Finite Group Databasebeta

← #29574124

Subgroups — #29574124

Order 15616.

464 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21446122C211yesmaximal, characteristic
2#21446023C211yesmaximal, characteristic, complemented
3#21446001C211yesmaximal, characteristic, complemented
4#21444310C211yesmaximal, characteristic
5#21443704C211yesmaximal, characteristic, complemented
6#21443666C211yesmaximal, characteristic, complemented
7C61×#2976C211yesmaximal, characteristic
8C22×#11758194C2211yescharacteristic
9C2×#14519712—21no—
10C2×#14519706—21no—
11C2×#14519704—21no—
12C2×C22⋊#92418C2211yescharacteristic
13C4×C22⋊Dic61C2211yescharacteristic
14C4×C22⋊Dic61C2211yescharacteristic
15#16839526—21no—
16#16839521—21no—
17#16839516—21no—
18#16839499C411yescharacteristic, complemented
19#16839499C411yescharacteristic, complemented
20#16839496C411yescharacteristic, complemented
21#16839496C411yescharacteristic, complemented
22C61×#525C2211yescharacteristic
23C61×#436—21no—
24C61×#435—21no—
25C61×#435—21no—
26C8×D4×C61—21no—
27C61×#433—21no—
28C61×#408—21no—
29C61×#407C2211yescharacteristic
30C2×C22⋊C8×C61C2211yescharacteristic
31C2×#11758194—21no—
32C2×#11758194—21no—
33C2×#11758194—42no—
34C22×#92418—21no—
35C22×#92418—21no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—21no—
38C2×C4×Dic61—21no—
39C2×C4×Dic61—21no—
40#14519712—42no—
41#14519706—42no—
42#14519704—42no—
43C22⋊#92418—21no—
44C22⋊#92418C2×C411yescharacteristic
45C22⋊#92418C2×C411yescharacteristic
46#14519684—21no—
47#14519684C2×C411yescharacteristic
48#14519684C2×C411yescharacteristic
49C2×#41×C61D411yescharacteristic
50C23×C4×C61C2311yescharacteristic
51C2×M16×C61—21no—
52C2×M16×C61D411yescharacteristic
53C2×M16×C61D411yescharacteristic
54C22×C8×C61—21no—
55C22×C8×C61D411yescharacteristic
56C22×C8×C61D411yescharacteristic
57#123×C61—21no—
58C22⋊Q8×C61C2×C411yescharacteristic
59C22⋊Q8×C61Q811yescharacteristic
60#121×C61C2×C411yescharacteristic
61#121×C61Q811yescharacteristic
62#120×C61—21no—
63C4×D4×C61—21no—
64C4×D4×C61—21no—
65#117×C61D411yescharacteristic
66C4⋊C8×C61—21no—
67C4⋊C8×C61—21no—
68C22⋊C8×C61—21no—
69C22⋊C8×C61—21no—
70C22⋊C8×C61—21no—
71C22⋊C8×C61—21no—
72C22⋊C8×C61—21no—
73C22⋊C8×C61—21no—
74#97×C61—21no—
75C4×C8×C61—21no—
76C24×C61#4111yescharacteristic
77#41×C61—41no—
78#41×C61—41no—
79C2×Q8×C61C4⋊C411yescharacteristic
80C2×D4×C61—21no—
81C2×D4×C61—21no—
82C2×D4×C61—21no—
83C2×D4×C61C4⋊C411yescharacteristic
84C22×C4×C61—21no—
85C22×C4×C61—41no—
86C22×C4×C61—41no—
87C22×C4×C61C22×C411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×Q811yescharacteristic
92C22×C4×C61#4111yescharacteristic
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61—41no—
96M16×C61—41no—
97M16×C61—41no—
98M16×C61—81no—
99C2×C8×C61—21no—
100C2×C8×C61—21no—
101C2×C8×C61—21no—
102C2×C8×C61—21no—
103C2×C8×C61—41no—
104C2×C8×C61—41no—
105C2×C8×C61—81no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61C4⋊C411yescharacteristic
109C4⋊C4×C61C4⋊C411yescharacteristic
110C22⋊C4×C61—21no—
111C22⋊C4×C61—21no—
112C22⋊C4×C61—21no—
113C22⋊C4×C61—21no—
114C22⋊C4×C61—21no—
115C42×C61—21no—
