Finite Group Databasebeta

← #29576403

Subgroups — #29576403

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×#16840043C211yesmaximal, characteristic
2M16×D244C211yesmaximal, characteristic
3#21446663C211yesmaximal, characteristic, complemented
4#21444490C211yesmaximal, characteristic, complemented
5#21444245C211yesmaximal, characteristic
6#21443694C211yesmaximal, characteristic, complemented
7C61×#1283C211yesmaximal, characteristic, complemented
8C2×M16×D61—21no—
9C2×#14519777C2211yescharacteristic
10C2×C4×D244C2211yescharacteristic
11#16840043—21no—
12#16840043—21no—
13#16840043—21no—
14#16840043—21no—
15#16839841—21no—
16#16839835—22no—
17C8×D244—21no—
18C2×#14519711C412yes—
19#16839786C2211yescharacteristic
20#16839786C2211yescharacteristic
21C2×#14519699C412yes—
22C2×#14519695—21no—
23C2×#14519695—21no—
24#16839725—21no—
25#16839507—22no—
26#16839491—21no—
27#16839491—21no—
28C61×#431C2211yescharacteristic
29C2×C4⋊C8×C61C2211yescharacteristic
30C4×M16×C61C2211yescharacteristic
31C22×Dic122—21no—
32C22×D244C2×C411yescharacteristic
33C22×C4×D61—21no—
34C2×#11758194—21no—
35C2×#11758194—21no—
36C2×#11758189—22no—
37M16×D61—42no—
38M16×D61—42no—
39C2×C8×D61—22no—
40#14519777—21no—
41#14519777—21no—
42#14519777—21no—
43#14519777—21no—
44C4×D244—21no—
45C4×D244—21no—
46C4×D244D411yescharacteristic
47C4×D244D411yescharacteristic
48C4×D244D411yescharacteristic
49C4×D244D411yescharacteristic
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61C2×C412yes—
52C2×C4⋊Dic61C2×C411yescharacteristic
53C2×#11758198—21no—
54#14519711—22no—
55#14519711—22no—
56C4⋊#92418—21no—
57C4⋊#92418—21no—
58C4⋊#92418—21no—
59C4⋊#92418D412yes—
60#14519699—22no—
61#14519699—22no—
62#14519695—42no—
63#14519695—42no—
64#14519689—22no—
65#14519689—22no—
66C2×M16×C61D411yescharacteristic
67C2×M16×C61D411yescharacteristic
68C22×C8×C61C2×C412yes—
69C2×C42×C61C2311yescharacteristic
70C4⋊C8×C61—21no—
71C4⋊C8×C61—21no—
72C4⋊C8×C61—21no—
73C22⋊C8×C61—21no—
74#97×C61—21no—
75C4×C8×C61—21no—
76C4×C8×C61—21no—
77C23×D61—21no—
78C22×C4×C61C22×C411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61#4111yescharacteristic
81C22×Dic61—21no—
82C22×Dic61—22no—
83C2×D244—22no—
84C2×D244—21no—
85C2×D244—41no—
86C2×D244C22⋊C411yescharacteristic
87C2×D244C22⋊C411yescharacteristic
88C2×C4×D61—21no—
89C2×C4×D61—21no—
90C2×C4×D61—21no—
91C2×C4×D61—21no—
92C2×C4×D61—41no—
93C2×C4×D61—41no—
94C2×C4×D61—41no—
95C2×C4×D61—41no—
96C2×Dic122—41no—
97C2×Dic122—41no—
98C2×Dic122—41no—
99C2×Dic122—41no—
100C2×Dic122—41no—
101C2×Dic122—41no—
102M16×C61—21no—
103M16×C61—21no—
104M16×C61—21no—
105M16×C61D812yes—
106C2×C8×C61—21no—
107C2×C8×C61—22no—
108C2×C8×C61—21no—
109C2×C8×C61—42no—
110C2×C8×C61—41no—
111C42×C61C2×D411yescharacteristic
112C42×C61C2×D411yescharacteristic
113C42×C61C2×D411yescharacteristic
114C42×C61#4111yescharacteristic
115#11758198—21no—
116#11758198—21no—
117#11758198—21no—
118#11758198—21no—
119C4⋊Dic61—21no—
120C4⋊Dic61—22no—
121C4⋊Dic61—22no—
122C4⋊Dic61—41no—
123C4⋊Dic61—41no—
124C4⋊Dic61C22⋊C411yescharacteristic
