Finite Group Databasebeta

← #29572741

Subgroups — #29572741

Order 15616.

658 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21445799C211yesmaximal, characteristic
2C2×#16839499C212yesmaximal
3#21444049C211yesmaximal, characteristic
4#21443697C212yesmaximal
5C4×C22⋊Q8×C61C211yesmaximal, characteristic
6C22×C4⋊Dic61—21no—
7C2×C4⋊#92418—21no—
8C2×C22⋊#92418C2212yes—
9C2×#14519684—21no—
10C2×#14519684C2212yes—
11#16839517—21no—
12#16839499—24no—
13C2×C22⋊Q8×C61C2211yescharacteristic
14C2×C4×Q8×C61C411yescharacteristic, complemented
15C22×C42×C61C2211yescharacteristic
16C61×#391C411yescharacteristic, complemented
17C61×#389C411yescharacteristic, complemented
18C61×#387—21no—
19C61×#385—21no—
20C61×#380C2211yescharacteristic
21C4×C4⋊C4×C61—21no—
22C4×C4⋊C4×C61C411yescharacteristic, complemented
23C4×C22⋊C4×C61—21no—
24C23×Dic61—22no—
25C22×#92418—22no—
26C2×C4⋊Dic61—24no—
27C2×C4⋊Dic61—42no—
28C2×C4⋊Dic61—42no—
29C4⋊#92418—42no—
30C22⋊#92418—22no—
31C22⋊#92418—22no—
32#14519684—22no—
33#14519684—22no—
34#14519684—42no—
35C22×Q8×C61C2×C411yescharacteristic
36C23×C4×C61D411yescharacteristic
37C23×C4×C61D411yescharacteristic
38C23×C4×C61C2311yescharacteristic
39C22⋊Q8×C61—21no—
40C22⋊Q8×C61—21no—
41C22⋊Q8×C61D412yes—
42C22⋊Q8×C61Q812yes—
43C4×Q8×C61—42no—
44C2×C4⋊C4×C61—21no—
45C2×C4⋊C4×C61—21no—
46C2×C4⋊C4×C61—21no—
47C2×C4⋊C4×C61C2×C411yescharacteristic
48C2×C4⋊C4×C61C2×C411yescharacteristic
49C2×C4⋊C4×C61C2×C411yescharacteristic
50C2×C22⋊C4×C61—21no—
51C2×C22⋊C4×C61—21no—
52C2×C42×C61—21no—
53C2×C42×C61—21no—
54C2×C42×C61—42no—
55C2×C42×C61C2×C411yescharacteristic
56C2×C42×C61C2×C411yescharacteristic
57C2×C42×C61D411yescharacteristic
58C2×C42×C61D411yescharacteristic
59#95×C61—21no—
60#95×C61—21no—
61#95×C61—21no—
62#95×C61—21no—
63#95×C61—21no—
64C24×C61C2×D411yescharacteristic
65C2×Q8×C61—21no—
66C2×Q8×C61—41no—
67C2×Q8×C61—41no—
68C2×Q8×C61C4⋊C412yescomplemented
69C22×C4×C61—22no—
70C22×C4×C61—22no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—41no—
78C22×C4×C61—41no—
79C22×C4×C61—41no—
80C22×C4×C61—41no—
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22⋊C411yescharacteristic
84C22×C4×C61C22⋊C411yescharacteristic
85C22×C4×C61SD1612yes—
86C22×C4×C61Q1612yes—
87C22×C4×C61C22×C411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×Q811yescharacteristic
91C22×C4×C61#4111yescharacteristic
92C22×C4×C61#4111yescharacteristic
93C22×Dic61—24no—
94C22×Dic61—42no—
95C22×Dic61—42no—
96C22×Dic61—42no—
97C22×Dic61—42no—
98C22×Dic61—42no—
99C22×Dic61—42no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—22no—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—42no—
104C4⋊C4×C61—42no—
105C4⋊C4×C61—41no—
106C4⋊C4×C61—41no—
107C4⋊C4×C61C4⋊C412yescomplemented
108C22⋊C4×C61—22no—
109C22⋊C4×C61—22no—
110C22⋊C4×C61—41no—
111C22⋊C4×C61—41no—
112C42×C61—22no—
113C42×C61—22no—
114C42×C61—41no—
115C42×C61—41no—
116C42×C61—42no—
117C42×C61—42no—
118C4⋊Dic61—48no—
119C2×#92418—42no—
120C2×#92418—42no—
121C2×#92418—42no—
122C23×C61—21no—
123C23×C61—21no—
124C23×C61—21no—
125C23×C61—21no—
126C23×C61C2×C22⋊C411yescharacteristic
127C23×C61#12111yescharacteristic
128C23×C61C22⋊Q811yescharacteristic
129C23×C61C2×SD1611yescharacteristic
130C23×C61C2×Q1611yescharacteristic
131C23×C61AGL(1,ℤ/8)11yescharacteristic
132C23×C61#13711yescharacteristic
