Finite Group Databasebeta

← #29575904

Subgroups — #29575904

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#21455299C211yesmaximal, characteristic
2#21454387C211yesmaximal, characteristic, complemented
3#21444486C211yesmaximal, characteristic, complemented
4C2×#16839505C211yesmaximal, characteristic, complemented
5#21443715C211yesmaximal, characteristic, complemented
6#21443694C211yesmaximal, characteristic, complemented
7C61×#1477C211yesmaximal, characteristic, complemented
8C2×#14519802—21no—
9C2×#14519784—21no—
10C2×#14519774C2211yescharacteristic
11#16839962—21no—
12#16839962C412yes—
13#16839957—21no—
14#16839957C412yes—
15#16839789—22no—
16#16839789—21no—
17#16839786—21no—
18#16839786C2211yescharacteristic
19#16839786C2211yescharacteristic
20C2×C4⋊#92418—21no—
21C2×C4⋊#92418C2211yescharacteristic
22#16839766—21no—
23#16839505—21no—
24#16839505—21no—
25#16839505—22no—
26C4×#41×C61C2211yescharacteristic
27C61×#431C2211yescharacteristic
28C2×C4⋊C8×C61C2211yescharacteristic
29C61×Q8⋊C8—21no—
30C61×D4⋊C8—21no—
31C22×D244—21no—
32C22×D244—21no—
33C22×D244—21no—
34#14519802—42no—
35#14519802—42no—
36C2×#11758201—22no—
37C2×#11758199—22no—
38#14519784—42no—
39#14519784—42no—
40C2×#11758194—22no—
41C2×#11758194—21no—
42C2×#11758194—21no—
43C22×#92418—22no—
44C22×#92418—21no—
45#14519774—21no—
46#14519774—21no—
47#14519774—21no—
48#14519774C2×C412yes—
49C4⋊#92418—22no—
50C4⋊#92418—22no—
51C4⋊#92418—21no—
52C4⋊#92418—21no—
53C4⋊#92418—21no—
54C4⋊#92418C2×C412yes—
55C4⋊#92418D411yescharacteristic
56C4⋊#92418D411yescharacteristic
57C4⋊#92418D412yes—
58#14519697—22no—
59#14519697—22no—
60C2×#41×C61D411yescharacteristic
61C22×C8×C61—21no—
62C4×Q8×C61C2×C411yescharacteristic
63C4×Q8×C61D411yescharacteristic
64C4×D4×C61—21no—
65C4×D4×C61—21no—
66C4×D4×C61C2×C411yescharacteristic
67C4×D4×C61D411yescharacteristic
68#117×C61—21no—
69#117×C61D411yescharacteristic
70C2×C42×C61—21no—
71C2×C42×C61C2311yescharacteristic
72C4⋊C8×C61—21no—
73C4⋊C8×C61—21no—
74C4⋊C8×C61—21no—
75C22⋊C8×C61—21no—
76C4×C8×C61—21no—
77C23×D61—41no—
78#41×C61—21no—
79#41×C61—21no—
80#41×C61D812yes—
81#41×C61SD1612yes—
82C2×Q8×C61C22⋊C411yescharacteristic
83C2×D4×C61—21no—
84C2×D4×C61C22⋊C411yescharacteristic
85C22×C4×C61—21no—
86C22×C4×C61—21no—
87C22×C4×C61—21no—
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61#4111yescharacteristic
90C22×C4×C61#4111yescharacteristic
91C2×D244—22no—
92C2×D244—22no—
93C2×D244—22no—
94C2×D244—41no—
95C2×D244—41no—
96C2×D244—41no—
97C2×D244—41no—
98C2×D244—41no—
99C2×D244—41no—
100C2×D244—41no—
101C2×D244—82no—
102C2×D244—81no—
103C2×D244—81no—
104C2×C8×C61—41no—
105C2×C8×C61—41no—
106C2×C8×C61—81no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61C22⋊C411yescharacteristic
110C4⋊C4×C61C22⋊C411yescharacteristic
111C22⋊C4×C61—21no—
112C22⋊C4×C61—21no—
113C22⋊C4×C61—21no—
114C42×C61—21no—
115C42×C61—21no—
116C42×C61—21no—
117C42×C61C22×C411yescharacteristic
118C42×C61C2×D411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61C2×D411yescharacteristic
121#11758201—44no—
122#11758199—44no—
123#11758194—44no—
124#11758194—41no—
125#11758194—41no—
126#11758194—41no—
127#11758194—41no—
128C2×#92418—22no—
129C2×#92418—21no—
130C2×#92418—22no—
131C2×#92418—21no—
132C2×#92418—44no—
