Finite Group Databasebeta

← #29583393

Subgroups — #29583393

Order 15616.

572 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450818C211yesmaximal, characteristic
2#21446625C211yesmaximal, characteristic
3#21446578C211yesmaximal, characteristic, complemented
4#21445824C211yesmaximal, characteristic, complemented
5C2×#16839505C211yesmaximal, characteristic
6#21443659C211yesmaximal, characteristic, complemented
7C2×C61×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×#14519774C2211yescharacteristic
9C2×#14519766—21no—
10C2×#14519711—21no—
11C2×C4⋊#92418C2211yescharacteristic
12C2×#14519701—21no—
13C2×#14519699—22no—
14#16839679—22no—
15#16839664—21no—
16#16839664—21no—
17#16839663—21no—
18#16839632C2211yescharacteristic
19C4×C4⋊Dic61C2211yescharacteristic
20#16839525—21no—
21#16839524—21no—
22#16839505—22no—
23C2×C4×Q8×C61C2211yescharacteristic
24C2×C4⋊C8×C61C2211yescharacteristic
25C2×C4×C8×C61C2211yescharacteristic
26C61×Q8⋊C8—22no—
27C22×D244—21no—
28C22×D244—21no—
29C22×D244D412yes—
30C22×#92418—21no—
31#14519774—21no—
32#14519774—21no—
33#14519774—21no—
34#14519774—21no—
35#14519766—42no—
36#14519766—42no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61D411yescharacteristic
41C2×C4⋊Dic61D411yescharacteristic
42C2×#11758198—22no—
43C2×#11758198—22no—
44C2×#11758198—22no—
45C2×#11758198—22no—
46C2×C4×Dic61—21no—
47#14519711—42no—
48C4⋊#92418—21no—
49C4⋊#92418—21no—
50#14519701—42no—
51#14519699—44no—
52#14519684—21no—
53C22×Q8×C61D411yescharacteristic
54C22×C8×C61—21no—
55C22×C8×C61—21no—
56C4×Q8×C61—21no—
57C4×Q8×C61—21no—
58C4×Q8×C61—21no—
59C4×Q8×C61—21no—
60C2×C4⋊C4×C61—21no—
61C2×C4⋊C4×C61D411yescharacteristic
62C2×C42×C61—21no—
63C2×C42×C61C2311yescharacteristic
64C4⋊C8×C61—21no—
65C4⋊C8×C61D412yes—
66C4×C8×C61—21no—
67C4×C8×C61—21no—
68C23×D61—22no—
69C2×Q8×C61—21no—
70C2×Q8×C61—21no—
71C2×Q8×C61—21no—
72C2×Q8×C61—21no—
73C2×Q8×C61—21no—
74C2×Q8×C61D812yes—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×Dic61—21no—
81C22×Dic61—21no—
82C22×Dic61—41no—
83C2×D244—22no—
84C2×D244—22no—
85C2×D244—42no—
86C2×D244—42no—
87C2×D244—41no—
88C2×D244—41no—
89C2×D244—41no—
90C2×D244—41no—
91C2×D244—41no—
92C2×D244—41no—
93C2×D244—41no—
94C2×D244—41no—
95C2×D244—41no—
96C2×D244—41no—
97C2×D244—41no—
98C2×D244—41no—
99C2×C8×C61—22no—
100C2×C8×C61—22no—
101C2×C8×C61—41no—
102C2×C8×C61—41no—
103C2×C8×C61—41no—
104C2×C8×C61—41no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—41no—
108C4⋊C4×C61—41no—
109C42×C61—41no—
110C42×C61—41no—
111C42×C61C2×D411yescharacteristic
112C42×C61#4111yescharacteristic
113C42×C61#4111yescharacteristic
114C42×C61#4111yescharacteristic
115#11758198—44no—
116#11758198—44no—
117#11758198—82no—
118#11758198—82no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61—21no—
122C4⋊Dic61—21no—
123C4⋊Dic61—42no—
124C4⋊Dic61—41no—
125C4⋊Dic61—41no—
126C4⋊Dic61—41no—
127C4⋊Dic61—41no—
128C4×Dic61—41no—
129C4×Dic61—41no—
130C2×#92418—41no—
131C2×#92418—41no—
132C2×#92418—41no—
133C23×C61#12011yescharacteristic
134C22×D61—42no—
135C22×D61—42no—
136C22×D61—42no—
137C22×D61—42no—
138C22×D61—42no—
139C22×D61—42no—
140C22×D61—42no—
141Q8×C61—22no—
142Q8×C61—22no—
143Q8×C61—41no—
144Q8×C61—41no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—22no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#12311yescharacteristic