133Q8×C61—42no—
134Q8×C61—42no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—24no—
142C2×C4×C61—24no—
143C2×C4×C61—22no—
144C2×C4×C61—22no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—42no—
156C2×C4×C61—42no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61#9912yes—
168C2×C4×C61#10312yes—
169C2×C4×C61#10312yes—
170C2×C4×C61#10412yes—
171C2×C4×C61C2×C4⋊C411yescharacteristic
172C2×C4×C61#11711yescharacteristic
173C2×C4×C61#12111yescharacteristic
174C2×C4×C61C22⋊Q811yescharacteristic
175C2×Dic61—44no—
176C2×Dic61—44no—
177C2×Dic61—44no—
178C2×Dic61—82no—
179C2×Dic61—82no—
180C2×Dic61—82no—
181C2×Dic61—82no—
182#92418—82no—
183#8572—611nomaximal, Sylow
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—21no—
196C22×C61#38011yescharacteristic
197C22×C61C2×#9911yescharacteristic
198C22×C61#41011yescharacteristic
199C22×C61C2×#10311yescharacteristic
200C22×C61#41711yescharacteristic
201C22×C61C2×#10411yescharacteristic
202C22×C61#42111yescharacteristic
203C22×C61#46011yescharacteristic
204C22×C61#46211yescharacteristic
205C22×C61#47511yescharacteristic
206C22×C61#47711yescharacteristic
207C4×C61—21no—
208C4×C61—21no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—22no—
212C4×C61—42no—
213C4×C61—42no—
214C4×C61—41no—
215C4×C61—41no—
216C4×C61—81no—
217C4×C61—81no—
218C4×C61#32814yescomplemented
219Dic61—84no—
220C4×C22⋊Q8D6111yescharacteristic
221#1797—611no—
222C2×#328—612no—
223C4⋊#95—611no—
224#1294—612no—
225C2×C61—21no—
226C2×C61—21no—
227C2×C61—21no—
228C2×C61—21no—
229C2×C61C2×#32811yescharacteristic
230C2×C61#150911yescharacteristic
231C2×C61#151311yescharacteristic
232C2×C61#151811yescharacteristic
233C2×C61#179911yescharacteristic
234C2×C61#180311yescharacteristic
235C2×C61#180611yescharacteristic
236C2×C22⋊Q8C2×D6111yescharacteristic
237C2×C4×Q8Dic6111yescharacteristic, complemented
238C22×C4⋊C4—1221no—
239C22×C42C2×D6111yescharacteristic
240C2×C4⋊C8—1221no—
241C2×C22⋊C8—612no—
242#391Dic6111yescharacteristic, complemented
243#389Dic6111yescharacteristic, complemented
244#387—21no—
245#385—21no—
246#380C2×D6111yescharacteristic
247C4×C4⋊C4—21no—
248C4×C4⋊C4Dic6111yescharacteristic, complemented
249C4×C22⋊C4—21no—
250C2×#95—612no—
251C2×#95—1221no—
252#336—1221no—
253#328—1224no—
254C61#857211yescharacteristic, complemented, Sylow
255C22×Q8C2×Dic6111yescharacteristic
256C23×C4—1222no—
257C23×C4C22⋊D6111yescharacteristic
258C23×C4C22⋊D6111yescharacteristic
259C23×C4C22×D6111yescharacteristic
260C22×C8—1222no—
261C22⋊Q8—21no—
262C22⋊Q8—21no—
263C22⋊Q8Dic12212yes—
264C22⋊Q8D24412yes—
265C4×Q8—42no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—1224no—
270C2×C4⋊C4—2442no—
271C2×C4⋊C4—2442no—
272C2×C4⋊C4C2×Dic6111yescharacteristic
273C2×C4⋊C4C2×Dic6111yescharacteristic
274C2×C4⋊C4C2×Dic6111yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—42no—
280C2×C42C2×Dic6111yescharacteristic
281C2×C42C2×Dic6111yescharacteristic
282C2×C42C22⋊D6111yescharacteristic
283C2×C42C22⋊D6111yescharacteristic
284C4⋊C8—2442no—
285C22⋊C8—1222no—
286C22⋊C8—1222no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—1222no—
293#95—1222no—
294#95—2442no—
295C24C2×C22⋊D6111yescharacteristic
296C2×Q8—21no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8C4⋊Dic6112yescomplemented
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—1224no—
