Finite Group Databasebeta

← #29662726

Subgroups — #29662726

Order 15616.

596 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456403C211yesmaximal, characteristic, complemented
2#21455600C212yesmaximal
3#21454782C211yesmaximal, characteristic
4#21445400C212yesmaximal, complemented
5#21445392C212yesmaximal, complemented
6#21445348C211yesmaximal, characteristic, complemented
7#21445342C211yesmaximal, characteristic, complemented
8#21445310C211yesmaximal, characteristic, complemented
9#334×D61C211yesmaximal, characteristic, complemented
10#21444475C212yesmaximal, complemented
11C61×#1689C211yesmaximal, characteristic, complemented
12#16840317C2212yes—
13C4×#11758224C2211yescharacteristic
14#16840167C2211yescharacteristic
15#16840112C2212yes—
16#16840068—22no—
17#16840060—22no—
18#16840058C2212yes—
19#16840052C2212yes—
20#16840044C2212yes—
21#16840000—21no—
22#16840000—21no—
23#16839892C2211yescharacteristic
24#16839889C2211yescharacteristic
25Q16×Dic61—21no—
26Q16×Dic61—21no—
27#16839840C2211yescharacteristic
28#16839836—21no—
29#16839831—21no—
30#16839814C2211yescharacteristic
31#16839812—21no—
32#16839799—22no—
33C4⋊C8×D61C2212yes—
34#16839755—22no—
35M16×Dic61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839513C2211yescharacteristic
38#16839512C2212yes—
39#16839510C2212yes—
40#16839510C2212yes—
41#16839508C2212yes—
42C61×#494C2211yescharacteristic
43C61×#474C2212yes—
44C61×#436C2211yescharacteristic
45C61×#334C2211yescharacteristic
46C61×#332C2212yes—
47C2×#11758224D411yescharacteristic
48C2×#11758194—21no—
49C2×C8×D61—22no—
50C2×C8×D61D411yescharacteristic
51C2×C8×D61D411yescharacteristic
52#14519777C2312yes—
53#14519768—22no—
54#14519757—22no—
55#14519753—22no—
56Q8×Dic61—22no—
57C4×Dic122C2311yescharacteristic
58#14519742—22no—
59#14519742D411yescharacteristic
60D4×Dic61D411yescharacteristic
61C4×C22⋊D61—21no—
62#14519737—21no—
63#14519735—22no—
64#14519735D411yescharacteristic
65#14519733—21no—
66C2×C4×Dic61—21no—
67C42×D61C2311yescharacteristic
68#14519711—21no—
69#14519711—21no—
70#14519711—22no—
71#14519708—22no—
72#14519708—22no—
73C4⋊#92418C2311yescharacteristic
74C4⋊#92418C2312yes—
75#14519706—21no—
76#14519706—22no—
77#14519706—21no—
78#14519702—22no—
79#14519701—22no—
80#14519701—22no—
81#14519700—22no—
82#14519698—22no—
83C22⋊#92418—21no—
84#14519689—21no—
85#14519687—21no—
86#14519687—21no—
87C4×#92418C2311yescharacteristic
88C8×Dic61—21no—
89C8×Dic61D411yescharacteristic
90C8×Dic61D411yescharacteristic
91C2×Q16×C61—21no—
92C2×Q16×C61—21no—
93C2×M16×C61—21no—
94C4⋊Q8×C61C2312yes—
95C4×D4×C61C2311yescharacteristic
96#107×C61—22no—
97C4⋊C8×C61C2311yescharacteristic
98C4⋊C8×C61C2312yes—
99#102×C61—22no—
100C22⋊C8×C61—21no—
101C4×C8×C61C2311yescharacteristic
102C2×Q8×C61—22no—
103C2×D4×C61C2×D411yescharacteristic
104C22×C4×C61—21no—
105C2×C22⋊D61—21no—
106C22×Dic61—21no—
107C22×Dic61—21no—
108#11758224—24no—
109C2×C4×D61C2×D411yescharacteristic
110C2×C4×D61C2×D411yescharacteristic
111C2×C4×D61#4111yescharacteristic
112C2×Dic122—22no—
113C2×Dic122C2×D411yescharacteristic
114Q16×C61—42no—
115Q16×C61—42no—
116M16×C61—42no—
117C2×C8×C61—21no—
118C2×C8×C61—22no—
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61C2×D411yescharacteristic