157C2×C4×C61#12311yescharacteristic
158C2×C4×C61#12411yescharacteristic
159C2×C4×C61C2×D811yescharacteristic
160C2×C4×C61#13511yescharacteristic
161C2×C4×C61#13511yescharacteristic
162C2×C4×C61#13511yescharacteristic
163C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C61#13711yescharacteristic
166C2×C4×C61#13711yescharacteristic
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—81no—
174C2×Dic61—81no—
175C2×Dic61—81no—
176D244—44no—
177D244—82no—
178D244—81no—
179D244—81no—
180D244—81no—
181D244—81no—
182D244—81no—
183D244—81no—
184D244—81no—
185D244—81no—
186C8×C61—42no—
187C8×C61—42no—
188#92418—81no—
189#10411—611nomaximal, Sylow
190C22×C61#39411yescharacteristic
191C22×C61#44711yescharacteristic
192C22×C61#44911yescharacteristic
193C22×C61#45211yescharacteristic
194C22×C61#45211yescharacteristic
195C22×C61#45311yescharacteristic
196C22×C61#45511yescharacteristic
197C2×D61—82no—
198C2×D61—82no—
199C2×D61—82no—
200C2×D61—82no—
201C2×D61—82no—
202C2×D61—82no—
203C2×D61—82no—
204C4×C61—21no—
205C4×C61—21no—
206C4×C61—42no—
207C4×C61—81no—
208C4×C61#46311yescharacteristic
209C4×C61#46411yescharacteristic
210C4×C61#46611yescharacteristic
211C4×C61#46811yescharacteristic
212C4×C61#48011yescharacteristic
213C4×C61#48211yescharacteristic
214C4×C61#48811yescharacteristic
215C4×C61#48911yescharacteristic
216Dic61—81no—
217Dic61—81no—
218Dic61—161no—
219#2548—611no—
220#1957—611no—
221#1947—611no—
222#1803—611no—
223C2×#331—611no—
224C2×Q8⋊C8D6111yescharacteristic, complemented
225#1279—611no—
226C2×C61#162111yescharacteristic
227C2×C61#163011yescharacteristic
228C2×C61#164011yescharacteristic
229C2×C61#164111yescharacteristic
230C2×C61#203311yescharacteristic
231C2×C61#203711yescharacteristic
232C2×C61#204211yescharacteristic
233D61—162no—
234C2×#127—611no—
235C2×#124—1221no—
236C2×C4×Q8C2×D6111yescharacteristic
237C2×#107—1221no—
238C2×C4⋊C8—611no—
239C2×C4⋊C8C2×D6111yescharacteristic
240C2×#103—1221no—
241C2×#102—1222no—
242C2×C4×C8C2×D6111yescharacteristic
243#394—1222no—
244#390—1221no—
245#390—1221no—
246#390—1221no—
247#384—611no—
248C4×C4⋊C4—611no—
249#340—1221no—
250#340—1221no—
251#331—1222no—
252Q8⋊C8—22no—
253C61#1041111yescharacteristic, complemented, Sylow
254C22×Q8D24411yescharacteristic
255C22×D4—612no—
256C22×D4—1221no—
257C22×D4—1221no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—1221no—
261#127—1221no—
262#127—1221no—
263#127—1221no—
264#127—1221no—
265#124—2442no—
266#124—2442no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—611no—
273C2×C4⋊C4—611no—
274C2×C4⋊C4—1221no—
275C2×C4⋊C4—1221no—
276C2×C4⋊C4—1221no—
277C2×C4⋊C4D24411yescharacteristic
278C2×C22⋊C4—1222no—
279C2×C22⋊C4—1222no—
280C2×C22⋊C4—1222no—
281C2×C22⋊C4—1222no—
282C2×C42—21no—
283C2×C42—1221no—
284C2×C42C22×D6111yescharacteristic
285#107—2442no—
286C4⋊C8—21no—
287C4⋊C8—1221no—
288C4⋊C8—1221no—
289C4⋊C8C22⋊D6112yes—
290#103—2442no—
291#102—2444no—
292C4×C8—21no—
293C4×C8—21no—
294#95—1221no—
295C24—1222no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8D48812yes—
302C2×D4—1222no—
303C2×D4—1222no—
304C2×D4—2442no—
305C2×D4—2442no—
306C2×D4—2441no—
307C2×D4—2441no—
308C2×D4—2441no—
309C2×D4—2441no—
310C2×D4—2441no—
311C2×D4—2441no—
312C2×D4—2441no—
313C2×D4—2441no—
314C2×D4—2441no—
315C2×D4—2441no—
316C2×D4—2441no—
