Finite Group Databasebeta

← #29576401

Subgroups — #29576401

Order 15616.

528 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840039C211yesmaximal, characteristic
2#21454554C211yesmaximal, characteristic
3#21446664C211yesmaximal, characteristic, complemented
4#21444245C211yesmaximal, characteristic
5#21443693C211yesmaximal, characteristic, complemented
6#21443668C211yesmaximal, characteristic, complemented
7C61×#1546C211yesmaximal, characteristic, complemented
8C2×M16×D61—21no—
9C2×C4×D244C2211yescharacteristic
10#16840039—21no—
11#16840039—21no—
12#16840039—21no—
13#16840039—21no—
14#16839839—21no—
15#16839832—22no—
16#16839827—21no—
17C2×#14519710C412yes—
18#16839786C2211yescharacteristic
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519697C412yes—
21C2×#14519696—21no—
22C2×#14519696—21no—
23#16839725—21no—
24C4×#11758194C2211yescharacteristic
25#16839491—21no—
26#16839491—21no—
27C2×C4⋊Q8×C61C2211yescharacteristic
28C61×#431C2211yescharacteristic
29C61×#423C2211yescharacteristic
30C61×#332—22no—
31C22×D244C2×C411yescharacteristic
32C22×C4×D61—21no—
33C2×#11758194—21no—
34C2×#11758194D411yescharacteristic
35C2×#11758194D411yescharacteristic
36C2×#11758189—22no—
37M16×D61—42no—
38M16×D61—42no—
39C22×#92418C2×C412yes—
40C2×C8×D61—22no—
41C4×D244—21no—
42C4×D244—21no—
43C4×D244D411yescharacteristic
44C4×D244D411yescharacteristic
45C4×D244D411yescharacteristic
46C4×D244D411yescharacteristic
47C2×C4⋊Dic61C2×C411yescharacteristic
48C2×#11758198—21no—
49#14519710—22no—
50#14519710—22no—
51C4⋊#92418—21no—
52C4⋊#92418—21no—
53C4⋊#92418—21no—
54C4⋊#92418—21no—
55#14519697—22no—
56#14519697—22no—
57#14519696—42no—
58#14519696—42no—
59#14519689—22no—
60#14519689—22no—
61#14519688—21no—
62C4×#92418—21no—
63C22×Q8×C61—21no—
64C2×M16×C61—21no—
65C4⋊Q8×C61—21no—
66C4⋊Q8×C61—21no—
67C4⋊Q8×C61—21no—
68C4⋊Q8×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61C2×C412yes—
71C2×C42×C61C2311yescharacteristic
72C4⋊C8×C61—21no—
73C4⋊C8×C61—21no—
74C4⋊C8×C61D412yes—
75C22⋊C8×C61—21no—
76C4×C8×C61—21no—
77C23×D61—21no—
78C2×Q8×C61—41no—
79C2×Q8×C61—41no—
80C2×Q8×C61—41no—
81C2×Q8×C61—41no—
82C2×Q8×C61—41no—
83C2×Q8×C61—41no—
84C22×C4×C61—22no—
85C22×C4×C61C22×C411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61#4111yescharacteristic
88C22×Dic61—21no—
89C2×D244—22no—
90C2×D244—21no—
91C2×D244—41no—
92C2×D244C22⋊C411yescharacteristic
93C2×D244C22⋊C411yescharacteristic
94C2×C4×D61—21no—
95C2×C4×D61—21no—
96C2×C4×D61—21no—
97C2×C4×D61—21no—
98C2×C4×D61—41no—
99C2×C4×D61—41no—
100C2×C4×D61—41no—
101C2×C4×D61—41no—
102M16×C61—41no—
103M16×C61—41no—
104C2×C8×C61—22no—
105C2×C8×C61—41no—
106C4⋊C4×C61—22no—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—41no—
109C4⋊C4×C61—41no—
110C42×C61C2×D411yescharacteristic
111C42×C61C2×D411yescharacteristic
112C42×C61C2×D411yescharacteristic
113C42×C61#4111yescharacteristic
114#11758198—21no—
115#11758198—21no—
116#11758198—21no—
117#11758198—21no—
118C4⋊Dic61—21no—
119C4⋊Dic61C22⋊C411yescharacteristic
120C4⋊Dic61C22⋊C411yescharacteristic
121#11758194—21no—
122#11758194—21no—
123#11758194—21no—
124#11758194—41no—
125#11758194—41no—
126#11758194D812yes—
127C2×#92418—21no—
128C2×#92418—22no—