116C22⋊Dic61—22no—
117C22⋊Dic61—42no—
118C22⋊Dic61—41no—
119C4×Dic61—41no—
120C4×Dic61—41no—
121C4×Dic61—41no—
122C4×Dic61—41no—
123#11758194—22no—
124#11758194—22no—
125#11758194—41no—
126#11758194—41no—
127C2×#92418—22no—
128C2×#92418—21no—
129C2×#92418—21no—
130C2×#92418—41no—
131C2×#92418—41no—
132C2×#92418—41no—
133C2×#92418—41no—
134C23×C61—21no—
135C23×C61—21no—
136C23×C61—21no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61C4×D411yescharacteristic
140C23×C61C22⋊Q811yescharacteristic
141C23×C61#12311yescharacteristic
142Q8×C61—41no—
143D4×C61—41no—
144D4×C61—41no—
145D4×C61—41no—
146D4×C61—81no—
147D4×C61—81no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61C2×C4⋊C411yescharacteristic
163C2×C4×C61C4×D411yescharacteristic
164C2×C4×C61#12011yescharacteristic
165C2×C4×C61C22⋊Q811yescharacteristic
166C2×Dic61—22no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—42no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—81no—
174C2×Dic61—81no—
175C8×C61—41no—
176C8×C61—41no—
177C8×C61—41no—
178C8×C61—41no—
179#92418—41no—
180#92418—41no—
181#92418—42no—
182#8875—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—41no—
192C22×C61—41no—
193C22×C61—41no—
194C22×C61—81no—
195C22×C61—81no—
196C22×C61#38511yescharacteristic
197C22×C61#44311yescharacteristic
198C22×C61#44411yescharacteristic
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—41no—
203C4×C61—41no—
204C4×C61—41no—
205C4×C61B(4,𝔽2)11yescharacteristic
206C4×C61#45811yescharacteristic
207Dic61—42no—
208Dic61—81no—
209Dic61—82no—
210#2976D6111yescharacteristic
211#1871—611no—
212#1855—611no—
213#1851—611no—
214#1501—611no—
215#1297—611no—
216#1282—611no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—41no—
220C2×C61—41no—
221C2×C61—81no—
222C2×C61#186111yescharacteristic
223C2×C61#186611yescharacteristic
224C2×C61#186711yescharacteristic
225C22×M16—611no—
226#525C2×D6111yescharacteristic
227#436—21no—
228#435—21no—
229#435—21no—
230C8×D4—21no—
231#433—21no—
232C2×#108—1221no—
233C2×C4⋊C8—1221no—
234C2×#104—1221no—
235#408—21no—
236#407C2×D6111yescharacteristic
237C2×C22⋊C8—611no—
238C2×C22⋊C8C2×D6111yescharacteristic
239C4×C22⋊C4—611no—
240C4×C22⋊C4—611no—
241#341—1221no—
242#339—1221no—
243#336—1221no—
244#328—611no—
245#328—611no—
246#327—611no—
247#327—611no—
248C61#887511yescharacteristic, complemented, Sylow
249C2×#41C22⋊D6111yescharacteristic
250C23×C4C22×D6111yescharacteristic
251C2×M16—21no—
252C2×M16—1221no—
253C2×M16—1221no—
254C2×M16—2442no—
255C2×M16C22⋊D6111yescharacteristic
256C2×M16C22⋊D6111yescharacteristic
257C22×C8—21no—
258C22×C8—1221no—
259C22×C8—1221no—
260C22×C8C22⋊D6111yescharacteristic
261C22×C8C22⋊D6111yescharacteristic
262#123—21no—
263C22⋊Q8Dic12211yescharacteristic
264C22⋊Q8C4×D6111yescharacteristic
265#121Dic12211yescharacteristic
266#121C4×D6111yescharacteristic
267#120—21no—
268C4×D4—21no—
269C4×D4—21no—
270#117C22⋊D6111yescharacteristic
271C2×C22⋊C4—1221no—
272C2×C22⋊C4—1221no—
273C2×C42—1221no—
274C2×C42—1221no—
275#108—2442no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—2442no—