317C2×D4—2441no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—1221no—
321C22×C4—1221no—
322C22×C4—2441no—
323C22×C4C2×D24411yescharacteristic
324C22×C4D4×D6111yescharacteristic
325C22×C4D4×D6111yescharacteristic
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—2441no—
333C2×C8—2441no—
334C2×C8—2441no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—1221no—
340C4⋊C4—1221no—
341C4⋊C4—1221no—
342C4⋊C4—1221no—
343C4⋊C4—2442no—
344C4⋊C4—2441no—
345C4⋊C4—2441no—
346C4⋊C4—2441no—
347C4⋊C4—2441no—
348C22⋊C4—4882no—
349C22⋊C4—4882no—
350C22⋊C4—2444no—
351C22⋊C4—2444no—
352C42—41no—
353C42—41no—
354C42—2441no—
355C42—2441no—
356C42#1175822211yescharacteristic
357C42#1175822411yescharacteristic
358C42#1175822411yescharacteristic
359C42C2×C22⋊D6111yescharacteristic
360C23—2442no—
361C23—2442no—
362C23—2442no—
363C23—2442no—
364C23—2442no—
365C23—2442no—
366C23—2442no—
367C23#1451974711yescharacteristic
368Q8—22no—
369Q8—22no—
370Q8—41no—
371Q8—41no—
372D4—4882no—
373D4—4881no—
374D4—4881no—
375D4—4881no—
376D4—4881no—
377D4—4881no—
378D4—4881no—
379D4—4881no—
380D4—4881no—
381D4—2444no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—4881no—
391C2×C4—4881no—
392C2×C4—4881no—
393C2×C4—2441no—
394C2×C4—2441no—
395C2×C4—2441no—
396C2×C4—2441no—
397C2×C4—2441no—
398C2×C4—2441no—
399C2×C4#1451974811yescharacteristic
400C2×C4#1451974911yescharacteristic
401C2×C4#1451975111yescharacteristic
402C2×C4#1451975911yescharacteristic
403C2×C4#1451976111yescharacteristic
404C2×C4#1451976311yescharacteristic
405C2×C4C2×D48811yescharacteristic
406C2×C4#1451979811yescharacteristic
407C2×C4#1451979911yescharacteristic
408C2×C4#1451980011yescharacteristic
409C2×C4#1451980411yescharacteristic
410C2×C4#1451980511yescharacteristic
411C2×C4#1451980911yescharacteristic
412C2×C4#1451981011yescharacteristic
413C8—42no—
414C8—42no—
415C8—4881no—
416C22—4882no—
417C22—4882no—
418C22—4882no—
419C22—4882no—
420C22—4882no—
421C22—4882no—
422C22—4882no—
423C22#1683967811yescharacteristic
424C22#1683989811yescharacteristic
425C22#1683991011yescharacteristic
426C22#1683992611yescharacteristic
427C22#1683992711yescharacteristic
428C22#1683993111yescharacteristic
429C22#1683994111yescharacteristic
430C4—21no—
431C4—21no—
432C4—42no—
433C4—81no—
434C4—9761no—
435C4—4881no—
436C4—4881no—
437C4#1683997911yescharacteristic
438C4#1683998611yescharacteristic
439C4#1683999811yescharacteristic
440C4#1684000811yescharacteristic
441C4#1684007611yescharacteristic
442C4#1684008711yescharacteristic
443C4#1684011811yescharacteristic
444C4#1684012311yescharacteristic
445C2—9762no—
446C2#2144488611yescharacteristic
447C2#2144494211yescharacteristic
448C2#2144500811yescharacteristic
449C2#2144501411yescharacteristic
450C2#2144700211yescharacteristic
451C2#2144702811yescharacteristic
452C2#2144705711yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC2×C61×Q8⋊C8C2
socleC23×C61#120
prime radicalC2×#14519774C22
hypercenterC2×Q8⋊C8D61
hypocenterC61#10411
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. #21446578C2semidirect
4. #21445824C2semidirect
6. #21443659C2semidirect
7. C2×C61×Q8⋊C8C2semidirect
224. C2×Q8⋊C8D61semidirect
253. C61#10411semidirect

Maximal subgroups

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

subgroupindex
#214508182
#214466252
#214465782
#214458242
C2×#168395052
#214436592
C2×C61×Q8⋊C82
#1041161