133C2×#92418—41no—
134C2×#92418—41no—
135C23×C61—21no—
136C23×C61#12311yescharacteristic
137C22×D61—42no—
138C22×D61—82no—
139C22×D61—82no—
140C22×D61—81no—
141C22×D61—81no—
142Q8×C61—21no—
143Q8×C61#10212yes—
144D4×C61—22no—
145D4×C61—22no—
146D4×C61—21no—
147D4×C61#10212yes—
148C2×C4×C61—21no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—81no—
159C2×C4×C61C2×C22⋊C411yescharacteristic
160C2×C4×C61C4×D411yescharacteristic
161C2×C4×C61C4×D411yescharacteristic
162C2×C4×C61#12011yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61C2×D811yescharacteristic
166C2×C4×C61C2×SD1611yescharacteristic
167C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C61#13711yescharacteristic
169D244—42no—
170D244—42no—
171D244—82no—
172D244—82no—
173D244—82no—
174D244—81no—
175D244—81no—
176D244—81no—
177D244—81no—
178C8×C61—81no—
179C8×C61—81no—
180#92418—42no—
181#92418—41no—
182#92418—42no—
183#92418—41no—
184#9173—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—22no—
188C22×C61—41no—
189C22×C61—41no—
190C22×C61#38611yescharacteristic
191C22×C61#48011yescharacteristic
192C22×C61#48111yescharacteristic
193C2×D61—82no—
194C2×D61—82no—
195C2×D61—82no—
196C2×D61—162no—
197C2×D61—162no—
198C4×C61—22no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—41no—
203C4×C61—41no—
204C4×C61C2×#10211yescharacteristic
205C4×C61#41711yescharacteristic
206C4×C61#44711yescharacteristic
207C4×C61#44811yescharacteristic
208C4×C61#47111yescharacteristic
209C4×C61#47211yescharacteristic
210#3038—611no—
211#2906—611no—
212#1545—611no—
213C2×#331—611no—
214#1477D6111yescharacteristic, complemented
215#1301—611no—
216#1292—611no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—42no—
220C2×C61#187411yescharacteristic
221C2×C61#190311yescharacteristic
222C2×C61#190711yescharacteristic
223D61—162no—
224C2×AGL(1,ℤ/8)—1221no—
225C2×#131—1221no—
226C2×#127—611no—
227C4×#41C2×D6111yescharacteristic
228#460—612no—
229#460—1221no—
230#459—612no—
231#459—1221no—
232#431C2×D6111yescharacteristic
233#424—1222no—
234#424—1221no—
235#423—611no—
236#423—611no—
237#423—1221no—
238C2×C4⋊C8—611no—
239C2×C4⋊C8—1221no—
240C2×C4⋊C8C2×D6111yescharacteristic
241#418—1221no—
242#331—1221no—
243#331—1221no—
244#331—1222no—
245Q8⋊C8—21no—
246D4⋊C8—21no—
247C61#917311yescharacteristic, complemented, Sylow
248C2×#41D24411yescharacteristic
249C22×D4—1221no—
250C22×D4—1221no—
251C22×D4—1221no—
252AGL(1,ℤ/8)—2442no—
253AGL(1,ℤ/8)—2442no—
254C2×SD16—1222no—
255C2×D8—1222no—
256#131—2442no—
257#131—2442no—
258C2×M16—1222no—
259C2×M16—1221no—
260C2×M16—1221no—
261C22×C8—21no—
262C22×C8—1222no—
263C22×C8—1221no—
264#127—612no—
265#127—1221no—
266#127—1221no—
267#127—1221no—
268C4×Q8C4×D6111yescharacteristic
269C4×Q8C22⋊D6111yescharacteristic
270C4×D4—21no—
271C4×D4—21no—
272C4×D4C4×D6111yescharacteristic
273C4×D4C22⋊D6111yescharacteristic
274#117—21no—
275#117D24411yescharacteristic
276C2×C42—21no—
277C2×C42C22×D6111yescharacteristic
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—612no—
282C4⋊C8—611no—
283C4⋊C8—611no—
284C4⋊C8—612no—
285C4⋊C8—1222no—
286C4⋊C8—1222no—
287C4⋊C8—1221no—
288C4⋊C8—1221no—
289C4⋊C8—1221no—
290#102—1222no—
291#102—1222no—
292C22⋊C8—21no—
293C4×C8—21no—
294C24—2441no—
295#41—21no—
296#41—21no—