125C4⋊Dic61C22⋊C411yescharacteristic
126#11758194—41no—
127#11758194—41no—
128#11758194—41no—
129#11758194—41no—
130C2×#92418—22no—
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—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—41no—
144C2×C4×C61C2×C22⋊C411yescharacteristic
145C2×C4×C61C4×D411yescharacteristic
146C2×C4×C61#12011yescharacteristic
147C2×C4×C61#12111yescharacteristic
148C2×C4×C61#12111yescharacteristic
149C2×C4×C61#12311yescharacteristic
150C2×C4×C61C2×D811yescharacteristic
151C2×C4×C61#13511yescharacteristic
152C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
154C2×Dic61—21no—
155C2×Dic61—21no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—42no—
159C2×Dic61—42no—
160C2×Dic61—42no—
161D244—21no—
162D244—21no—
163D244—21no—
164D244—42no—
165D244#10112yescomplemented
166C4×D61—41no—
167C4×D61—41no—
168C4×D61—41no—
169C4×D61—41no—
170C4×D61—41no—
171C4×D61—41no—
172C4×D61—41no—
173C4×D61—41no—
174Dic122—81no—
175Dic122—81no—
176Dic122—81no—
177Dic122—81no—
178C8×C61—22no—
179C8×C61—42no—
180C8×C61—41no—
181C8×C61—81no—
182#92418—42no—
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×C61C2×#10111yescharacteristic
202C4×C61#41311yescharacteristic
203C4×C61#44711yescharacteristic
204C4×C61#44911yescharacteristic
205C4×C61#45911yescharacteristic
206C4×C61#46311yescharacteristic
207Dic61—41no—
208Dic61—41no—
209Dic61—82no—
210C2×#474—611no—
211D4×M16—611no—
212#1966—611no—
213#1546—611no—
214#1486—611no—
215#1292—611no—
216#1283D6111yescharacteristic, complemented
217C2×C61—21no—
218C2×C61—21no—
219C2×C61#187911yescharacteristic
220C2×C61#190111yescharacteristic
221C2×C61#190211yescharacteristic
222D61—42no—
223D61—81no—
224C22×M16—1221no—
225C2×C4⋊Q8—611no—
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#423—611no—
238C2×C4⋊C8C2×D6111yescharacteristic
239C2×#102—612no—
240C2×#101—1221no—
241C2×#101—1221no—
242#407—1221no—
243C4×M16C2×D6111yescharacteristic
244#332—1222no—
245D4⋊C8—1221no—
246D4⋊C8—1221no—
247C61#925111yescharacteristic, complemented, Sylow
248C22×Q8—1221no—
249C22×D4—611no—
250C23×C4—1221no—
251C2×M16—1222no—
252C2×M16—1221no—
253C2×M16—1221no—
254C2×M16—2442no—
255C2×M16—2442no—
256C2×M16D24411yescharacteristic
257C2×M16D24411yescharacteristic
258C22×C8—1222no—
259C22×C8C4×D6112yes—
260C4⋊Q8—1221no—
261C4⋊Q8—1221no—
262C4⋊Q8—1221no—
263C4⋊Q8—1221no—
264C4×D4—611no—
265C4×D4—611no—
266C4×D4—611no—
267C4×D4—611no—
268C4×D4—1221no—
269C4×D4—1221no—
270C2×C4⋊C4—611no—
271C2×C4⋊C4—612no—
272C2×C4⋊C4—1221no—
273C2×C22⋊C4—1221no—
274C2×C42C22×D6111yescharacteristic
275#107—1222no—
276#107—1222no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—612no—
281C4⋊C8—1221no—
282C4⋊C8—1221no—
283C4⋊C8—1221no—
284#102—1222no—
285#102—1222no—
286#101—2442no—
287#101—2442no—
288C22⋊C8—21no—
289C22⋊C8—1222no—
290C22⋊C8—1222no—
291#97—21no—
292C4×C8—21no—
293C4×C8—21no—
294C24—1221no—
295C2×Q8—2441no—
296C2×Q8—2441no—
297C2×Q8—2441no—
298C2×Q8—2441no—
299C2×Q8—2441no—
300C2×Q8—2441no—
301C2×D4—611no—
302C2×D4—611no—
303C2×D4—1222no—
304C2×D4—1221no—