121C4⋊C4×C61—22no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61C2×D411yescharacteristic
124C22⋊C4×C61—21no—
125C42×C61C2411yescharacteristic
126C22⋊Dic61—21no—
127#11758198—21no—
128#11758198—42no—
129C4⋊Dic61—22no—
130C4⋊Dic61—22no—
131C4⋊Dic61—42no—
132C4⋊Dic61C2×D411yescharacteristic
133#11758196—21no—
134#11758196—21no—
135#11758196—42no—
136C4×Dic61—21no—
137C4×Dic61—21no—
138C4×Dic61—21no—
139C4×Dic61—42no—
140C4×Dic61C2×D411yescharacteristic
141C4×Dic61C2×D411yescharacteristic
142C4×Dic61#4111yescharacteristic
143#11758194—42no—
144C2×#92418—21no—
145C2×#92418—22no—
146C2×#92418C2×D411yescharacteristic
147C2×#92418C2×D411yescharacteristic
148C8×D61—22no—
149C8×D61—41no—
150C8×D61—44no—
151C23×C61—21no—
152C22×D61#12111yescharacteristic
153Q8×C61—42no—
154Q8×C61—42no—
155D4×C61—22no—
156C2×C4×C61—21no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—42no—
160C2×C4×C61C22×D411yescharacteristic
161C2×C4×C61C22×D411yescharacteristic
162C2×C4×C61C2×#4111yescharacteristic
163C22⋊D61—42no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—42no—
167C2×Dic61—42no—
168C2×Dic61—42no—
169C2×Dic61#12111yescharacteristic
170C2×Dic61#12111yescharacteristic
171C2×Dic61#12111yescharacteristic
172C4×D61—21no—
173C4×D61—21no—
174C4×D61—41no—
175C4×D61#13714yes—
176Dic122—22no—
177Dic122—82no—
178C8×C61—21no—
179C8×C61—41no—
180C8×C61—42no—
181C8×C61#13512yes—
182#92418—21no—
183#92418—21no—
184#92418—42no—
185#92418—41no—
186#19158—611nomaximal, Sylow
187C22×C61—41no—
188C22×C61—41no—
189C22×C61—41no—
190C22×C61C2×#12111yescharacteristic
191C2×D61—21no—
192C2×D61—21no—
193C2×D61#47012yes—
194C4×C61—21no—
195C4×C61—41no—
196C4×C61—82no—
197C4×C61C2×#13511yescharacteristic
198C4×C61C2×#13712yes—
199C4×C61#57511yescharacteristic
200Dic61—21no—
201Dic61—21no—
202Dic61—41no—
203Dic61—41no—
204Dic61—82no—
205Dic61#47012yes—
206#3170—611no—
207#3076—612no—
208#2965—611no—
209#1704—612no—
210#1703—612no—
211#1696—611no—
212#1695—611no—
213#1690—611no—
214#1689D6111yescharacteristic, complemented
215C2×#334—611no—
216#1542—612no—
217C2×C61—81no—
218C2×C61#303511yescharacteristic
219C2×C61C2×#47011yescharacteristic
220C2×C61#306711yescharacteristic
221D61—22no—
222#533—612no—
223C4×#41—611no—
224#500—611no—
225#494C2×D6111yescharacteristic
226#487—612no—
227#479—1222no—
228#478—1222no—
229#477—612no—
230#476—612no—
231#475—612no—
232#474C2×D6112yes—
233#467—1221no—
234#467—1221no—
235#446—611no—
236#446—611no—
237C4×Q16—1221no—
238C4×Q16—1221no—
239#436—611no—
240#436C2×D6111yescharacteristic
241#435—1221no—
242#435—1221no—
243#431—611no—
244#431—1221no—
245#427—1222no—
246C2×C4⋊C8—612no—
247#416—1222no—
248C4×M16—1221no—
249C2×C4×C8—611no—
250#334—612no—
251#334—611no—
252#334C2×D6111yescharacteristic
253#333—612no—
254#333—612no—
255#332—612no—
256#332C2×D6112yes—
257C61#1915811yescharacteristic, complemented, Sylow
258C2×#41—611no—
259C2×Q16—21no—
260C2×Q16—21no—
261C2×M16—21no—
262C2×M16—1221no—
263C22×C8—611no—
264C22×C8—611no—
265C22×C8—1222no—
266C4⋊Q8—612no—
267C4⋊Q8C22×D6112yes—
268#125—1222no—
269C22⋊Q8—1222no—
270C22⋊Q8—1222no—
271C4×Q8—611no—
272C4×Q8—611no—
273C4×Q8—1222no—
274C4×Q8—1222no—
275C4×D4—611no—
276C4×D4—1221no—
277C4×D4—1221no—
278C4×D4C22×D6111yescharacteristic
279#117—611no—
280#117—1221no—