129C2×#92418—21no—
130C2×#92418—42no—
131C2×#92418—41no—
132#11758189—44no—
133C8×D61—44no—
134C23×C61C4×D411yescharacteristic
135C22×D61—21no—
136C22×D61—21no—
137C22×D61—42no—
138C22×D61—41no—
139C22×D61—41no—
140C22×D61—81no—
141Q8×C61—81no—
142Q8×C61—81no—
143Q8×C61—81no—
144Q8×C61—81no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—41no—
148C2×C4×C61—42no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61C2×C22⋊C411yescharacteristic
152C2×C4×C61C4×D411yescharacteristic
153C2×C4×C61#12011yescharacteristic
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#12311yescharacteristic
157C2×C4×C61C2×D811yescharacteristic
158C2×C4×C61#13511yescharacteristic
159C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165D244—21no—
166D244—21no—
167D244—21no—
168D244—42no—
169D244#10112yescomplemented
170C4×D61—41no—
171C4×D61—41no—
172C4×D61—41no—
173C4×D61—41no—
174C4×D61—41no—
175C4×D61—41no—
176C4×D61—41no—
177C4×D61—41no—
178C8×C61—42no—
179C8×C61—81no—
180#92418—22no—
181#92418—42no—
182#92418—41no—
183#92418—81no—
184#9251—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61#38611yescharacteristic
188C22×C61C4×D811yescharacteristic
189C22×C61#44211yescharacteristic
190C2×D61—22no—
191C2×D61—41no—
192C2×D61—41no—
193C2×D61—41no—
194C2×D61—41no—
195C2×D61—41no—
196C2×D61—82no—
197C2×D61—82no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—82no—
202C4×C61C2×#10111yescharacteristic
203C4×C61#41311yescharacteristic
204C4×C61#44711yescharacteristic
205C4×C61#44911yescharacteristic
206C4×C61#45911yescharacteristic
207C4×C61#46311yescharacteristic
208Dic61—41no—
209Dic61—41no—
210C2×#474—611no—
211D4×M16—611no—
212#1966—611no—
213#1546D6111yescharacteristic, complemented
214#1486—611no—
215#1292—611no—
216#1283—611no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61#187911yescharacteristic
220C2×C61#190111yescharacteristic
221C2×C61#190211yescharacteristic
222D61—42no—
223D61—81no—
224C22×M16—1221no—
225C2×C4⋊Q8C2×D6111yescharacteristic
226C2×C4×D4—611no—
227#474—1221no—
228#474—1221no—
229#474—1221no—
230#474—1221no—
231#436—1221no—
232#435—1222no—
233C8×D4—1221no—
234#431C2×D6111yescharacteristic
235C2×#107—612no—
236#423—611no—
237#423C2×D6111yescharacteristic
238C2×C4⋊C8—611no—
239C2×#102—612no—
240C2×#101—1221no—
241C2×#101—1221no—
242#407—1221no—
243C4×M16—611no—
244#332—22no—
245D4⋊C8—1221no—
246D4⋊C8—1221no—
247C61#925111yescharacteristic, complemented, Sylow
248C22×Q8—21no—
249C22×D4—611no—
250C23×C4—1221no—
251C2×M16—21no—
252C2×M16—611no—
253C2×M16—611no—
254C2×M16—1222no—
255C2×M16—1221no—
256C2×M16—2442no—
257C2×M16—2442no—
258C22×C8—612no—
259C22×C8—1222no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4×D4—611no—
265C4×D4—611no—
266C4×D4—611no—
267C4×D4—611no—
268C4×D4—1221no—
269C4×D4—1221no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—611no—
272C2×C4⋊C4C4×D6112yes—
273C2×C22⋊C4—1221no—
274C2×C42C22×D6111yescharacteristic
275#107—1222no—
276#107—1222no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—1221no—
280C4⋊C8—1221no—
281C4⋊C8—1221no—
282C4⋊C8—1221no—
283C4⋊C8C22⋊D6112yes—
284#102—1222no—
285#102—1222no—
286#101—2442no—
287#101—2442no—
288C22⋊C8—21no—
289C22⋊C8—1222no—
