Finite Group Databasebeta

← #29662727

Subgroups — #29662727

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#21455601C212yesmaximal
2#21454785C211yesmaximal, characteristic
3#21445398C212yesmaximal, complemented
4#21445391C212yesmaximal, complemented
5#21445344C211yesmaximal, characteristic, complemented
6#21445339C211yesmaximal, characteristic, complemented
7#21445310C211yesmaximal, characteristic, complemented
8#21445305C211yesmaximal, characteristic, complemented
9C2×#16839513C211yesmaximal, characteristic, complemented
10#21444476C212yesmaximal, complemented
11C61×#3170C211yesmaximal, characteristic, complemented
12C4×#11758222C2211yescharacteristic
13#16840066—22no—
14#16840059—22no—
15#16840054C2212yes—
16#16840049C2212yes—
17#16840044C2212yes—
18#16840039C2212yes—
19#16839892C2211yescharacteristic
20#16839892C2211yescharacteristic
21#16839841C2211yescharacteristic
22#16839841C2211yescharacteristic
23#16839833—21no—
24#16839833—21no—
25#16839815C2211yescharacteristic
26#16839813—21no—
27#16839798—22no—
28C2×C4⋊#92418C2212yes—
29#16839756—22no—
30C4×#11758189—21no—
31#16839513C2212yes—
32#16839513C2211yescharacteristic
33#16839513C2211yescharacteristic
34#16839511C2212yes—
35#16839511C2212yes—
36#16839508C2212yes—
37#16839508C2212yes—
38C61×#533C2212yes—
39C61×#500C2211yescharacteristic
40C61×#494C2211yescharacteristic
41C61×#487C2212yes—
42C61×#467—21no—
43C61×#467—21no—
44C4×Q16×C61—21no—
45C4×Q16×C61—21no—
46C2×C4×C8×C61C2211yescharacteristic
47C2×#11758222D411yescharacteristic
48C2×#11758189—21no—
49C2×#11758189—21no—
50C22×#92418—22no—
51C4×Dic122D411yescharacteristic
52C4×Dic122C2311yescharacteristic
53C4×D244D411yescharacteristic
54C4×D244C2311yescharacteristic
55C4×C22⋊D61—21no—
56C4×C22⋊D61—21no—
57#14519736D411yescharacteristic
58#14519734—21no—
59C42×D61—21no—
60#14519710—22no—
61#14519710—22no—
62#14519708—22no—
63#14519708—22no—
64C4⋊#92418—22no—
65C4⋊#92418C2311yescharacteristic
66C4⋊#92418C2311yescharacteristic
67C4⋊#92418C2312yes—
68C4⋊#92418C2312yes—
69#14519706—21no—
70#14519706—21no—
71#14519700—22no—
72#14519700—22no—
73#14519697—22no—
74#14519697—22no—
75C22⋊#92418—21no—
76#14519689—21no—
77#14519689—21no—
78#14519687—21no—
79#14519687—21no—
80C4×#92418—21no—
81C2×Q16×C61—21no—
82C2×Q16×C61—21no—
83C22×C8×C61D411yescharacteristic
84C22×C8×C61D411yescharacteristic
85C4⋊Q8×C61C2312yes—
86C4⋊Q8×C61C2312yes—
87#125×C61—22no—
88C22⋊Q8×C61—22no—
89C22⋊Q8×C61—22no—
90C4×Q8×C61—22no—
91C4×Q8×C61—22no—
92#117×C61—22no—
93C2×C42×C61C2311yescharacteristic
94#107×C61—21no—
95#107×C61—21no—
96#103×C61—22no—
97#103×C61—22no—
98C4×C8×C61D411yescharacteristic
99C4×C8×C61D411yescharacteristic
100C4×C8×C61C2311yescharacteristic
101C4×C8×C61C2311yescharacteristic
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61#4111yescharacteristic
107C2×C22⋊D61—21no—
108#11758222—24no—
109C2×D244C2×D411yescharacteristic
110C2×C4×D61—21no—
111C2×C4×D61—21no—
112C2×C4×D61—21no—
113C2×Dic122C2×D411yescharacteristic
114Q16×C61—42no—
115Q16×C61—42no—
116C2×C8×C61—22no—
117C2×C8×C61—41no—
118C2×C8×C61C2×D411yescharacteristic
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61C2×D411yescharacteristic
121C2×C8×C61C2×D411yescharacteristic