297#41#1175819012yes—
298#41D48812yes—
299C2×Q8#1175819811yescharacteristic
300C2×D4—21no—
301C2×D4—1222no—
302C2×D4—1222no—
303C2×D4—1222no—
304C2×D4—4881no—
305C2×D4—4881no—
306C2×D4—4882no—
307C2×D4—2441no—
308C2×D4—2441no—
309C2×D4—2441no—
310C2×D4—2441no—
311C2×D4—2441no—
312C2×D4—2441no—
313C2×D4—2441no—
314C2×D4#1175819811yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4C2×D24411yescharacteristic
319C22×C4#1175822411yescharacteristic
320C22×C4#1175822411yescharacteristic
321SD16—2444no—
322D8—2444no—
323M16—2444no—
324M16—2441no—
325M16—2441no—
326M16—2441no—
327M16—2441no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—81no—
331C2×C8—1222no—
332C2×C8—1221no—
333C2×C8—1221no—
334C2×C8—1222no—
335C2×C8—2444no—
336C2×C8—2441no—
337C2×C8—2441no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4#1175819811yescharacteristic
341C4⋊C4#1175819811yescharacteristic
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42C2×C4×D6111yescharacteristic
349C42D4×D6111yescharacteristic
350C42D4×D6111yescharacteristic
351C42C2×C22⋊D6111yescharacteristic
352C23—21no—
353C23—4882no—
354C23—4882no—
355C23—4881no—
356C23—4881no—
357C23—2442no—
358C23#1451976111yescharacteristic
359Q8—21no—
360Q8#1451969912yes—
361D4—22no—
362D4—22no—
363D4—21no—
364D4—4882no—
365D4—4882no—
366D4—4882no—
367D4—4881no—
368D4—4881no—
369D4—4881no—
370D4—4881no—
371D4—2442no—
372D4—2442no—
373D4#1451969912yes—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—81no—
385C2×C4C2×#1175819811yescharacteristic
386C2×C4#1451973711yescharacteristic
387C2×C4#1451973711yescharacteristic
388C2×C4#1451974711yescharacteristic
389C2×C4#1451974911yescharacteristic
390C2×C4#1451974911yescharacteristic
391C2×C4C2×D48811yescharacteristic
392C2×C4C2×#1175819011yescharacteristic
393C2×C4#1451980511yescharacteristic
394C2×C4#1451981011yescharacteristic
395C8—81no—
396C8—81no—
397C8—2442no—
398C8—2441no—
399C8—2442no—
400C8—2441no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—9762no—
407C22—9762no—
408C22—4882no—
409C22—4882no—
410C22—4882no—
411C22#1683963911yescharacteristic
412C22#1684007611yescharacteristic
413C22#1684008111yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—22no—
418C4—41no—
419C4—41no—
420C4C2×#1451969911yescharacteristic
421C4#1683976411yescharacteristic
422C4#1683989811yescharacteristic
423C4#1683990311yescharacteristic
424C4#1684002711yescharacteristic
425C4#1684003011yescharacteristic
426C2—21no—
427C2—21no—
428C2—42no—
429C2—9762no—
430C2#2144613411yescharacteristic
431C2#2144629711yescharacteristic
432C2#2144632511yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1477C2
socleC22×C61#386
prime radical#21455299C2
cosocleC2×C42×C61C23
hypercenter#1477D61
hypocenterC61#9173
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1477

Complements

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

NH ≅ G/Ntype
2. #21454387C2semidirect
3. #21444486C2semidirect
4. C2×#16839505C2semidirect
5. #21443715C2semidirect
6. #21443694C2semidirect
7. C61×#1477C2semidirect
214. #1477D61semidirect
247. C61#9173semidirect

Maximal subgroups

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

subgroupindex
#214552992
#214543872
#214444862
C2×#168395052
#214437152
#214436942
C61×#14772
#917361