Finite Group Databasebeta

← #29575903

Subgroups — #29575903

Order 15616.

562 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
1#21455295C211yesmaximal, characteristic
2#21454388C211yesmaximal, characteristic, complemented
3C2×#16839506C211yesmaximal, characteristic, complemented
4#21444207C211yesmaximal, characteristic, complemented
5#21443715C211yesmaximal, characteristic, complemented
6#21443693C211yesmaximal, characteristic, complemented
7C61×#1545C211yesmaximal, characteristic, complemented
8C2×#14519803—21no—
9C2×#14519785—21no—
10C4×#11758222C2211yescharacteristic
11#16839959—21no—
12#16839959C412yes—
13#16839954—21no—
14#16839954C412yes—
15#16839790—21no—
16#16839787—22no—
17#16839786C2211yescharacteristic
18#16839784—21no—
19C2×C4⋊#92418C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C4⋊C8×D61—21no—
22#16839767—21no—
23#16839506—21no—
24#16839506—21no—
25#16839494—21no—
26#16839491—21no—
27C2×#127×C61C2211yescharacteristic
28C61×#431C2211yescharacteristic
29C61×#423C2211yescharacteristic
30C61×#331—22no—
31C2×#11758222D411yescharacteristic
32#14519803—42no—
33#14519803—42no—
34C2×#11758200—22no—
35C2×#11758199—22no—
36#14519785—42no—
37#14519785—42no—
38C2×#11758194—21no—
39C2×#11758189—22no—
40C22×#92418—21no—
41C22×#92418—21no—
42C2×C8×D61—22no—
43C4×Dic122C2×C411yescharacteristic
44C4×Dic122D411yescharacteristic
45C4×D244C2×C411yescharacteristic
46C4×D244D411yescharacteristic
47C4×C22⋊D61—21no—
48C4×C22⋊D61—21no—
49#14519736D411yescharacteristic
50#14519734—21no—
51C42×D61—21no—
52C4⋊#92418—21no—
53C4⋊#92418—21no—
54C4⋊#92418—21no—
55C4⋊#92418—21no—
56C4⋊#92418C2×C412yes—
57C4⋊#92418D411yescharacteristic
58C4⋊#92418D411yescharacteristic
59#14519706—22no—
60#14519706—22no—
61#14519698—22no—
62#14519698—22no—
63C22×D4×C61—21no—
64C22×D4×C61—21no—
65C22×D4×C61—21no—
66C2×M16×C61—21no—
67#127×C61—21no—
68#127×C61—21no—
69#127×C61—21no—
70#127×C61C2×C412yes—
71C2×C42×C61C2311yescharacteristic
72C4⋊C8×C61—21no—
73C4⋊C8×C61—21no—
74C4⋊C8×C61D412yes—
75C22⋊C8×C61—21no—
76C4×C8×C61—21no—
77C24×C61—41no—
78C2×D4×C61—22no—
79C2×D4×C61—22no—
80C2×D4×C61—22no—
81C2×D4×C61—41no—
82C2×D4×C61—41no—
83C2×D4×C61—41no—
84C2×D4×C61—41no—
85C2×D4×C61—41no—
86C2×D4×C61—41no—
87C2×D4×C61—41no—
88C2×D4×C61—81no—
89C2×D4×C61—81no—
90C2×D4×C61—82no—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61#4111yescharacteristic
93C22×C4×C61#4111yescharacteristic
94C2×C22⋊D61—21no—
95#11758222—21no—
96#11758222—21no—
97#11758222D812yes—
98#11758222SD1612yes—
99C2×D244C22⋊C411yescharacteristic
100C2×C4×D61—21no—
101C2×C4×D61—21no—
102C2×C4×D61—21no—
103C2×Dic122C22⋊C411yescharacteristic
104M16×C61—41no—
105M16×C61—41no—
106C2×C8×C61—22no—
107C2×C8×C61—41no—
108C42×C61C22×C411yescharacteristic
109C42×C61C2×D411yescharacteristic
110C42×C61C2×D411yescharacteristic
111C42×C61C2×D411yescharacteristic
112C22⋊Dic61—21no—
113#11758200—44no—
114#11758199—44no—
115#11758198—21no—
116#11758198—21no—
117C4⋊Dic61C22⋊C411yescharacteristic
118C4⋊Dic61C22⋊C411yescharacteristic
119#11758196—21no—
120#11758196—21no—
121C4×Dic61—21no—
122C4×Dic61—21no—
123C4×Dic61—21no—
124#11758194—41no—
125#11758194—41no—
126C2×#92418—22no—
127C2×#92418—21no—
128C2×#92418—21no—
129C2×#92418—41no—
130C2×#92418—41no—
131C2×#92418—41no—