305C2×D4—2441no—
306C22×C4—1221no—
307C22×C4—1221no—
308C22×C4—1221no—
309C22×C4—1221no—
310C22×C4—1221no—
311C22×C4—1222no—
312C22×C4—2441no—
313C22×C4—2441no—
314C22×C4—2441no—
315C22×C4—2441no—
316C22×C4C2×C4×D6111yescharacteristic
317C22×C4C2×D24411yescharacteristic
318C22×C4#1175822211yescharacteristic
319M16—21no—
320M16—21no—
321M16—21no—
322M16—2444no—
323M16—2441no—
324M16—2441no—
325M16—2441no—
326M16—2441no—
327M16D48812yes—
328C2×C8—21no—
329C2×C8—22no—
330C2×C8—21no—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—1222no—
334C2×C8—2444no—
335C2×C8—2441no—
336C4⋊C4—611no—
337C4⋊C4—611no—
338C4⋊C4—1221no—
339C4⋊C4—1222no—
340C4⋊C4—1222no—
341C4⋊C4—2441no—
342C4⋊C4—2441no—
343C22⋊C4—1221no—
344C22⋊C4—1221no—
345C22⋊C4—1221no—
346C22⋊C4—1221no—
347C42D4×D6111yescharacteristic
348C42D4×D6111yescharacteristic
349C42D4×D6111yescharacteristic
350C42#1175822611yescharacteristic
351C23—1221no—
352C23—1221no—
353C23—4881no—
354C23—2442no—
355C23—2441no—
356C23—2441no—
357C23C4×D24411yescharacteristic
358Q8—4881no—
359Q8—4881no—
360Q8—4881no—
361Q8—4881no—
362D4—612no—
363D4—1221no—
364D4—1221no—
365D4—1221no—
366D4—2442no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—1221no—
371C2×C4—1221no—
372C2×C4—2442no—
373C2×C4—2442no—
374C2×C4—2442no—
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×C4#1451973911yescharacteristic
387C2×C4#1451974711yescharacteristic
388C2×C4#1451975011yescharacteristic
389C2×C4#1451975211yescharacteristic
390C2×C4#1451976211yescharacteristic
391C2×C4C2×D48811yescharacteristic
392C2×C4#1451979911yescharacteristic
393C2×C4#1451980511yescharacteristic
394C2×C4#1451980511yescharacteristic
395C8—22no—
396C8—42no—
397C8—41no—
398C8—81no—
399C8—4881no—
400C8—2442no—
401C22—21no—
402C22—21no—
403C22—1222no—
404C22—4882no—
405C22—4882no—
406C22—2441no—
407C22—2441no—
408C22—2441no—
409C22—2441no—
410C22—2441no—
411C22#1683964011yescharacteristic
412C22C4×D48811yescharacteristic
413C22#1683987511yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—4882no—
418C4—2441no—
419C4—2441no—
420C4#101×D6111yescharacteristic
421C4#1683974311yescharacteristic
422C4#1683989811yescharacteristic
423C4#1683990911yescharacteristic
424C4#1683995811yescharacteristic
425C4#1683997811yescharacteristic
426C2—21no—
427C2—21no—
428C2—4881no—
429C2—2442no—
430C2#2144616611yescharacteristic
431C2#2144628611yescharacteristic
432C2#2144629111yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1283C2
socleC22×C61#386
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#1283D61
hypocenterC61#9251
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1283

Complements

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

NH ≅ G/Ntype
3. #21446663C2semidirect
4. #21444490C2semidirect
6. #21443694C2semidirect
7. C61×#1283C2semidirect
165. D244#101semidirect
216. #1283D61semidirect
247. C61#9251semidirect

Maximal subgroups

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

subgroupindex
C2×#168400432
M16×D2442
#214466632
#214444902
#214442452
#214436942
C61×#12832
#925161