Finite Group Databasebeta

← #29575560

Subgroups — #29575560

Order 15616.

560 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454845C211yesmaximal, characteristic
2#21448488C211yesmaximal, characteristic
3#21444135C211yesmaximal, characteristic
4C2×#16839488C211yesmaximal, characteristic, complemented
5#21443666C211yesmaximal, characteristic, complemented
6#21443665C211yesmaximal, characteristic, complemented
7C2×C61×#324C211yesmaximal, characteristic, complemented
8C2×#14519760C411yescharacteristic
9C2×C4×C22⋊D61C2211yescharacteristic
10#16839833—22no—
11#16839830—22no—
12#16839816—21no—
13#16839724—21no—
14C2×C22⋊#92418C2211yescharacteristic
15C2×C22⋊#92418C2211yescharacteristic
16#16839657—21no—
17#16839652—21no—
18#16839643—21no—
19#16839641C411yescharacteristic
20#16839635C411yescharacteristic
21#16839633—21no—
22#16839625C411yescharacteristic
23C4×C22⋊Dic61C2211yescharacteristic
24#16839488—22no—
25#16839487—21no—
26#16839485—21no—
27C22×C4⋊C4×C61C2211yescharacteristic
28C2×C22⋊C8×C61C2211yescharacteristic
29C2×C22⋊C8×C61C2211yescharacteristic
30C61×#324—22no—
31C22×C22⋊D61C2×C411yescharacteristic
32C22×C4×D61D411yescharacteristic
33C2×#11758189—22no—
34C22×#92418—21no—
35C22×#92418—21no—
36#14519760—22no—
37#14519760—22no—
38C4×C22⋊D61—21no—
39C4×C22⋊D61—21no—
40C4×C22⋊D61—21no—
41C4×C22⋊D61—21no—
42C2×#11758196—21no—
43C2×#11758196—21no—
44C2×#11758196D411yescharacteristic
45C2×C22⋊Dic61C2×C411yescharacteristic
46C2×C22⋊Dic61C2×C411yescharacteristic
47C2×C22⋊Dic61C2×C411yescharacteristic
48C2×#11758198—21no—
49C2×#11758198—21no—
50C2×#11758198D411yescharacteristic
51C2×C4×Dic61—21no—
52C2×C4×Dic61D411yescharacteristic
53#14519706—22no—
54C22⋊#92418—21no—
55C22⋊#92418—21no—
56C22⋊#92418—21no—
57C22⋊#92418D412yes—
58#14519689—22no—
59#14519687—22no—
60#14519684—21no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684—21no—
65C23×C4×C61C2×C411yescharacteristic
66C23×C4×C61C2×C411yescharacteristic
67C23×C4×C61C2311yescharacteristic
68C22×C8×C61—21no—
69C22×C8×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61—21no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61—21no—
76C22⋊C8×C61—21no—
77C22⋊C8×C61—21no—
78C22⋊C8×C61—21no—
79C22⋊C8×C61D412yes—
80C24×C61C22×C411yescharacteristic
81C23×D61C22⋊C411yescharacteristic
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—21no—
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61—81no—
92C22×C4×C61—81no—
93C22×C4×C61M1612yescomplemented
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61#4111yescharacteristic
99C22×C4×C61#4111yescharacteristic
100C2×C22⋊D61—21no—
101C2×C22⋊D61—21no—
102C2×C22⋊D61—21no—
103C2×C22⋊D61—21no—
104C2×C22⋊D61—21no—
105C2×C22⋊D61—21no—
106C22×Dic61—21no—
107C22×Dic61—21no—
108C22×Dic61C22⋊C411yescharacteristic
109C22×Dic61C22⋊C411yescharacteristic
110C22×Dic61C22⋊C411yescharacteristic
111C2×C4×D61—21no—
112C2×C4×D61—21no—
113C2×C4×D61—81no—
114C2×C8×C61—22no—
115C2×C8×C61—41no—
116C2×C8×C61—81no—
117C2×C8×C61—81no—
118C4⋊C4×C61—41no—
119C4⋊C4×C61—41no—
120C4⋊C4×C61—41no—
121C4⋊C4×C61—41no—
122C22⋊Dic61—21no—
123C22⋊Dic61—21no—
124C22⋊Dic61—21no—
125C22⋊Dic61—21no—
126C22⋊Dic61—21no—
127C22⋊Dic61M1612yescomplemented
128#11758198—24no—
129#11758198—21no—
130#11758198—21no—
131#11758198—81no—
132#11758196—24no—
133#11758196—21no—
134#11758196—21no—
135#11758196—81no—
136C4×Dic61—21no—
137C4×Dic61—21no—