132C2×#92418—81no—
133#11758189—44no—
134C8×D61—44no—
135C23×C61—42no—
136C23×C61—82no—
137C23×C61—82no—
138C23×C61—81no—
139C23×C61—81no—
140C23×C61#12311yescharacteristic
141C22×D61—21no—
142D4×C61—42no—
143D4×C61—42no—
144D4×C61—82no—
145D4×C61—82no—
146D4×C61—82no—
147D4×C61—81no—
148D4×C61—81no—
149D4×C61—81no—
150D4×C61—81no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—41no—
154C2×C4×C61C2×C22⋊C411yescharacteristic
155C2×C4×C61C4×D411yescharacteristic
156C2×C4×C61C4×D411yescharacteristic
157C2×C4×C61#12011yescharacteristic
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61C2×D811yescharacteristic
161C2×C4×C61C2×SD1611yescharacteristic
162C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C61#13711yescharacteristic
164C22⋊D61—22no—
165C22⋊D61—22no—
166C2×Dic61—21no—
167C2×Dic61—21no—
168C2×Dic61—21no—
169D244—21no—
170D244#10212yes—
171C4×D61—22no—
172C4×D61—22no—
173C4×D61—41no—
174C4×D61—41no—
175C4×D61—81no—
176Dic122—21no—
177Dic122#10212yes—
178C8×C61—42no—
179C8×C61—81no—
180#92418—42no—
181#92418—41no—
182#92418—41no—
183#92418—81no—
184#9173—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—82no—
188C22×C61—82no—
189C22×C61—82no—
190C22×C61—162no—
191C22×C61—162no—
192C22×C61#38611yescharacteristic
193C22×C61#48011yescharacteristic
194C22×C61#48111yescharacteristic
195C2×D61—22no—
196C2×D61—41no—
197C2×D61—41no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61C2×#10211yescharacteristic
202C4×C61#41711yescharacteristic
203C4×C61#44711yescharacteristic
204C4×C61#44811yescharacteristic
205C4×C61#47111yescharacteristic
206C4×C61#47211yescharacteristic
207Dic61—22no—
208Dic61—41no—
209Dic61—41no—
210#3038—611no—
211#2906—611no—
212#1545D6111yescharacteristic, complemented
213C2×#331—611no—
214#1477—611no—
215#1301—611no—
216#1292—611no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—162no—
220C2×C61#187411yescharacteristic
221C2×C61#190311yescharacteristic
222C2×C61#190711yescharacteristic
223D61—42no—
224C2×AGL(1,ℤ/8)—1221no—
225C2×#131—1221no—
226C2×#127C2×D6111yescharacteristic
227C4×#41—611no—
228#460—612no—
229#460—1221no—
230#459—612no—
231#459—1221no—
232#431C2×D6111yescharacteristic
233#424—1221no—
234#424—1222no—
235#423—611no—
236#423—1221no—
237#423C2×D6111yescharacteristic
238C2×C4⋊C8—611no—
239C2×C4⋊C8—611no—
240C2×C4⋊C8—1221no—
241#418—1221no—
242#331—22no—
243#331—1221no—
244#331—1221no—
245Q8⋊C8—1221no—
246D4⋊C8—1221no—
247C61#917311yescharacteristic, complemented, Sylow
248C2×#41—611no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—21no—
252AGL(1,ℤ/8)—2442no—
253AGL(1,ℤ/8)—2442no—
254C2×SD16—1222no—
255C2×D8—1222no—
256#131—2442no—
257#131—2442no—
258C2×M16—21no—
259C2×M16—1222no—
260C2×M16—1221no—
261C22×C8—1222no—
262C22×C8—1221no—
263C22×C8—1221no—
264#127—21no—
265#127—21no—
266#127—21no—
267#127C4×D6112yes—
268C4×Q8—611no—
269C4×Q8—611no—
270C4×D4—611no—
271C4×D4—611no—
272C4×D4—1221no—
273C4×D4—1221no—
274#117—611no—
275#117—1221no—
276C2×C42—1221no—
277C2×C42C22×D6111yescharacteristic
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—612no—
281C4⋊C8—611no—
282C4⋊C8—611no—
283C4⋊C8—1222no—
284C4⋊C8—1222no—
285C4⋊C8—1221no—
286C4⋊C8—1221no—
287C4⋊C8—1221no—
288C4⋊C8—1221no—
289C4⋊C8C22⋊D6112yes—
290#102—1222no—
291#102—1222no—
