Finite Group Databasebeta

← #29594997

Subgroups — #29594997

Order 15616.

746 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
1C22×#14519696C211yesmaximal, characteristic
2#21448405C211yesmaximal, characteristic
3#21443945C214yesmaximal, complemented
4C22×#101×C61C211yesmaximal, characteristic
5C22×#11758194—21no—
6C2×#14519736C411yescharacteristic
7C2×#14519736C411yescharacteristic
8C2×#14519696—21no—
9C2×#14519696—22no—
10C2×#14519696C414yescomplemented
11C2×#14519696C2212yes—
12#16839669—21no—
13#16839669—21no—
14#16839669C412yes—
15#16839669C2212yes—
16#16839625—21no—
17#16839625—21no—
18C23×Q8×C61C2211yescharacteristic
19C22×M16×C61—21no—
20C2×#101×C61—21no—
21C2×#101×C61—22no—
22C2×#101×C61C414yescomplemented
23C2×#101×C61C2212yes—
24C2×#11758194—24no—
25C2×#11758194—22no—
26C2×#11758194—42no—
27C2×#11758194—41no—
28C22×#92418—41no—
29#14519736—41no—
30#14519736—41no—
31#14519736—42no—
32C2×C4⋊Dic61—21no—
33C2×C4⋊Dic61C2×C412yes—
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×C4×Dic61—21no—
37C2×C4×Dic61C2×C412yes—
38#14519696—24no—
39#14519696—22no—
40#14519696C2×C414yes—
41#14519684—22no—
42#14519684—22no—
43#14519684—22no—
44#14519684—22no—
45C22×Q8×C61—41no—
46C22×Q8×C61—41no—
47C22×Q8×C61—42no—
48C22×Q8×C61C2×C412yes—
49C22×Q8×C61C2×C411yescharacteristic
50C22×Q8×C61C2×C412yes—
51C22×Q8×C61D411yescharacteristic
52C22×Q8×C61D412yes—
53C22×Q8×C61D411yescharacteristic
54C22×Q8×C61Q812yes—
55C22×Q8×C61C2311yescharacteristic
56C23×C4×C61C2×C411yescharacteristic
57C23×C4×C61D411yescharacteristic
58C23×C4×C61D411yescharacteristic
59C2×M16×C61—24no—
60C2×M16×C61—22no—
61C2×M16×C61—42no—
62C2×M16×C61—41no—
63C22×C8×C61—41no—
64#101×C61—24no—
65#101×C61—22no—
66#101×C61C2×C414yes—
67C24×C61C22⋊C411yescharacteristic
68C2×Q8×C61—42no—
69C2×Q8×C61—42no—
70C2×Q8×C61—42no—
71C2×Q8×C61—42no—
72C2×Q8×C61—41no—
73C2×Q8×C61—41no—
74C2×Q8×C61—42no—
75C2×Q8×C61—42no—
76C2×Q8×C61—41no—
77C2×Q8×C61—41no—
78C2×Q8×C61—41no—
79C2×Q8×C61—41no—
80C2×Q8×C61—42no—
81C2×Q8×C61—42no—
82C2×Q8×C61—41no—
83C2×Q8×C61—41no—
84C2×Q8×C61C4212yes—
85C2×Q8×C61C22⋊C412yes—
86C2×Q8×C61C22⋊C412yes—
87C2×Q8×C61C4⋊C412yes—
88C2×Q8×C61C4⋊C412yes—
89C2×Q8×C61C4⋊C412yes—
90C2×Q8×C61C22×C411yescharacteristic
91C2×Q8×C61C22×C411yescharacteristic
92C2×Q8×C61C2×D411yescharacteristic
93C2×Q8×C61C2×Q811yescharacteristic
94C22×C4×C61—21no—
95C22×C4×C61—22no—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—22no—
100C22×C4×C61—21no—
101C22×C4×C61—22no—
102C22×C4×C61—21no—
103C22×C4×C61C4212yes—
104C22×C4×C61C22⋊C411yescharacteristic
105C22×C4×C61C22⋊C411yescharacteristic
106C22×C4×C61C22⋊C411yescharacteristic
107C22×C4×C61C22⋊C412yes—
108C22×C4×C61C22⋊C412yes—
109C22×C4×C61C4⋊C412yes—
110C22×C4×C61C4⋊C412yes—
111C22×C4×C61C4⋊C412yes—
112C22×C4×C61C22×C411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×Dic61—22no—
116C22×Dic61—22no—
117M16×C61—24no—
118M16×C61—44no—
119M16×C61—42no—
120C2×C8×C61—44no—
121C2×C8×C61—42no—
122C22⋊Dic61—41no—
123C22⋊Dic61—41no—
124C22⋊Dic61—42no—
125C4⋊Dic61—42no—
