Finite Group Databasebeta

← #29663975

Subgroups — #29663975

Order 15616.

670 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840112C211yesmaximal, characteristic, complemented
2#21455628C211yesmaximal, characteristic
3#21455307C211yesmaximal, characteristic
4#21454775C211yesmaximal, characteristic
5#21445248C214yesmaximal, complemented
6#21445241C214yesmaximal, complemented
7#21444574C211yesmaximal, characteristic, complemented
8#21444490C211yesmaximal, characteristic, complemented
9C61×#1549C211yesmaximal, characteristic, complemented
10C2×#14519810—21no—
11C2×#14519777C2211yescharacteristic
12C4×#11758222C2211yescharacteristic
13#16840112—21no—
14#16840112—21no—
15#16840112C2214yes—
16#16840054—21no—
17#16840049—21no—
18#16840044C2212yes—
19#16840039C2212yes—
20#16839973—21no—
21#16839973C2212yes—
22#16839968—21no—
23#16839968C2212yes—
24#16839886C2212yes—
25#16839827C2212yes—
26#16839825—22no—
27#16839813—21no—
28#16839802—21no—
29#16839786C2211yescharacteristic
30#16839772—21no—
31#16839761—21no—
32C2×#14519701—22no—
33C2×C4×#92418C2211yescharacteristic
34C4×C8×D61—21no—
35#16839512C2214yes—
36#16839507C2214yes—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#423C2211yescharacteristic
40C61×#332C2214yes—
41C2×#11758222D411yescharacteristic
42C22×Dic122—21no—
43#14519810—44no—
44C2×Dic244—22no—
45C2×#11758190—22no—
46C2×#11758194—21no—
47C22×#92418D412yes—
48C2×C8×D61—22no—
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52#14519777C2312yes—
53C4×Dic122D411yescharacteristic
54C4×Dic122C2311yescharacteristic
55C4×D244D411yescharacteristic
56C4×D244C2311yescharacteristic
57C4×C22⋊D61—21no—
58C4×C22⋊D61—21no—
59#14519736D411yescharacteristic
60#14519734—21no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C42×D61—21no—
64#14519710—22no—
65#14519708—22no—
66C4⋊#92418—21no—
67C4⋊#92418C2312yes—
68#14519706—22no—
69#14519702—22no—
70#14519702—22no—
71#14519701—24no—
72#14519701—24no—
73#14519700—22no—
74#14519697—22no—
75#14519689—22no—
76C4×#92418D411yescharacteristic
77C4×#92418D411yescharacteristic
78C4×#92418C2312yes—
79C8×Dic61—22no—
80C22×Q8×C61—21no—
81C2×M16×C61—21no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61—21no—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61C2312yes—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C42×C61C2311yescharacteristic
89C4⋊C8×C61—21no—
90C4⋊C8×C61C2312yes—
91C4⋊C8×C61C2312yes—
92C22⋊C8×C61—21no—
93C4×C8×C61—21no—
94C2×Q8×C61—22no—
95C2×Q8×C61—44no—
96C2×Q8×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61#4111yescharacteristic
101C2×C22⋊D61—21no—
102C22×Dic61—41no—
103#11758222—22no—
104#11758222SD1614yes—
105C2×D244C2×D411yescharacteristic
106C2×C4×D61—21no—
107C2×C4×D61—21no—
108C2×C4×D61—21no—
109C2×Dic122—22no—
110C2×Dic122—42no—
111C2×Dic122—42no—
112C2×Dic122—41no—
113C2×Dic122C2×D411yescharacteristic
114M16×C61—42no—
115C2×C8×C61—22no—
116C2×C8×C61—22no—
117C4⋊C4×C61—22no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—41no—
120C4⋊C4×C61—41no—
121C42×C61C2×D411yescharacteristic
122C42×C61C2×D411yescharacteristic
123C42×C61#4111yescharacteristic
124C42×C61C2411yescharacteristic
