Finite Group Databasebeta

← #29657835

Subgroups — #29657835

Order 15616.

930 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457037C212yesmaximal, complemented
2C2×#16840141C211yesmaximal, characteristic, complemented
3#21444964C212yesmaximal, complemented
4#21444959C212yesmaximal, complemented
5#21444953C214yesmaximal, complemented
6#21444545C211yesmaximal, characteristic, complemented
7#21444279C212yesmaximal, complemented
8C61×#3544C211yesmaximal, characteristic
9C22×#11758202—22no—
10C2×#14519777C2211yescharacteristic
11#16840172—22no—
12#16840141—22no—
13#16840141C2212yes—
14#16840141C2212yes—
15#16840121—22no—
16#16840090—22no—
17#16840073—22no—
18#16840057C2214yes—
19#16840052C2214yes—
20#16839819C2212yes—
21C2×C4×#92418C2211yescharacteristic
22#16839511C2212yes—
23#16839511C2212yes—
24Q8⋊#92418C2214yes—
25D4⋊#92418C2214yes—
26C61×#563—22no—
27Q82×C61C2211yescharacteristic
28C61×#555C2211yescharacteristic
29C61×#554C2212yes—
30C61×#552—22no—
31C61×#547—22no—
32C61×#544—22no—
33C61×#542—22no—
34C61×#536—22no—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C61×#529C2212yes—
37C22×Dic122—21no—
38C2×#11758202—24no—
39C2×#11758202—24no—
40C2×#11758202—48no—
41C22×#92418—21no—
42#14519777—22no—
43#14519777—21no—
44#14519777C2312yes—
45C2×C4⋊Dic61—22no—
46#14519710—24no—
47#14519708—24no—
48C4⋊#92418C2314yes—
49#14519701—24no—
50#14519698—24no—
51C22⋊#92418—22no—
52#14519688—22no—
53C4×#92418—21no—
54C4×#92418C2312yes—
55C2×#41×C61—22no—
56C22×Q8×C61D412yes—
57C4⋊Q8×C61—22no—
58C4⋊Q8×C61—22no—
59C4⋊Q8×C61—22no—
60C4⋊Q8×C61—22no—
61C4⋊Q8×C61D412yes—
62C4⋊Q8×C61C2311yescharacteristic
63C4⋊Q8×C61C2311yescharacteristic
64#126×C61—24no—
65#126×C61—22no—
66#125×C61—24no—
67#125×C61—22no—
68#125×C61D412yes—
69#124×C61—22no—
70#124×C61—22no—
71#124×C61D412yes—
72#123×C61—24no—
73#123×C61—24no—
74#123×C61—22no—
75C22⋊Q8×C61—24no—
76C22⋊Q8×C61—24no—
77C22⋊Q8×C61—24no—
78C22⋊Q8×C61—22no—
79C22⋊Q8×C61D412yes—
80#121×C61—22no—
81#121×C61D412yes—
82C4×Q8×C61—22no—
83C4×Q8×C61C2312yes—
84C4×D4×C61—22no—
85C4×D4×C61—22no—
86C4×D4×C61C2312yes—
87#117×C61—22no—
88#117×C61—22no—
89#117×C61—22no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61—22no—
92C2×C42×C61C2311yescharacteristic
93#41×C61—84no—
94C2×Q8×C61—22no—
95C2×Q8×C61—22no—
96C2×Q8×C61—44no—
97C2×Q8×C61C2×D412yes—
98C2×Q8×C61C2×D412yes—
99C2×Q8×C61C2×D412yes—
100C2×D4×C61—22no—
101C2×D4×C61—22no—
102C2×D4×C61C2×D412yes—
103C22×C4×C61—22no—
104C22×C4×C61—24no—
105C22×C4×C61—22no—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D412yes—
108C22×Dic61—41no—
109C2×Dic122—22no—
110C2×Dic122—44no—
111C2×Dic122—41no—
112C4⋊C4×C61—24no—
113C4⋊C4×C61—24no—
114C4⋊C4×C61—24no—
115C4⋊C4×C61—24no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61—22no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61C2×D412yes—
123C4⋊C4×C61C2×D412yes—
124C22⋊C4×C61—24no—
125C22⋊C4×C61—24no—
126C22⋊C4×C61—22no—
127C22⋊C4×C61—22no—
128C22⋊C4×C61—22no—
129C22⋊C4×C61—22no—
130C22⋊C4×C61—22no—
131C42×C61—22no—
132C42×C61—22no—
133C42×C61—22no—
134C42×C61C2×D412yes—
135C42×C61C2×D411yescharacteristic
136C42×C61C2411yescharacteristic
137#11758202—48no—
138#11758202—48no—
139C4⋊Dic61—24no—
140C4⋊Dic61—42no—
141C2×#92418—24no—
142C2×#92418—22no—
143C2×#92418—24no—
144C23×C61—22no—
145C23×C61#12011yescharacteristic
146Q8×C61—24no—
147Q8×C61—42no—
148Q8×C61—42no—
149Q8×C61—82no—
150Q8×C61—82no—
151D4×C61—24no—
152D4×C61—44no—
153D4×C61—82no—
154C2×C4×C61—22no—
155C2×C4×C61—24no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—22no—
159C2×C4×C61—22no—
160C2×C4×C61—22no—
161C2×C4×C61—44no—
162C2×C4×C61—44no—
163C2×C4×C61—84no—
164C2×C4×C61#12012yes—
165C2×C4×C61#12011yescharacteristic
166C2×C4×C61C22×D411yescharacteristic
167C2×C4×C61C22×D412yes—
168C2×Dic61—42no—
169C2×Dic61—82no—
170Dic122—44no—
171Dic122—82no—
172#92418—24no—
173#92418—44no—
174#18685—611nomaximal, Sylow
175C22×C61—22no—
176C22×C61—42no—
177C22×C61—42no—
178C22×C61—42no—
179C22×C61#45312yes—
180C22×C61C2×#12011yescharacteristic
181C4×C61—22no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—42no—