126C4⋊Dic61—42no—
127C4×Dic61—42no—
128C4×Dic61—42no—
129#11758194—24no—
130#11758194—44no—
131#11758194—42no—
132C2×#92418—44no—
133C2×#92418—42no—
134C23×C61—42no—
135C23×C61#9512yes—
136C23×C61#9911yescharacteristic
137C23×C61#9911yescharacteristic
138C23×C61#9912yes—
139C23×C61C2×C22⋊C411yescharacteristic
140Q8×C61—42no—
141Q8×C61—42no—
142Q8×C61—42no—
143Q8×C61—42no—
144Q8×C61—42no—
145Q8×C61—42no—
146Q8×C61—41no—
147Q8×C61—41no—
148Q8×C61—41no—
149Q8×C61—41no—
150C2×C4×C61—22no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—22no—
154C2×C4×C61—22no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—22no—
159C2×C4×C61—22no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61#9512yes—
175C2×C4×C61#9512yes—
176C2×C4×C61#9512yes—
177C2×C4×C61C2×C4211yescharacteristic
178C2×C4×C61C2×C22⋊C411yescharacteristic
179C2×C4×C61C2×C22⋊C411yescharacteristic
180C2×C4×C61C2×C4⋊C411yescharacteristic
181C2×C4×C61C2×C4⋊C411yescharacteristic
182C2×C4×C61C2×C4⋊C411yescharacteristic
183C2×Dic61—44no—
184C2×Dic61—44no—
185C2×Dic61—42no—
186C2×Dic61—42no—
187C8×C61—44no—
188#92418—44no—
189#12586—611nomaximal, Sylow
190C22×C61—22no—
191C22×C61—22no—
192C22×C61—22no—
193C22×C61—42no—
194C22×C61—42no—
195C22×C61#34212yes—
196C22×C61#34212yes—
197C22×C61C2×#9511yescharacteristic
198C22×C61C2×#9911yescharacteristic
199C22×C61C2×#9911yescharacteristic
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—22no—
203C4×C61—22no—
204C4×C61—22no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—21no—
208C4×C61—21no—
209Dic61—82no—
210Dic61—82no—
211C22×#101—611no—
212C22×#101D6111yescharacteristic
213#2310—611no—
214#1394—614no—
215C2×C61—21no—
216C2×C61—22no—
217C2×C61—22no—
218C2×C61—21no—
219C2×C61C2×#34211yescharacteristic
220C2×C61#213312yes—
221C23×Q8C2×D6111yescharacteristic
222C22×M16—21no—
223C22×M16—1221no—
224C2×#117—611no—
225C2×#117—611no—
226C2×#101—21no—
227C2×#101—22no—
228C2×#101—614no—
229C2×#101—612no—
230C2×#101—1221no—
231C2×#101—1222no—
232C2×#101Dic6114yescomplemented
233C2×#101C2×D6112yes—
234#391—612no—
235#391—612no—
236#391—1221no—
237#391—1221no—
238#381—1221no—
239#381—1221no—
240C61#1258611yescharacteristic, complemented, Sylow
241C22×Q8—41no—
242C22×Q8—41no—
243C22×Q8—42no—
244C22×Q8Dic12212yes—
245C22×Q8C4×D6112yes—
246C22×Q8C4×D6111yescharacteristic
247C22×Q8D24411yescharacteristic
248C22×Q8C2×Dic6112yes—
249C22×Q8C22⋊D6112yes—
250C22×Q8C22⋊D6111yescharacteristic
251C22×Q8C22×D6111yescharacteristic
252C23×C4C4×D6111yescharacteristic
253C23×C4D24411yescharacteristic
254C23×C4C22⋊D6111yescharacteristic
255C2×M16—24no—
256C2×M16—22no—
257C2×M16—41no—
258C2×M16—42no—
259C2×M16—1224no—
260C2×M16—1222no—
261C2×M16—2441no—
262C2×M16—2442no—
263C22×C8—41no—
264C22×C8—2441no—
265#117—2442no—
266#117—2441no—
267#117—2441no—
268C2×C4⋊C4—612no—
269C2×C4⋊C4—1221no—
270C2×C22⋊C4—1221no—
271C2×C22⋊C4—1221no—
272C2×C42—612no—
273C2×C42—1221no—
274#101—24no—
275#101—22no—
276#101—614no—
277#101—1224no—
278#101—1222no—
279#101C2×Dic6114yes—
280#95—1222no—