281#117—1222no—
282C2×C42—611no—
283C2×C42—1221no—
284#107—22no—
285#107—1222no—
286#107—1221no—
287#107—1221no—
288#106—1222no—
289#106—1222no—
290C4⋊C8—611no—
291C4⋊C8—612no—
292C4⋊C8—1222no—
293C4⋊C8—1221no—
294C4⋊C8—1221no—
295C4⋊C8C22×D6111yescharacteristic
296C4⋊C8C22×D6112yes—
297#103—1222no—
298#103—1222no—
299#103—1222no—
300#103—1222no—
301#102—22no—
302#102—1222no—
303C22⋊C8—21no—
304C22⋊C8—1221no—
305C22⋊C8—1221no—
306#97—1221no—
307#97—1221no—
308C4×C8—611no—
309C4×C8—611no—
310C4×C8—611no—
311C4×C8—1221no—
312C4×C8C22×D6111yescharacteristic
313#41—1224no—
314C2×Q8—22no—
315C2×Q8—611no—
316C2×Q8—1222no—
317C2×D4—1221no—
318C2×D4D4×D6111yescharacteristic
319C22×C4—21no—
320C22×C4—611no—
321C22×C4—611no—
322C22×C4—611no—
323C22×C4—1221no—
324C22×C4—1221no—
325Q16—42no—
326Q16—42no—
327M16—42no—
328M16—2442no—
329C2×C8—21no—
330C2×C8—22no—
331C2×C8—611no—
332C2×C8—611no—
333C2×C8—1222no—
334C2×C8—1221no—
335C2×C8—1222no—
336C2×C8—2441no—
337C2×C8—2444no—
338C2×C8D4×D6111yescharacteristic
339C2×C8D4×D6111yescharacteristic
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—611no—
343C4⋊C4—1221no—
344C4⋊C4—1221no—
345C4⋊C4—1222no—
346C4⋊C4—1222no—
347C4⋊C4—2442no—
348C4⋊C4—2442no—
349C4⋊C4D4×D6111yescharacteristic
350C22⋊C4—21no—
351C22⋊C4—1221no—
352C22⋊C4—1221no—
353C22⋊C4—2442no—
354C42—611no—
355C42—611no—
356C42—611no—
357C42—1221no—
358C42—1221no—
359C42—1221no—
360C42—2442no—
361C42C23×D6111yescharacteristic
362C23—21no—
363C23—611no—
364Q8—42no—
365Q8—42no—
366Q8—1222no—
367Q8—4882no—
368D4—22no—
369D4—2442no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—611no—
375C2×C4—614no—
376C2×C4—611no—
377C2×C4—611no—
378C2×C4—1221no—
379C2×C4—1221no—
380C2×C4—1221no—
381C2×C4—1221no—
382C2×C4—2441no—
383C2×C4—2442no—
384C2×C4—2442no—
385C2×C4—2442no—
386C2×C4C2×D4×D6111yescharacteristic
387C2×C4C2×D4×D6111yescharacteristic
388C2×C4#41×D6111yescharacteristic
389C8—21no—
390C8—41no—
391C8—42no—
392C8—1221no—
393C8—1221no—
394C8—2442no—
395C8—2441no—
396C8#1451979612yes—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—612no—
401C22—1221no—
402C22—1221no—
403C22#121×D6111yescharacteristic
404C4—21no—
405C4—41no—
406C4—82no—
407C4—612no—
408C4—1221no—
409C4—1221no—
410C4—4882no—
411C4—2441no—
412C4—2441no—
413C4C2×#1451979611yescharacteristic
414C4#137×D6112yes—
415C4#1684055811yescharacteristic
416C2—81no—
417C2—1222no—
418C2#2145526711yescharacteristic
419C2#470×D6111yescharacteristic
420C2#2145554111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1689C2
socleC22×C61C2×#121
prime radicalC2×#11758224D4
hypercenter#1689D61
hypocenterC61#19158
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1689

Complements

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

NH ≅ G/Ntype
1. #21456403C2semidirect
4. #21445400C2semidirect
5. #21445392C2semidirect
6. #21445348C2semidirect
7. #21445342C2semidirect
8. #21445310C2semidirect
9. #334×D61C2semidirect
10. #21444475C2semidirect
11. C61×#1689C2semidirect
214. #1689D61semidirect
257. C61#19158semidirect

Maximal subgroups

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

subgroupindex
#214564032
#214556002
#214547822
#214454002
#214453922
#214453482
#214453422
#214453102
#334×D612
#214444752
C61×#16892
#1915861