290C22⋊C8—1222no—
291#97—1221no—
292C4×C8—21no—
293C4×C8—1221no—
294C24—1221no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×D4—611no—
302C2×D4—611no—
303C2×D4—1222no—
304C2×D4—1221no—
305C2×D4—2441no—
306C22×C4—22no—
307C22×C4—1221no—
308C22×C4—1221no—
309C22×C4—1221no—
310C22×C4—1221no—
311C22×C4—1221no—
312C22×C4—2441no—
313C22×C4—2441no—
314C22×C4—2441no—
315C22×C4—2441no—
316C22×C4C2×C4×D6111yescharacteristic
317C22×C4D4×D6111yescharacteristic
318C22×C4#1175822411yescharacteristic
319M16—41no—
320M16—41no—
321M16—612no—
322M16—1221no—
323M16—1221no—
324M16—1221no—
325M16—2444no—
326M16—2441no—
327M16—2441no—
328C2×C8—22no—
329C2×C8—41no—
330C2×C8—1221no—
331C2×C8—1222no—
332C2×C8—1221no—
333C2×C8—2444no—
334C2×C8—2441no—
335C2×C8—2442no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—611no—
341C4⋊C4—611no—
342C4⋊C4—1221no—
343C22⋊C4—1221no—
344C22⋊C4—1221no—
345C22⋊C4—1221no—
346C22⋊C4—1221no—
347C42#1175822211yescharacteristic
348C42D4×D6111yescharacteristic
349C42D4×D6111yescharacteristic
350C42C2×C22⋊D6111yescharacteristic
351C23—1221no—
352C23—1221no—
353C23—4881no—
354C23—2442no—
355C23—2441no—
356C23—2441no—
357C23#1451973711yescharacteristic
358Q8—81no—
359Q8—81no—
360Q8—81no—
361Q8—81no—
362D4—612no—
363D4—1221no—
364D4—1221no—
365D4—1221no—
366D4—2442no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—1221no—
374C2×C4—1221no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4—2441no—
382C2×C4—2441no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4C22⋊C4×D6111yescharacteristic
386C2×C4C4×C22⋊D6111yescharacteristic
387C2×C4#1451974511yescharacteristic
388C2×C4#1451974811yescharacteristic
389C2×C4#1451975011yescharacteristic
390C2×C4#1451975911yescharacteristic
391C2×C4D8×D6111yescharacteristic
392C2×C4#1451979811yescharacteristic
393C2×C4#1451980111yescharacteristic
394C2×C4#1451980411yescharacteristic
395C8—42no—
396C8—81no—
397C8—1222no—
398C8—4881no—
399C8—2442no—
400C8—2441no—
401C22—21no—
402C22—21no—
403C22—1222no—
404C22—4882no—
405C22—4882no—
406C22—2441no—
407C22—2441no—
408C22—2441no—
409C22—2441no—
410C22—2441no—
411C22#1683963811yescharacteristic
412C22#1683984211yescharacteristic
413C22#1683987111yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—82no—
418C4—2441no—
419C4—2441no—
420C4#101×D6111yescharacteristic
421C4#1683974511yescharacteristic
422C4#1683989411yescharacteristic
423C4#1683990611yescharacteristic
424C4#1683995611yescharacteristic
425C4#1683997611yescharacteristic
426C2—21no—
427C2—21no—
428C2—4881no—
429C2—2442no—
430C2#2144616411yescharacteristic
431C2#2144628411yescharacteristic
432C2#2144628911yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1546C2
socleC22×C61#386
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#1546D61
hypocenterC61#9251
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
3. #21446664C2semidirect
5. #21443693C2semidirect
6. #21443668C2semidirect
7. C61×#1546C2semidirect
169. D244#101semidirect
213. #1546D61semidirect
247. C61#9251semidirect

Maximal subgroups

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

subgroupindex
C2×#168400392
#214545542
#214466642
#214442452
#214436932
#214436682
C61×#15462
#925161