122C4⋊C4×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—42no—
127C4⋊C4×C61—42no—
128C22⋊C4×C61—42no—
129C42×C61—42no—
130C42×C61C2×D411yescharacteristic
131C42×C61C2×D411yescharacteristic
132C42×C61#4111yescharacteristic
133C42×C61C2411yescharacteristic
134C22⋊Dic61—21no—
135#11758198—21no—
136#11758198—21no—
137C4⋊Dic61C2×D411yescharacteristic
138C4⋊Dic61C2×D411yescharacteristic
139#11758196—21no—
140#11758196—21no—
141C4×Dic61—21no—
142C4×Dic61—21no—
143C4×Dic61—21no—
144C2×#92418—21no—
145C2×#92418—21no—
146C2×#92418—22no—
147C2×#92418—22no—
148C2×#92418—44no—
149#11758189—42no—
150#11758189—42no—
151C23×C61#12111yescharacteristic
152C22×D61—21no—
153Q8×C61—42no—
154Q8×C61—42no—
155Q8×C61—82no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—41no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#13714yes—
165C2×C4×C61C22×D411yescharacteristic
166C2×C4×C61C22×D411yescharacteristic
167C2×C4×C61C2×#4111yescharacteristic
168C22⋊D61—42no—
169C2×Dic61—21no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172D244—22no—
173C4×D61—42no—
174C4×D61—41no—
175C4×D61—41no—
176C4×D61—42no—
177Dic122—22no—
178C8×C61—21no—
179C8×C61—21no—
180C8×C61—21no—
181C8×C61#13512yes—
182#92418—41no—
183#92418—42no—
184#92418—42no—
185#92418—41no—
186#19158—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61#47012yes—
190C22×C61C2×#12111yescharacteristic
191C2×D61—41no—
192C2×D61—41no—
193C2×D61—41no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—82no—
198C4×C61—82no—
199C4×C61#47012yes—
200C4×C61C2×#13511yescharacteristic
201C4×C61C2×#13712yes—
202C4×C61#57511yescharacteristic
203Dic61—41no—
204Dic61—41no—
205Dic61—41no—
206#3170D6111yescharacteristic, complemented
207#3076—612no—
208#2965—611no—
209#1704—612no—
210#1703—612no—
211#1696—611no—
212#1695—611no—
213#1690—611no—
214#1689—611no—
215C2×#334—611no—
216#1542—612no—
217C2×C61—22no—
218C2×C61#303511yescharacteristic
219C2×C61C2×#47011yescharacteristic
220C2×C61#306711yescharacteristic
221D61—81no—
222#533C2×D6112yes—
223C4×#41—611no—
224#500C2×D6111yescharacteristic
225#494C2×D6111yescharacteristic
226#487C2×D6112yes—
227#479—1222no—
228#478—1222no—
229#477—612no—
230#476—612no—
231#475—612no—
232#474—612no—
233#467—21no—
234#467—21no—
235#446—611no—
236#446—611no—
237C4×Q16—21no—
238C4×Q16—21no—
239#436—611no—
240#436—611no—
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×C8C2×D6111yescharacteristic
250#334—611no—
251#334—611no—
252#334—612no—
253#333—612no—
254#333—612no—
255#332—612no—
256#332—612no—
257C61#1915811yescharacteristic, complemented, Sylow
258C2×#41—611no—
259C2×Q16—21no—
260C2×Q16—21no—
261C2×M16—1221no—
262C2×M16—1221no—
263C22×C8—1222no—
264C22×C8C22⋊D6111yescharacteristic
265C22×C8C22⋊D6111yescharacteristic
266C4⋊Q8C22×D6112yes—
267C4⋊Q8C22×D6112yes—
268#125—22no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—611no—
274C4×Q8—611no—
275C4×D4—611no—
276C4×D4—611no—
277C4×D4—1221no—
278C4×D4—1221no—
279#117—22no—
280#117—611no—
281#117—1221no—
282C2×C42—1221no—
283C2×C42C22×D6111yescharacteristic
284#107—21no—
285#107—21no—
286#107—1222no—
287#107—1222no—
288#106—1222no—
289#106—1222no—
290C4⋊C8—611no—
291C4⋊C8—611no—
292C4⋊C8—612no—
293C4⋊C8—612no—
294C4⋊C8—1221no—
295C4⋊C8—1221no—
296C4⋊C8—1222no—
297#103—22no—
