61—41no—
144C23×C61—41no—
145C23×C61C4×D411yescharacteristic
146C23×C61#12011yescharacteristic
147C23×C61#12111yescharacteristic
148Q8×C61—42no—
149Q8×C61—81no—
150D4×C61—42no—
151D4×C61—81no—
152D4×C61—82no—
153D4×C61—82no—
154C2×C4×C61—22no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
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—81no—
166C2×C4×C61—81no—
167C2×C4×C61C2×C22⋊C411yescharacteristic
168C2×C4×C61C4×D411yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12311yescharacteristic
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—42no—
177C2×Dic61—41no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61—81no—
185C2×Dic61—81no—
186C2×Dic61—84no—
187C8×C61—42no—
188#92418—81no—
189#9341—611nomaximal, Sylow
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—21no—
198C22×C61—42no—
199C22×C61—42no—
200C22×C61—42no—
201C22×C61—81no—
202C22×C61—81no—
203C22×C61—81no—
204C22×C61—81no—
205C22×C61—81no—
206C22×C61—81no—
207C22×C61#38611yescharacteristic
208C22×C61#45511yescharacteristic
209C22×C61#45611yescharacteristic
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—42no—
215C4×C61—81no—
216C4×C61—81no—
217Dic61—81no—
218Dic61—81no—
219Dic61—81no—
220Dic61—81no—
221Dic61—82no—
222#3193D6111yescharacteristic
223#2377—611no—
224#1996—611no—
225#1511—611no—
226#1507—611no—
227#1305—611no—
228#1284—611no—
229C2×C61—21no—
230C2×C61—21no—
231C2×C61—41no—
232C2×C61—41no—
233C2×C61—82no—
234C2×C61#188211yescharacteristic
235C2×C61#190411yescharacteristic
236C2×C61#191311yescharacteristic
237C2×C22⋊Q8C2×D6111yescharacteristic
238C2×#121C2×D6111yescharacteristic
239#484—21no—
240#480Dic6112yescomplemented
241#468—22no—
242#450—21no—
243#448Dic6112yescomplemented
244#407—611no—
245#407C2×D6111yescharacteristic
246#390—1221no—
247#388—1221no—
248#387—1221no—
249#386—1221no—
250#386—1221no—
251#385—1221no—
252#381—611no—
253#380—611no—
254C4×C22⋊C4—611no—
255#342—1221no—
256#342—1221no—
257#328—1221no—
258#328—1221no—
259#327—1222no—
260C61#934111yescharacteristic, complemented, Sylow
261C22×Q8C2×Dic6111yescharacteristic, complemented
262C22×D4—21no—
263C22×D4C22⋊D6111yescharacteristic
264C23×C4—1221no—
265C23×C4C22×D6111yescharacteristic
266C2×SD16—22no—
267C2×M16—21no—
268C2×M16—1221no—
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271C22⋊Q8—21no—
272C22⋊Q8—21no—
273#121—21no—
274#121—21no—
275#121C2×Dic6112yes—
276#121C22⋊D6112yes—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—1221no—
279C2×C4⋊C4—1221no—
280C2×C4⋊C4C2×Dic6111yescharacteristic, complemented
281C2×C4⋊C4C22⋊D6111yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—611no—
285C2×C22⋊C4—611no—
286C2×C22⋊C4—1221no—
287C2×C22⋊C4—1221no—
288C2×C22⋊C4—1221no—
289C2×C22⋊C4—1221no—
290C2×C22⋊C4—1221no—
291C2×C42—1221no—
292C2×C42—1221no—
293#107—22no—
294#103—22no—
295#102—22no—
296C22⋊C8—21no—
297C22⋊C8—1221no—
298C22⋊C8—1221no—
299C22⋊C8C2×Dic6112yescomplemented
300#95—611no—
301#95—611no—
302#95—1221no—
303#95—1221no—
304#95—1221no—
305#95—1221no—
306#95—1221no—
307C24—21no—
308C24D4×D6111yescharacteristic
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—81no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—21no—
315C2×D4—81no—
316C2×D4—81no—
317C2×D4—81no—
318C2×D4C22⋊Dic6112yes—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—81no—
323C22×C4—1221no—
324C22×C4—1221no—
325C22×C4—1221no—
326C22×C4—1221no—
327C22×C4—4881no—
328C22×C4—4881no—
329C22×C4—4881no—
330C22×C4—2441no—
331C22×C4D4×D6111yescharacteristic
332C22×C4#1175822411yescharacteristic
333C22×C4#1175822411yescharacteristic
334C22×C4C22×Dic6111yescharacteristic
335C22×C4C2×C22⋊D6111yescharacteristic
336C22×C4C2×C22⋊D6111yescharacteristic
337SD16—82no—
338M16—81no—
339M16—4881no—
340C2×C8—22no—
341C2×C8—2441no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—4881no—
348C4⋊C4—4881no—
349C4⋊C4C22⋊Dic6112yes—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—1222no—
355C22⋊C4—4881no—
356C22⋊C4—4881no—
357C22⋊C4—4881no—
358C22⋊C4—4881no—
359C22⋊C4—2441no—
360C22⋊C4—2441no—
361C22⋊C4—2441no—
362C42—4881no—
363C42—4881no—
364C23—21no—
365C23—21no—
366C23—22no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23D4×Dic6111yescharacteristic
376C23#1451974511yescharacteristic
377C23#1451974911yescharacteristic
378Q8—42no—
379Q8—81no—
380D4—42no—
381D4—81no—
382D4—82no—
383D4—82no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—4881no—
398C2×C4—4881no—
399C2×C4—4884no—
400C2×C4—2441no—
401C2×C4—2441no—
402C2×C4—2442no—
403C2×C4—2441no—
404C2×C4—2441no—
405C2×C4—2441no—
406C2×C4—2441no—
407C2×C4—2441no—
408C2×C4—2441no—
409C2×C4—2441no—
410C2×C4—2441no—
411C2×C4—2441no—
412C2×C4—2441no—
413C2×C4C2×C22⋊Dic6111yescharacteristic
414C2×C4D4×Dic6111yescharacteristic
415C2×C4#1451974911yescharacteristic
416C2×C4#1451975811yescharacteristic
417C8—42no—
418C8—4881no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22#1683964211yescharacteristic
437C22#1683993811yescharacteristic
438C22#1683994311yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—42no—
444C4—81no—
445C4—81no—
446C4—4881no—
447C4—4881no—
448C4—4881no—
449C4—4881no—
450C4—4882no—
451C2—21no—
452C2—21no—
453C2—41no—
454C2—41no—
455C2—82no—
456C2#2144619011yescharacteristic
457C2#2144630811yescharacteristic
458C2#2144636211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#3193C2
socleC22×C61#386
prime radicalC2×#121×C61C22
cosocleC23×C4×C61C23
hypercenter#3193D61
hypocenterC61#9341
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3193

Complements

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

NH ≅ G/Ntype
2. #21446805C2semidirect
3. #21444352C2semidirect
5. #21443726C2semidirect
6. #21443673C2semidirect
26. C61×#480C4semidirect
29. C61×#448C4semidirect
53. C22×Q8×C61C2×C4semidirect
68. C2×C4⋊C4×C61C2×C4semidirect
76. C22⋊C8×C61C2×C4semidirect
240. #480Dic61semidirect
243. #448Dic61semidirect
260. C61#9341semidirect
261. C22×Q8C2×Dic61semidirect
280. C2×C4⋊C4C2×Dic61semidirect
299. C22⋊C8C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214489762
#214468052
#214443522
#214443352
#214437262
#214436732
C61×#31932
#934161