138C4×Dic61—81no—
139C2×#92418—22no—
140C2×#92418—41no—
141C2×#92418—81no—
142C2×#92418—81no—
143#11758189—82no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61C2×C22⋊C411yescharacteristic
147C23×C61C4×D411yescharacteristic
148C23×C61C4×D411yescharacteristic
149C23×C61C2×M1611yescharacteristic
150C23×C61C2×M1611yescharacteristic
151C23×C61#13111yescharacteristic
152C23×C61#13111yescharacteristic
153C22×D61—21no—
154C22×D61—21no—
155C22×D61—21no—
156C22×D61—41no—
157C22×D61—41no—
158C2×C4×C61—22no—
159C2×C4×C61—22no—
160C2×C4×C61—22no—
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—81no—
175C2×C4×C61—81no—
176C2×C4×C61#12011yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61#12111yescharacteristic
179C2×C4×C61#12311yescharacteristic
180C22⋊D61—41no—
181C22⋊D61—41no—
182C22⋊D61—41no—
183C22⋊D61—41no—
184C2×Dic61—21no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61—22no—
189C2×Dic61—21no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C4×D61—42no—
198C4×D61—81no—
199C8×C61—42no—
200C8×C61—81no—
201#92418—42no—
202#92418—81no—
203#9110—611nomaximal, Sylow
204C22×C61—21no—
205C22×C61—21no—
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—41no—
211C22×C61—41no—
212C22×C61—41no—
213C22×C61—41no—
214C22×C61#38611yescharacteristic
215C22×C61#40711yescharacteristic
216C22×C61#40811yescharacteristic
217C22×C61#43511yescharacteristic
218C22×C61#43511yescharacteristic
219C22×C61#43611yescharacteristic
220C22×C61#43611yescharacteristic
221C2×D61—42no—
222C2×D61—41no—
223C2×D61—41no—
224C2×D61—41no—
225C2×D61—81no—
226C4×C61—21no—
227C4×C61—21no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—42no—
231C4×C61—81no—
232Dic61—42no—
233Dic61—41no—
234Dic61—41no—
235Dic61—41no—
236Dic61—81no—
237#2973—611no—
238#2322—611no—
239#1459—611no—
240C2×#324—611no—
241C2×#324D6111yescharacteristic, complemented
242#1282—611no—
243#1282—611no—
244C2×C61—21no—
245C2×C61—21no—
246C2×C61—21no—
247C2×C61—21no—
248C2×C61—21no—
249C2×C61—21no—
250C2×C61#187011yescharacteristic
251C2×C61#191311yescharacteristic
252C2×C61#191511yescharacteristic
253D61—81no—
254C2×#123—611no—
255C2×C4×D4—611no—
256C22×C4⋊C4C2×D6111yescharacteristic
257#435—1222no—
258#435—1222no—
259#432—1221no—
260#407—1221no—
261C2×C22⋊C8—611no—
262C2×C22⋊C8—611no—
263C2×C22⋊C8C2×D6111yescharacteristic
264C2×C22⋊C8C2×D6111yescharacteristic
265#389—1221no—
266#388—1221no—
267#387—1221no—
268#386—611no—
269#385—611no—
270#385—1221no—
271#381—611no—
272C4×C22⋊C4—611no—
273#324—22no—
274#324—1221no—
275#324—1222no—
276#323—1221no—
277C61#911011yescharacteristic, complemented, Sylow
278C22×D4—611no—
279C23×C4—611no—
280C23×C4C4×D6111yescharacteristic
281C23×C4C4×D6111yescharacteristic
282C23×C4C22×D6111yescharacteristic
283C2×M16—1222no—
284C22×C8—21no—
285C22×C8—21no—
286C22×C8—1221no—
287C22×C8—1221no—
288#123—1222no—
289#123—1222no—
290C4×D4—1221no—
291C4×D4—1221no—
292C4×D4—1221no—
293C4×D4—1221no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—611no—
301C2×C4⋊C4—1221no—
302C2×C4⋊C4—1221no—
303C2×C22⋊C4—611no—
304C2×C22⋊C4—611no—
305C2×C22⋊C4—611no—
306C2×C22⋊C4—611no—
307C2×C22⋊C4—1221no—
308C2×C22⋊C4—1221no—
309C2×C42—611no—
310C2×C42—1221no—
311C4⋊C8—1222no—
312C22⋊C8—21no—
313C22⋊C8—21no—
314C22⋊C8—21no—
315C22⋊C8—612no—
316C22⋊C8—1222no—
317C22⋊C8—1221no—
318C22⋊C8—1221no—
319C22⋊C8—1221no—
320C22⋊C8C22⋊D6112yes—
321#97—1222no—
322#95—1221no—
