Finite Group Databasebeta

← #29663271

Subgroups — #29663271

Order 15616.

570 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457162C212yesmaximal
2#21456330C211yesmaximal, characteristic
3#21454750C211yesmaximal, characteristic
4#21445348C211yesmaximal, characteristic, complemented
5#21445347C211yesmaximal, characteristic, complemented
6#21445334C211yesmaximal, characteristic, complemented
7#21445200C212yesmaximal, complemented
8#21445199C212yesmaximal, complemented
9#334×D61C211yesmaximal, characteristic
10#21444508C212yesmaximal
11C61×#1694C211yesmaximal, characteristic, complemented
12#16840317C2212yes—
13#16840316C2211yescharacteristic
14#16840167C2211yescharacteristic
15#16840163—21no—
16#16840163—21no—
17#16840158C2212yes—
18#16840058C2212yes—
19#16840057C2212yes—
20#16840055—22no—
21#16840002—22no—
22#16840000—22no—
23#16839970—22no—
24#16839969C2212yes—
25#16839892C2211yescharacteristic
26#16839890—21no—
27Q8×#92418C2211yescharacteristic
28#16839886—21no—
29#16839886C2211yescharacteristic
30#16839799—21no—
31#16839799—21no—
32#16839787—21no—
33#16839784C2211yescharacteristic
34C4⋊C8×D61C2212yes—
35#16839719—21no—
36C4×C8×D61C2211yescharacteristic
37#16839513C2211yescharacteristic
38#16839512C2212yes—
39#16839510C2212yes—
40#16839510C2212yes—
41#16839508C2212yes—
42C61×#500C2211yescharacteristic
43C61×#462C2212yes—
44C61×#446C2211yescharacteristic
45C61×#334C2211yescharacteristic
46C61×#332C2212yes—
47C2×#11758202—22no—
48C2×#11758189—21no—
49C2×C8×D61—22no—
50C2×C8×D61D411yescharacteristic
51C2×C8×D61D411yescharacteristic
52#14519777C2312yes—
53#14519776D411yescharacteristic
54#14519775D411yescharacteristic
55#14519768—22no—
56#14519767—21no—
57#14519757—22no—
58#14519757D411yescharacteristic
59#14519755—21no—
60#14519753—22no—
61#14519753D411yescharacteristic
62Q8×Dic61—22no—
63C4×Dic122—21no—
64C4×Dic122C2311yescharacteristic
65#14519742—22no—
66#14519735—22no—
67#14519734—21no—
68C42×D61C2311yescharacteristic
69#14519711—21no—
70#14519711—21no—
71#14519711—22no—
72#14519710—22no—
73#14519708—22no—
74#14519708—22no—
75C4⋊#92418—21no—
76C4⋊#92418C2312yes—
77C4⋊#92418C2311yescharacteristic
78#14519706—21no—
79#14519706—22no—
80#14519706—21no—
81#14519706—21no—
82#14519702—22no—
83#14519701—22no—
84#14519701—22no—
85#14519687—21no—
86C4×#92418—21no—
87C4×#92418C2311yescharacteristic
88C8×Dic61—21no—
89C8×Dic61D411yescharacteristic
90C8×Dic61D411yescharacteristic
91C2×Q16×C61—22no—
92C4⋊Q8×C61C2312yes—
93C4×Q8×C61C2311yescharacteristic
94#107×C61—21no—
95#107×C61—21no—
96C4⋊C8×C61—21no—
97C4⋊C8×C61C2311yescharacteristic
98C4⋊C8×C61C2312yes—
99#103×C61—22no—
100#97×C61—21no—
101C4×C8×C61C2311yescharacteristic
102C2×Q8×C61—22no—
103C2×Q8×C61C2×D411yescharacteristic
104C2×C4×D61C2×D411yescharacteristic
105C2×C4×D61C2×D411yescharacteristic
106C2×C4×D61#4111yescharacteristic
107C2×Dic122—21no—
108C2×Dic122—22no—
109C2×Dic122C2×D411yescharacteristic
110Q16×C61—82no—
111C2×C8×C61—21no—
112C2×C8×C61—22no—
113C2×C8×C61C2×D411yescharacteristic
114C2×C8×C61C2×D411yescharacteristic
115C4⋊C4×C61—21no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61—22no—