313C22×C4—2442no—
314C22×C4—2442no—
315C22×C4—2442no—
316C22×C4—2442no—
317C22×C4—2442no—
318C22×C4—2442no—
319C22×C4#1175820112yes—
320C22×C4#1175820212yes—
321C22×C4C22⋊Dic6111yescharacteristic
322C22×C4C22⋊Dic6111yescharacteristic
323C22×C4C22⋊Dic6111yescharacteristic
324C22×C4C22⋊Dic6111yescharacteristic
325C22×C4C2×Dic12211yescharacteristic
326C22×C4C2×D24411yescharacteristic
327C22×C4#1175822211yescharacteristic
328C22×C4#1175822211yescharacteristic
329C22×C4C22×Dic6111yescharacteristic
330C22×C4C2×C22⋊D6111yescharacteristic
331C2×C8—2442no—
332C2×C8—2442no—
333C2×C8—2442no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—2448no—
342C4⋊C4C4⋊Dic6112yescomplemented
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C42—22no—
348C42—22no—
349C42—41no—
350C42—41no—
351C42—42no—
352C42—42no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23C2×C22⋊Dic6111yescharacteristic
358C23#1451975111yescharacteristic
359C23#1451975611yescharacteristic
360C23C2×#1175820111yescharacteristic
361C23C2×#1175820211yescharacteristic
362C23#1451980211yescharacteristic
363C23#1451980711yescharacteristic
364Q8—42no—
365Q8—42no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—4882no—
399C2×C4—4882no—
400C2×C4—4882no—
401C2×C4—4882no—
402C2×C4—2444no—
403C2×C4—2444no—
404C2×C4—2444no—
405C2×C4#1451969212yes—
406C2×C4#1451970112yes—
407C2×C4#1451970112yes—
408C2×C4#1451970412yes—
409C2×C4C2×C4⋊Dic6111yescharacteristic
410C2×C4#1451973611yescharacteristic
411C2×C4#1451975111yescharacteristic
412C2×C4#1451975611yescharacteristic
413C8—4882no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22#1683962211yescharacteristic
427C22C2×#1451969211yescharacteristic
428C22#1683973611yescharacteristic
429C22C2×#1451970111yescharacteristic
430C22#1683976311yescharacteristic
431C22C2×#1451970411yescharacteristic
432C22#1683977911yescharacteristic
433C22#1683996211yescharacteristic
434C22#1683997211yescharacteristic
435C22#1684004711yescharacteristic
436C22#1684005711yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—22no—
442C4—41no—
443C4—41no—
444C4—42no—
445C4—42no—
446C4—81no—
447C4—81no—
448C4—4884no—
449C4#1683949914yescomplemented
450C2—21no—
451C2—21no—
452C2—21no—
453C2—21no—
454C2C2×#1683949911yescharacteristic
455C2#2144434411yescharacteristic
456C2#2144436211yescharacteristic
457C2#2144438311yescharacteristic
458C2#2144580711yescharacteristic
459C2#2144582411yescharacteristic
460C2#2144583911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic61
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC4×C22⋊Q8×C61C2
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenterC4×C22⋊Q8D61
hypocenterC61#8572
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
14. C2×C4×Q8×C61C4semidirect
16. C61×#391C4semidirect
17. C61×#389C4semidirect
22. C4×C4⋊C4×C61C4semidirect
68. C2×Q8×C61C4⋊C4semidirect
107. C4⋊C4×C61C4⋊C4semidirect
218. C4×C61#328semidirect
237. C2×C4×Q8Dic61semidirect
242. #391Dic61semidirect
243. #389Dic61semidirect
248. C4×C4⋊C4Dic61semidirect
254. C61#8572semidirect
299. C2×Q8C4⋊Dic61semidirect
342. C4⋊C4C4⋊Dic61semidirect
449. C4#16839499semidirect

Maximal subgroups

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

subgroupindex
#214457992
C2×#168394992
#214440492
#214436972
C4×C22⋊Q8×C612
#857261