323#95—1221no—
324#95—1221no—
325#95—1221no—
326#95—1221no—
327C24—611no—
328C24C2×C4×D6111yescharacteristic
329C2×D4—1221no—
330C2×D4—1221no—
331C2×D4—1221no—
332C2×D4—1221no—
333C2×D4—1221no—
334C2×D4—1221no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—81no—
345C22×C4—81no—
346C22×C4—611no—
347C22×C4—611no—
348C22×C4—611no—
349C22×C4—1221no—
350C22×C4—1221no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—4881no—
354C22×C4#1175818912yescomplemented
355C22×C4#1175822211yescharacteristic
356C22×C4D4×D6111yescharacteristic
357C22×C4D4×D6111yescharacteristic
358C22×C4D4×D6111yescharacteristic
359C22×C4#1175822411yescharacteristic
360C22×C4C2×C22⋊D6111yescharacteristic
361M16—4882no—
362C2×C8—22no—
363C2×C8—41no—
364C2×C8—81no—
365C2×C8—81no—
366C2×C8—1222no—
367C2×C8—4881no—
368C2×C8—4881no—
369C2×C8—2441no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—1224no—
375C4⋊C4—1221no—
376C4⋊C4—1221no—
377C4⋊C4—4881no—
378C22⋊C4—612no—
379C22⋊C4—1224no—
380C22⋊C4—1221no—
381C22⋊C4—1221no—
382C22⋊C4—1221no—
383C22⋊C4—1221no—
384C22⋊C4—1221no—
385C22⋊C4—1221no—
386C22⋊C4—1221no—
387C22⋊C4—4881no—
388C42—1221no—
389C42—1221no—
390C42—4881no—
391C23—41no—
392C23—41no—
393C23—1221no—
394C23—1221no—
395C23—1221no—
396C23—2441no—
397C23—2441no—
398C23C22⋊C4×D6111yescharacteristic
399C23#1451973711yescharacteristic
400C23C4×C22⋊D6111yescharacteristic
401C23M16×D6111yescharacteristic
402C23C2×#1175818911yescharacteristic
403C23#1451978311yescharacteristic
404C23#1451978511yescharacteristic
405D4—2441no—
406D4—2441no—
407D4—2441no—
408D4—2441no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—1222no—
428C2×C4—1221no—
429C2×C4—1221no—
430C2×C4—1221no—
431C2×C4—1221no—
432C2×C4—1221no—
433C2×C4—4881no—
434C2×C4—2442no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4—2441no—
439C2×C4—2441no—
440C2×C4—2441no—
441C2×C4—2441no—
442C2×C4#1451974511yescharacteristic
443C2×C4#1451974811yescharacteristic
444C2×C4#1451975011yescharacteristic
445C2×C4#1451975911yescharacteristic
446C8—42no—
447C8—81no—
448C8—4881no—
449C8—2442no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—4881no—
461C22—2442no—
462C22—2441no—
463C22—2441no—
464C22—2441no—
465C22#1683963811yescharacteristic
466C22#1683972411yescharacteristic
467C22#1683972711yescharacteristic
468C22#1683983011yescharacteristic
469C22#1683983311yescharacteristic
470C22#1683983711yescharacteristic
471C22#1683983811yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—42no—
477C4—81no—
478C4—4881no—
479C4—2442no—
480C4—2441no—
481C4—2441no—
482C4—2441no—
483C2—21no—
484C2—21no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—4881no—
490C2#2144611311yescharacteristic
491C2#2144635811yescharacteristic
492C2#2144636811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#324C2
socleC22×C61#386
prime radicalC22×C22⋊D61C2×C4
cosocleC23×C4×C61C23
hypercenterC2×#324D61
hypocenterC61#9110
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
4. C2×#16839488C2semidirect
5. #21443666C2semidirect
6. #21443665C2semidirect
7. C2×C61×#324C2semidirect
93. C22×C4×C61M16semidirect
127. C22⋊Dic61M16semidirect
241. C2×#324D61semidirect
277. C61#9110semidirect
354. C22×C4#11758189semidirect

Maximal subgroups

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

subgroupindex
#214548452
#214484882
#214441352
C2×#168394882
#214436662
#214436652
C2×C61×#3242
#911061