118C4⋊C4×C61C2×D411yescharacteristic
119C42×C61—21no—
120C42×C61C2411yescharacteristic
121#11758202—82no—
122#11758198—21no—
123#11758198—21no—
124#11758198—42no—
125C4⋊Dic61—21no—
126C4⋊Dic61—22no—
127C4⋊Dic61—22no—
128C4⋊Dic61—42no—
129C4⋊Dic61C2×D411yescharacteristic
130#11758196—21no—
131#11758196—21no—
132#11758196—21no—
133#11758196—21no—
134#11758196—21no—
135#11758196—42no—
136C4×Dic61—21no—
137C4×Dic61—42no—
138C4×Dic61C2×D411yescharacteristic
139C4×Dic61C2×D411yescharacteristic
140C4×Dic61#4111yescharacteristic
141C2×#92418—21no—
142C2×#92418—22no—
143C2×#92418C2×D411yescharacteristic
144C2×#92418C2×D411yescharacteristic
145#11758189—42no—
146C8×D61—22no—
147C8×D61—82no—
148C8×D61Q1614yescomplemented
149C22×D61#12111yescharacteristic
150Q8×C61—22no—
151Q8×C61—42no—
152Q8×C61—42no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—42no—
156C2×C4×C61C22×D411yescharacteristic
157C2×C4×C61C22×D411yescharacteristic
158C2×C4×C61C2×#4111yescharacteristic
159C2×Dic61—21no—
160C2×Dic61—21no—
161C2×Dic61—42no—
162C2×Dic61#12111yescharacteristic
163C2×Dic61#12111yescharacteristic
164C2×Dic61#12111yescharacteristic
165C4×D61—22no—
166C4×D61—41no—
167C4×D61C2×Q1612yes—
168C4×D61#13712yes—
169Dic122—22no—
170Dic122—42no—
171Dic122—82no—
172C8×C61—21no—
173C8×C61—41no—
174C8×C61—42no—
175C8×C61C2×Q1612yes—
176#92418—21no—
177#92418—41no—
178#92418—42no—
179#92418C2×Q1612yes—
180#19210—611nomaximal, Sylow
181C22×C61C2×#12111yescharacteristic
182C2×D61—21no—
183C2×D61—21no—
184C2×D61#46212yes—
185C4×C61—21no—
186C4×C61—41no—
187C4×C61—41no—
188C4×C61—82no—
189C4×C61C22×Q1611yescharacteristic
190C4×C61C2×#13711yescharacteristic
191C4×C61#57712yes—
192Dic61—21no—
193Dic61—21no—
194Dic61—41no—
195Dic61—41no—
196Dic61—82no—
197Dic61#46212yes—
198#3249—612no—
199#3160—611no—
200#2960—611no—
201#1696—611no—
202#1696—611no—
203#1694—611no—
204#1694D6111yescharacteristic, complemented
205#1672—612no—
206#1672—612no—
207C2×#334—611no—
208#1550—612no—
209C2×C61C2×#46211yescharacteristic
210C2×C61#306011yescharacteristic
211C2×C61#306711yescharacteristic
212D61—22no—
213#533—611no—
214#533—612no—
215#500—611no—
216#500C2×D6111yescharacteristic
217#499—1221no—
218#499—1221no—
219#498—612no—
220#477—612no—
221#477—612no—
222#477—1222no—
223#467—1222no—
224#467—1222no—
225#462—612no—
226#462—1222no—
227#462C2×D6112yes—
228#446—611no—
229#446—1221no—
230#446C2×D6111yescharacteristic
231C8×Q8—611no—
232C8×Q8—611no—
233C8×Q8—1221no—
234#427—1221no—
235#427—1221no—
236#424—1221no—
237#423—611no—
238C2×C4⋊C8—612no—
239#405—1221no—
240C2×C4×C8—611no—
241#334—612no—
242#334—611no—
243#334C2×D6111yescharacteristic
244#333—612no—
245#333—612no—
246#332—612no—
247#332C2×D6112yes—
248C61#1921011yescharacteristic, complemented, Sylow
249C2×Q16—22no—
250C2×Q16—1222no—
251C2×M16—1221no—
252C22×C8—611no—
253C22×C8—611no—
254C22×C8—1222no—
255C4⋊Q8—611no—
256C4⋊Q8—611no—
257C4⋊Q8—612no—
258C4⋊Q8C22×D6112yes—
259#125—1221no—
260#125—1222no—
261C22⋊Q8—611no—
262C22⋊Q8—611no—
263C22⋊Q8—1221no—
264C22⋊Q8—1222no—
265C22⋊Q8—1222no—
266C4×Q8—611no—
267C4×Q8—1221no—
268C4×Q8—1222no—