292C22⋊C8—21no—
293C4×C8—21no—
294C24—41no—
295#41—612no—
296#41—612no—
297#41—1221no—
298#41—1221no—
299C2×Q8—611no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—81no—
311C2×D4—81no—
312C2×D4—82no—
313C2×D4—611no—
314C2×D4—1221no—
315C22×C4—1221no—
316C22×C4—1221no—
317C22×C4—1221no—
318C22×C4#1175822211yescharacteristic
319C22×C4D4×D6111yescharacteristic
320C22×C4#1175822411yescharacteristic
321SD16—2444no—
322D8—2444no—
323M16—41no—
324M16—41no—
325M16—2444no—
326M16—2441no—
327M16—2441no—
328C2×C8—22no—
329C2×C8—41no—
330C2×C8—1222no—
331C2×C8—1221no—
332C2×C8—1221no—
333C2×C8—4881no—
334C2×C8—2444no—
335C2×C8—2441no—
336C2×C8—2441no—
337C2×C8—2441no—
338C4⋊C4—611no—
339C4⋊C4—611no—
340C4⋊C4—1221no—
341C4⋊C4—1221no—
342C22⋊C4—1221no—
343C22⋊C4—1221no—
344C22⋊C4—1221no—
345C42—1221no—
346C42—1221no—
347C42—1221no—
348C42C2×C4×D6111yescharacteristic
349C42D4×D6111yescharacteristic
350C42D4×D6111yescharacteristic
351C42C2×C22⋊D6111yescharacteristic
352C23—42no—
353C23—82no—
354C23—82no—
355C23—81no—
356C23—81no—
357C23—1221no—
358C23#1451975911yescharacteristic
359Q8—612no—
360Q8—1221no—
361D4—42no—
362D4—42no—
363D4—82no—
364D4—82no—
365D4—82no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—612no—
371D4—1222no—
372D4—1222no—
373D4—1221no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—1221no—
378C2×C4—1222no—
379C2×C4—1222no—
380C2×C4—1221no—
381C2×C4—1221no—
382C2×C4—4881no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4C22⋊C4×D6111yescharacteristic
386C2×C4#1451973711yescharacteristic
387C2×C4C4×C22⋊D6111yescharacteristic
388C2×C4#1451974511yescharacteristic
389C2×C4#1451974811yescharacteristic
390C2×C4#1451975011yescharacteristic
391C2×C4D8×D6111yescharacteristic
392C2×C4SD16×D6111yescharacteristic
393C2×C4#1451980111yescharacteristic
394C2×C4#1451980611yescharacteristic
395C8—42no—
396C8—81no—
397C8—4881no—
398C8—2442no—
399C8—2441no—
400C8—2441no—
401C22—21no—
402C22—21no—
403C22—82no—
404C22—82no—
405C22—82no—
406C22—162no—
407C22—162no—
408C22—1222no—
409C22—2441no—
410C22—2441no—
411C22#1683963811yescharacteristic
412C22#1684007411yescharacteristic
413C22#1684007911yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—1222no—
418C4—2441no—
419C4—2441no—
420C4#102×D6111yescharacteristic
421C4#1683976011yescharacteristic
422C4#1683989411yescharacteristic
423C4#1683989911yescharacteristic
424C4#1684002411yescharacteristic
425C4#1684003111yescharacteristic
426C2—21no—
427C2—21no—
428C2—162no—
429C2—2442no—
430C2#2144613311yescharacteristic
431C2#2144629611yescharacteristic
432C2#2144632411yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1545C2
socleC22×C61#386
prime radical#21455295C2
cosocleC2×C42×C61C23
hypercenter#1545D61
hypocenterC61#9173
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1545

Complements

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

NH ≅ G/Ntype
2. #21454388C2semidirect
3. C2×#16839506C2semidirect
4. #21444207C2semidirect
5. #21443715C2semidirect
6. #21443693C2semidirect
7. C61×#1545C2semidirect
212. #1545D61semidirect
247. C61#9173semidirect

Maximal subgroups

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

subgroupindex
#214552952
#214543882
C2×#168395062
#214442072
#214437152
#214436932
C61×#15452
#917361