279#104—2442no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—611no—
287C22⋊C8—611no—
288C22⋊C8—1221no—
289#97—21no—
290C4×C8—21no—
291#95—611no—
292#95—611no—
293#95—1221no—
294C24#1175822211yescharacteristic
295#41—41no—
296#41—41no—
297C2×Q8#1175819611yescharacteristic
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4#1175819611yescharacteristic
302C22×C4—21no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—1221no—
306C22×C4—1221no—
307C22×C4—2441no—
308C22×C4C2×Dic12211yescharacteristic
309C22×C4C2×C4×D6111yescharacteristic
310C22×C4#1175822211yescharacteristic
311C22×C4C2×C22⋊D6111yescharacteristic
312C22×C4C2×C22⋊D6111yescharacteristic
313C22×C4C2×C22⋊D6111yescharacteristic
314M16—41no—
315M16—41no—
316M16—81no—
317M16—1222no—
318M16—1222no—
319M16—2441no—
320M16—2441no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—81no—
328C2×C8—1222no—
329C2×C8—1221no—
330C2×C8—1221no—
331C2×C8—2441no—
332C2×C8—2441no—
333C2×C8—2441no—
334C2×C8—2441no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4#1175819611yescharacteristic
338C4⋊C4#1175819611yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—1222no—
345C22⋊C4—2442no—
346C22⋊C4—2441no—
347C42—21no—
348C42—2441no—
349C42—2441no—
350C42—2441no—
351C42—2441no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—41no—
357C23C4×C22⋊D6111yescharacteristic
358C23#1451975611yescharacteristic
359C23#1451976011yescharacteristic
360Q8—41no—
361D4—41no—
362D4—41no—
363D4—41no—
364D4—81no—
365D4—81no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—1222no—
381C2×C4—4881no—
382C2×C4—4881no—
383C2×C4—2441no—
384C2×C4—2442no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4C2×#1175819611yescharacteristic
390C2×C4C4×C22⋊D6111yescharacteristic
391C2×C4#1451974611yescharacteristic
392C2×C4#1451975611yescharacteristic
393C8—41no—
394C8—41no—
395C8—41no—
396C8—41no—
397C8—2441no—
398C8—2441no—
399C8—2442no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—81no—
412C22—81no—
413C22#1683963511yescharacteristic
414C22#1683987711yescharacteristic
415C22#1683988211yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—4881no—
423C4—4882no—
424C4—2442no—
425C4#1683994911yescharacteristic
426C4#1683995211yescharacteristic
427C2—21no—
428C2—21no—
429C2—41no—
430C2—41no—
431C2—81no—
432C2#2144605911yescharacteristic
433C2#2144609411yescharacteristic
434C2#2144610011yescharacteristic
subgroupG/H
centerC2×C4#14519746
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2976C2
socleC22×C61#385
prime radicalC61×#525C22
cosocleC23×C4×C61C23
hypercenter#2976D61
hypocenterC61#8875
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2976

Complements

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

NH ≅ G/Ntype
2. #21446023C2semidirect
3. #21446001C2semidirect
5. #21443704C2semidirect
6. #21443666C2semidirect
18. #16839499C4semidirect
19. #16839499C4semidirect
20. #16839496C4semidirect
21. #16839496C4semidirect
248. C61#8875semidirect

Maximal subgroups

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

subgroupindex
#214461222
#214460232
#214460012
#214443102
#214437042
#214436662
C61×#29762
#887561