281#95—1222no—
282#95—1222no—
283#95—1222no—
284C24#1175819811yescharacteristic
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8C4×Dic6112yes—
302C2×Q8#1175819612yes—
303C2×Q8#1175819612yes—
304C2×Q8C4⋊Dic6112yes—
305C2×Q8#1175819812yes—
306C2×Q8C22⋊Dic6112yes—
307C2×Q8C2×Dic12211yescharacteristic
308C2×Q8C2×C4×D6111yescharacteristic
309C2×Q8C22×Dic6111yescharacteristic
310C2×Q8C2×C22⋊D6111yescharacteristic
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—1222no—
321C22×C4—1222no—
322C22×C4C4×Dic6112yes—
323C22×C4#1175819612yes—
324C22×C4#1175819612yes—
325C22×C4C4⋊Dic6112yes—
326C22×C4#1175819811yescharacteristic
327C22×C4#1175819811yescharacteristic
328C22×C4#1175819811yescharacteristic
329C22×C4#1175819812yes—
330C22×C4C22⋊Dic6112yes—
331C22×C4C2×C4×D6111yescharacteristic
332C22×C4C2×D24411yescharacteristic
333C22×C4C2×C22⋊D6111yescharacteristic
334M16—24no—
335M16—42no—
336M16—44no—
337M16—1224no—
338M16—2442no—
339M16—2444no—
340C2×C8—42no—
341C2×C8—44no—
342C2×C8—2442no—
343C2×C8—2444no—
344C4⋊C4—2442no—
345C4⋊C4—2442no—
346C22⋊C4—2442no—
347C22⋊C4—2441no—
348C22⋊C4—2441no—
349C42—2442no—
350C42—2442no—
351C23—42no—
352C23#1451968412yes—
353C23#1451969111yescharacteristic
354C23#1451969111yescharacteristic
355C23#1451969112yes—
356C23C2×#1175819811yescharacteristic
357Q8—42no—
358Q8—42no—
359Q8—42no—
360Q8—42no—
361Q8—42no—
362Q8—42no—
363Q8—41no—
364Q8—41no—
365Q8—41no—
366Q8—41no—
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—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—2444no—
392C2×C4—2444no—
393C2×C4—2442no—
394C2×C4—2442no—
395C2×C4#1451968412yes—
396C2×C4#1451968412yes—
397C2×C4#1451968412yes—
398C2×C4C2×C4×Dic6111yescharacteristic
399C2×C4C2×#1175819811yescharacteristic
400C2×C4C2×C22⋊Dic6111yescharacteristic
401C2×C4C2×#1175819611yescharacteristic
402C2×C4C2×#1175819611yescharacteristic
403C2×C4C2×C4⋊Dic6111yescharacteristic
404C8—44no—
405C8—2444no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—42no—
410C22—42no—
411C22#1683952812yes—
412C22#1683952812yes—
413C22C2×#1451968411yescharacteristic
414C22C2×#1451969111yescharacteristic
415C22C2×#1451969111yescharacteristic
416C4—21no—
417C4—21no—
418C4—22no—
419C4—22no—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—4882no—
426C4—4882no—
427C2—21no—
428C2—22no—
429C2—22no—
430C2—21no—
431C2C2×#1683952811yescharacteristic
432C2#2144750612yes—
subgroupG/H
centerC22C2×#14519691
commutatorC2×C4×C61C2×C42
FrattiniC22×Q8C22×D61
FittingC22×#101×C61C2
socleC22×C61C2×#99
prime radicalC24×C61C22⋊C4
cosocleC22×Q8×C61C23
hypercenterC22×#101D61
hypocenterC61#12586
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ C22×#101

Complements

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

NH ≅ G/Ntype
3. #21443945C2semidirect
10. C2×#14519696C4semidirect
22. C2×#101×C61C4semidirect
232. C2×#101Dic61semidirect
240. C61#12586semidirect

Maximal subgroups

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

subgroupindex
C22×#145196962
#214484052
#214439452
C22×#101×C612
#1258661