125C22⋊Dic61—21no—
126#11758198—21no—
127#11758198—21no—
128C4⋊Dic61—22no—
129C4⋊Dic61—22no—
130C4⋊Dic61—41no—
131C4⋊Dic61—41no—
132C4⋊Dic61C2×D411yescharacteristic
133C4⋊Dic61C2×D411yescharacteristic
134#11758196—21no—
135#11758196—21no—
136C4×Dic61—21no—
137C4×Dic61—21no—
138C4×Dic61—21no—
139#11758194—42no—
140C2×#92418—22no—
141C2×#92418—42no—
142C2×#92418C2×D412yes—
143C2×#92418C2×D412yes—
144Dic244—44no—
145#11758190—44no—
146C8×D61—44no—
147C23×C61#12111yescharacteristic
148C22×D61—21no—
149Q8×C61—44no—
150Q8×C61—82no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—42no—
154C2×C4×C61—82no—
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#12111yescharacteristic
158C2×C4×C61C2×SD1612yes—
159C2×C4×C61#13512yes—
160C2×C4×C61C22×D411yescharacteristic
161C2×C4×C61C22×D411yescharacteristic
162C2×C4×C61C2×#4111yescharacteristic
163C22⋊D61—24no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—21no—
167C2×Dic61—42no—
168C2×Dic61—82no—
169D244—21no—
170D244C2×SD1612yes—
171C4×D61—24no—
172C4×D61—42no—
173C4×D61—42no—
174Dic122—21no—
175Dic122—42no—
176Dic122—42no—
177Dic122—82no—
178Dic122C2×SD1612yes—
179C8×C61—42no—
180C8×C61—42no—
181#92418—22no—
182#92418—22no—
183#92418—42no—
184#19275—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61#46512yes—
188C22×C61C2×#12111yescharacteristic
189C2×D61—22no—
190C2×D61—41no—
191C2×D61—41no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—82no—
195C4×C61#46512yes—
196C4×C61#47312yes—
197C4×C61C22×SD1611yescharacteristic
198C4×C61C2×#13511yescharacteristic
199C4×C61#57512yes—
200Dic61—22no—
201Dic61—41no—
202Dic61—41no—
203Dic61—82no—
204C2×#487—611no—
205#3079—611no—
206#3039—611no—
207C8×#41—611no—
208#1680—614no—
209#1679—614no—
210#1566—611no—
211#1549D6111yescharacteristic, complemented
212#1546—611no—
213C2×C61—22no—
214C2×C61#304011yescharacteristic
215C2×C61C2×#46511yescharacteristic
216C2×C61C2×#47311yescharacteristic
217D61—42no—
218C2×#137—1221no—
219C2×C4⋊Q8—611no—
220C2×C4⋊Q8C2×D6111yescharacteristic
221C4×#41—611no—
222#487—614no—
223#487—1221no—
224#487—1221no—
225#477—1221no—
226#476—1221no—
227#475—612no—
228#474—612no—
229#462—612no—
230#462—1221no—
231#461—612no—
232#461—1221no—
233C8×Q8—612no—
234C8×D4—612no—
235C8×D4—1222no—
236#431—1221no—
237#431C2×D6111yescharacteristic
238#428—1221no—
239#423—611no—
240#423C2×D6111yescharacteristic
241#419—1221no—
242#417—1221no—
243C2×#103—1222no—
244C2×C4×C8—611no—
245C2×C4×C8—1221no—
246#334—614no—
247#332—614no—
248#332C2×D6114yes—
249C61#1927511yescharacteristic, complemented, Sylow
250C2×#41—611no—
251C22×Q8—21no—
252C22×Q8—1221no—
253#137—2444no—
254C2×Q16—1222no—
255C2×SD16—1222no—
256C2×M16—21no—
257C2×M16—1221no—
258C22×C8—612no—
259C22×C8—1222no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—612no—
264C4⋊Q8—1221no—
265C4⋊Q8—1221no—
266C4⋊Q8—1221no—
267C4⋊Q8C22×D6112yes—
268C4×Q8—611no—
269C4×Q8—611no—
270C4×D4—611no—
271C4×D4—611no—
272C4×D4—1221no—
273C4×D4—1221no—
274#117—611no—
275#117—1221no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—1221no—
279C2×C4⋊C4—1221no—
280C2×C42—1221no—
281C2×C42C22×D6111yescharacteristic
282#107—1222no—