185C4×C61—42no—
186C4×C61—42no—
187C4×C61—42no—
188C4×C61—42no—
189C4×C61#57814yes—
190Dic61—82no—
191#3544D6111yescharacteristic
192#3238—612no—
193C2×#494—611no—
194#1634—612no—
195#1633—612no—
196#1632—614no—
197#1558—611no—
198#1494—612no—
199C2×C61—22no—
200C2×C61—82no—
201C2×C61#301112yes—
202C2×C61C2×#45311yescharacteristic
203C22×Q16—1222no—
204#563—22no—
205Q82C2×D6111yescharacteristic
206#555C2×D6111yescharacteristic
207#554C2×D6112yes—
208#552—22no—
209#547—22no—
210#544—22no—
211#542—22no—
212#536—22no—
213C2×C4⋊Q8—611no—
214C2×C4⋊Q8C2×D6111yescharacteristic
215#529C2×D6112yes—
216#501—1222no—
217#494—612no—
218#494—612no—
219#494—1222no—
220#489—1222no—
221#483—1222no—
222#480—1222no—
223#477—614no—
224#476—614no—
225#432—612no—
226C2×C4×C8—611no—
227#333—612no—
228#333—612no—
229Q8⋊C8—614no—
230D4⋊C8—614no—
231C61#1868511yescharacteristic, complemented, Sylow
232C2×#41—22no—
233C22×Q8—1221no—
234C22×Q8C22⋊D6112yes—
235C2×Q16—1224no—
236C2×Q16—1224no—
237C2×Q16—2448no—
238C22×C8—1221no—
239C4⋊Q8—22no—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243C4⋊Q8—612no—
244C4⋊Q8—1222no—
245C4⋊Q8—1221no—
246C4⋊Q8C22⋊D6112yes—
247C4⋊Q8C22×D6111yescharacteristic
248C4⋊Q8C22×D6111yescharacteristic
249#126—24no—
250#126—22no—
251#125—24no—
252#125—22no—
253#125C22⋊D6112yes—
254#124—22no—
255#124—22no—
256#124C22⋊D6112yes—
257#123—24no—
258#123—24no—
259#123—22no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263C22⋊Q8—22no—
264C22⋊Q8C22⋊D6112yes—
265#121—22no—
266#121C22⋊D6112yes—
267C4×Q8—22no—
268C4×Q8C22×D6112yes—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4C22×D6112yes—
272#117—22no—
273#117—22no—
274#117—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—1222no—
278C2×C42C22×D6111yescharacteristic
279#107—1224no—
280#106—1224no—
281C4⋊C8—614no—
282#103—1224no—
283#102—1224no—
284C22⋊C8—1222no—
285#97—1222no—
286C4×C8—612no—
287C4×C8—1221no—
288#41—84no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—44no—
292C2×Q8—1222no—
293C2×Q8—2444no—
294C2×Q8—2441no—
295C2×Q8C2×C22⋊D6112yes—
296C2×Q8C2×C22⋊D6112yes—
297C2×Q8C2×C22⋊D6112yes—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4C2×C22⋊D6112yes—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—2441no—
305C22×C4C2×C22⋊D6112yes—
306C22×C4C2×C22⋊D6111yescharacteristic
307Q16—2448no—
308Q16—2448no—
309C2×C8—1224no—
310C2×C8—1222no—
311C2×C8—1224no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—1224no—
323C4⋊C4—2442no—
324C4⋊C4C2×C22⋊D6112yes—
325C4⋊C4C2×C22⋊D6112yes—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42C2×C22⋊D6111yescharacteristic
337C42C2×C22⋊D6112yes—
338C42C23×D6111yescharacteristic
339C23—22no—
340C23#1451974611yescharacteristic
341Q8—24no—
342Q8—42no—
343Q8—42no—
344Q8—82no—
345Q8—82no—
346Q8—4882no—
347Q8—2444no—
348D4—24no—
349D4—44no—
350D4—82no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—84no—
361C2×C4—4882no—
362C2×C4—2442no—
363C2×C4#1451974612yes—
364C2×C4#1451974611yescharacteristic
365C2×C4C22×C22⋊D6111yescharacteristic
366C2×C4C22×C22⋊D6112yes—
367C8—1224no—
368C8—2444no—
369C22—22no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22#1683993012yes—
374C22C2×#1451974611yescharacteristic
375C4—22no—
376C4—21no—
377C4—21no—
378C4—42no—
379C4—42no—
380C4—42no—
381C4—42no—
382C4—42no—
383C4—4882no—
384C4#1684057714yes—
385C2—22no—
386C2—82no—
387C2#2145511412yes—
388C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3544C2
socleC22×C61C2×#120
hypercenter#3544D61
hypocenterC61#18685
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3544

Complements

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

NH ≅ G/Ntype
1. #21457037C2semidirect
2. C2×#16840141C2semidirect
3. #21444964C2semidirect
4. #21444959C2semidirect
5. #21444953C2semidirect
6. #21444545C2semidirect
7. #21444279C2semidirect
231. C61#18685semidirect

Maximal subgroups

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

subgroupindex
#214570372
C2×#168401412
#214449642
#214449592
#214449532
#214445452
#214442792
C61×#35442
#1868561