298#103—22no—
299#103—1222no—
300#103—1222no—
301#102—1222no—
302#102—1222no—
303C22⋊C8—1221no—
304C22⋊C8—1221no—
305C22⋊C8—1221no—
306#97—1221no—
307#97—1221no—
308C4×C8—1221no—
309C4×C8C22⋊D6111yescharacteristic
310C4×C8C22⋊D6111yescharacteristic
311C4×C8C22×D6111yescharacteristic
312C4×C8C22×D6111yescharacteristic
313#41—1224no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—611no—
317C2×D4—611no—
318C2×D4—1221no—
319C22×C4—1221no—
320C22×C4—1221no—
321C22×C4—1221no—
322C22×C4D4×D6111yescharacteristic
323C22×C4#1175822411yescharacteristic
324C22×C4C2×C22⋊D6111yescharacteristic
325Q16—42no—
326Q16—42no—
327M16—2442no—
328M16—2442no—
329C2×C8—22no—
330C2×C8—41no—
331C2×C8—1222no—
332C2×C8—1222no—
333C2×C8—1221no—
334C2×C8—1221no—
335C2×C8—2444no—
336C2×C8C2×C22⋊D6111yescharacteristic
337C2×C8C2×C22⋊D6111yescharacteristic
338C2×C8C2×C22⋊D6111yescharacteristic
339C2×C8C2×C22⋊D6111yescharacteristic
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—611no—
347C4⋊C4—611no—
348C4⋊C4—1221no—
349C4⋊C4—1221no—
350C22⋊C4—42no—
351C22⋊C4—1221no—
352C22⋊C4—1221no—
353C22⋊C4—1221no—
354C42—42no—
355C42—1221no—
356C42—1221no—
357C42—1221no—
358C42D4×D6111yescharacteristic
359C42#1175822411yescharacteristic
360C42C2×C22⋊D6111yescharacteristic
361C42C23×D6111yescharacteristic
362C23—1221no—
363C23#1451974811yescharacteristic
364Q8—42no—
365Q8—42no—
366Q8—82no—
367Q8—1222no—
368D4—1222no—
369D4—2442no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—1221no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—2442no—
379C2×C4—2442no—
380C2×C4—2441no—
381C2×C4—2441no—
382C2×C4#1451974811yescharacteristic
383C2×C4#1451974811yescharacteristic
384C2×C4#1451974811yescharacteristic
385C2×C4#1451980914yes—
386C2×C4C2×D4×D6111yescharacteristic
387C2×C4C22×C22⋊D6111yescharacteristic
388C2×C4C2×#1175822411yescharacteristic
389C8—21no—
390C8—21no—
391C8—21no—
392C8—2441no—
393C8—2442no—
394C8—2442no—
395C8—2441no—
396C8#1451979712yes—
397C22—21no—
398C22—21no—
399C22—2441no—
400C22—2441no—
401C22—2441no—
402C22#1684001712yes—
403C22C2×#1451974811yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—82no—
408C4—82no—
409C4—2441no—
410C4—2441no—
411C4—2441no—
412C4#1684001712yes—
413C4C2×#1451979711yescharacteristic
414C4C2×#1451980912yes—
415C4#1684056011yescharacteristic
416C2—22no—
417C2—4881no—
418C2#2145526811yescharacteristic
419C2C2×#1684001711yescharacteristic
420C2#2145553511yescharacteristic
subgroupG/H
centerC2×C4#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3170C2
socleC22×C61C2×#121
prime radicalC2×#11758222D4
hypercenter#3170D61
hypocenterC61#19158
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3170

Complements

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

NH ≅ G/Ntype
3. #21445398C2semidirect
4. #21445391C2semidirect
5. #21445344C2semidirect
6. #21445339C2semidirect
7. #21445310C2semidirect
8. #21445305C2semidirect
9. C2×#16839513C2semidirect
10. #21444476C2semidirect
11. C61×#3170C2semidirect
206. #3170D61semidirect
257. C61#19158semidirect

Maximal subgroups

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

subgroupindex
#214556012
#214547852
#214453982
#214453912
#214453442
#214453392
#214453102
#214453052
C2×#168395132
#214444762
C61×#31702
#1915861