283#106—1222no—
284C4⋊C8—21no—
285C4⋊C8—612no—
286C4⋊C8—1222no—
287C4⋊C8—1221no—
288C4⋊C8C22×D6112yes—
289C4⋊C8C22×D6112yes—
290#103—1222no—
291#103—1222no—
292#103—1222no—
293#103—1224no—
294#103—1224no—
295#102—1222no—
296C22⋊C8—21no—
297C22⋊C8—1222no—
298C4×C8—21no—
299C4×C8—611no—
300C4×C8—611no—
301C4×C8—612no—
302C4×C8—1222no—
303#41—614no—
304#41—1222no—
305C2×Q8—22no—
306C2×Q8—44no—
307C2×Q8—41no—
308C2×Q8—611no—
309C2×Q8—1222no—
310C2×Q8—2442no—
311C2×Q8—2442no—
312C2×Q8—2441no—
313C2×D4—611no—
314C2×D4—1221no—
315C22×C4—41no—
316C22×C4—1221no—
317C22×C4—1221no—
318C22×C4—1221no—
319C22×C4—2441no—
320C22×C4#1175822211yescharacteristic
321C22×C4D4×D6111yescharacteristic
322C22×C4D4×D6111yescharacteristic
323Q16—2444no—
324SD16—2444no—
325M16—42no—
326M16—2442no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—612no—
330C2×C8—612no—
331C2×C8—1222no—
332C2×C8—2444no—
333C2×C8—2442no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—611no—
339C4⋊C4—611no—
340C4⋊C4—1221no—
341C4⋊C4—1221no—
342C4⋊C4—1222no—
343C4⋊C4—1222no—
344C4⋊C4—2441no—
345C4⋊C4—2441no—
346C22⋊C4—1221no—
347C22⋊C4—1221no—
348C22⋊C4—1221no—
349C42—1221no—
350C42—1221no—
351C42—1221no—
352C42#1175822211yescharacteristic
353C42D4×D6111yescharacteristic
354C42D4×D6111yescharacteristic
355C42C23×D6111yescharacteristic
356C23—1221no—
357C23#1451975011yescharacteristic
358Q8—44no—
359Q8—82no—
360Q8—612no—
361Q8—1221no—
362Q8—4882no—
363Q8—2442no—
364Q8—2442no—
365D4—612no—
366D4—1224no—
367D4—1221no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—42no—
371C2×C4—82no—
372C2×C4—1221no—
373C2×C4—1224no—
374C2×C4—1221no—
375C2×C4—1221no—
376C2×C4—4882no—
377C2×C4—2442no—
378C2×C4—2442no—
379C2×C4—2442no—
380C2×C4#1451975011yescharacteristic
381C2×C4#1451975011yescharacteristic
382C2×C4#1451975011yescharacteristic
383C2×C4SD16×D6112yes—
384C2×C4#1451980012yes—
385C2×C4C2×D4×D6111yescharacteristic
386C2×C4C2×D4×D6111yescharacteristic
387C2×C4C2×#1175822211yescharacteristic
388C8—42no—
389C8—42no—
390C8—1222no—
391C8—1222no—
392C8—2442no—
393C22—21no—
394C22—21no—
395C22—1222no—
396C22—2441no—
397C22—2441no—
398C22#1683999412yes—
399C22C2×#1451975011yescharacteristic
400C4—21no—
401C4—21no—
402C4—82no—
403C4—1222no—
404C4—4882no—
405C4—2441no—
406C4—2441no—
407C4#1683999412yes—
408C4#1684003612yes—
409C4C2×SD16×D6111yescharacteristic
410C4C2×#1451980011yescharacteristic
411C4#1684055812yes—
412C2—22no—
413C2—2442no—
414C2#2145531211yescharacteristic
415C2C2×#1683999411yescharacteristic
416C2C2×#1684003611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1549C2
socleC22×C61C2×#121
prime radicalC2×#11758222D4
hypercenter#1549D61
hypocenterC61#19275
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
1. C2×#16840112C2semidirect
5. #21445248C2semidirect
6. #21445241C2semidirect
7. #21444574C2semidirect
8. #21444490C2semidirect
9. C61×#1549C2semidirect
211. #1549D61semidirect
249. C61#19275semidirect

Maximal subgroups

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

subgroupindex
C2×#168401122
#214556282
#214553072
#214547752
#214452482
#214452412
#214445742
#214444902
C61×#15492
#1927561