269C4×Q8—1222no—
270C4×Q8C22×D6111yescharacteristic
271#117—1221no—
272#117—1222no—
273C2×C42—611no—
274#107—21no—
275#107—21no—
276#107—1222no—
277#107—1221no—
278#107—1221no—
279#107—1222no—
280#106—1222no—
281#106—1222no—
282C4⋊C8—21no—
283C4⋊C8—611no—
284C4⋊C8—612no—
285C4⋊C8—1221no—
286C4⋊C8—1222no—
287C4⋊C8—1221no—
288C4⋊C8—1221no—
289C4⋊C8—1221no—
290C4⋊C8C22×D6111yescharacteristic
291C4⋊C8C22×D6112yes—
292#103—22no—
293#103—1222no—
294#103—1222no—
295#103—1222no—
296#97—21no—
297#97—1221no—
298C4×C8—611no—
299C4×C8—611no—
300C4×C8—611no—
301C4×C8—1221no—
302C4×C8—1221no—
303C4×C8C22×D6111yescharacteristic
304C2×Q8—22no—
305C2×Q8—611no—
306C2×Q8—1221no—
307C2×Q8—1222no—
308C2×Q8D4×D6111yescharacteristic
309C22×C4—611no—
310C22×C4—611no—
311C22×C4—611no—
312Q16—82no—
313Q16—4882no—
314M16—2442no—
315C2×C8—21no—
316C2×C8—22no—
317C2×C8—611no—
318C2×C8—614no—
319C2×C8—611no—
320C2×C8—1222no—
321C2×C8—1221no—
322C2×C8—1222no—
323C2×C8—4882no—
324C2×C8D4×D6111yescharacteristic
325C2×C8D4×D6111yescharacteristic
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—611no—
330C4⋊C4—1221no—
331C4⋊C4—1221no—
332C4⋊C4—1221no—
333C4⋊C4—1221no—
334C4⋊C4—1221no—
335C4⋊C4—1221no—
336C4⋊C4—1222no—
337C4⋊C4—1222no—
338C4⋊C4—2442no—
339C4⋊C4—2442no—
340C4⋊C4D4×D6111yescharacteristic
341C22⋊C4—1221no—
342C22⋊C4—1221no—
343C22⋊C4—2442no—
344C42—21no—
345C42—611no—
346C42—611no—
347C42—611no—
348C42—1221no—
349C42—2442no—
350C42C23×D6111yescharacteristic
351C23—611no—
352Q8—22no—
353Q8—42no—
354Q8—42no—
355Q8—1222no—
356Q8—4882no—
357Q8—2442no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—611no—
362C2×C4—612no—
363C2×C4—612no—
364C2×C4—611no—
365C2×C4—611no—
366C2×C4—1222no—
367C2×C4—1221no—
368C2×C4—1221no—
369C2×C4—2441no—
370C2×C4—2442no—
371C2×C4C2×D4×D6111yescharacteristic
372C2×C4C2×D4×D6111yescharacteristic
373C2×C4#41×D6111yescharacteristic
374C8—21no—
375C8—41no—
376C8—42no—
377C8—612no—
378C8—1221no—
379C8—2441no—
380C8—2442no—
381C8Q16×D6112yes—
382C22—612no—
383C22—1221no—
384C22—1221no—
385C22#121×D6111yescharacteristic
386C4—21no—
387C4—41no—
388C4—41no—
389C4—82no—
390C4—612no—
391C4—1221no—
392C4—1221no—
393C4—4882no—
394C4—2441no—
395C4—2441no—
396C4C2×Q16×D6111yescharacteristic
397C4#137×D6111yescharacteristic
398C4#1684057212yes—
399C2—1222no—
400C2#462×D6111yescharacteristic
401C2#2145547911yescharacteristic
402C2#2145554111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1694C2
socleC22×C61C2×#121
prime radicalC22×D61#121
hypercenter#1694D61
hypocenterC61#19210
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1694

Complements

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

NH ≅ G/Ntype
4. #21445348C2semidirect
5. #21445347C2semidirect
6. #21445334C2semidirect
7. #21445200C2semidirect
8. #21445199C2semidirect
11. C61×#1694C2semidirect
148. C8×D61Q16semidirect
204. #1694D61semidirect
248. C61#19210semidirect

Maximal subgroups

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

subgroupindex
#214571622
#214563302
#214547502
#214453482
#214453472
#214453342
#214452002
#214451992
#334×D612
#214445082
C61×#16942
#1921061