Finite Group Databasebeta

← #29725139

Subgroups — #29725139

Order 15616.

656 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×#16840033C211yesmaximal, characteristic
2#21454672C211yesmaximal, characteristic, complemented
3#21447277C212yesmaximal, complemented
4#21447094C212yesmaximal, complemented
5#21446727C211yesmaximal, characteristic, complemented
6#21446305C212yesmaximal, complemented
7#21445944C211yesmaximal, characteristic, complemented
8#21445929C212yesmaximal, complemented
9#21445874C211yesmaximal, characteristic, complemented
10#21445620C211yesmaximal, characteristic, complemented
11C61×#1953C211yesmaximal, characteristic, complemented
12C2×#14519805C2212yes—
13C22×D488C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15#16840236C2212yes—
16#16840033—22no—
17#16840033—22no—
18#16839875—22no—
19C4×D488—22no—
20#16839800C2211yescharacteristic
21#16839778C2212yes—
22#16839768C2212yes—
23#16839768C2212yes—
24#16839744C2211yescharacteristic
25C2×#14519693C2212yes—
26C2×#14519693C2212yes—
27#16839735C2211yescharacteristic
28#16839717C2211yescharacteristic
29#16839534C2212yes—
30#16839534C2211yescharacteristic
31#16839533C2212yes—
32#16839532C2211yescharacteristic
33#16839519C2212yes—
34C61×#430C2212yes—
35C2×#108×C61C2211yescharacteristic
36C2×#108×C61C2211yescharacteristic
37C61×#405C2211yescharacteristic
38C61×#344C2212yes—
39C2×#11758222C2311yescharacteristic
40C22×D244C2312yes—
41#14519805—24no—
42#14519805—42no—
43C2×#11758190—22no—
44C2×D488—21no—
45C2×D488—21no—
46C2×D488—21no—
47C2×D488—21no—
48C2×D488—22no—
49C2×D488D412yes—
50C2×D488D412yes—
51#14519783—21no—
52#14519783—21no—
53#14519783—21no—
54#14519783—21no—
55C2×#11758194C2311yescharacteristic
56C2×#11758194C2311yescharacteristic
57C2×#11758189—21no—
58C2×C8×D61—21no—
59C4×D244—24no—
60#14519739—42no—
61#14519736C2311yescharacteristic
62#14519736C2312yes—
63C22⋊C4×D61—24no—
64#14519711—21no—
65#14519709—21no—
66#14519703—42no—
67#14519699—24no—
68#14519699—42no—
69#14519695—21no—
70#14519693—24no—
71#14519693—21no—
72#14519693—42no—
73#14519691—22no—
74#14519687—21no—
75C8×Dic61—21no—
76C2×M16×C61C2312yes—
77C2×M16×C61C2312yes—
78C2×M16×C61C2311yescharacteristic
79C22×C8×C61C2311yescharacteristic
80#117×C61C2311yescharacteristic
81#108×C61—21no—
82#108×C61—21no—
83#108×C61—21no—
84#108×C61—21no—
85#108×C61—42no—
86#97×C61—21no—
87C4×C8×C61—21no—
88C23×D61—22no—
89C22×C4×C61C2411yescharacteristic
90C2×C22⋊D61—21no—
91#11758222—21no—
92#11758222—21no—
93#11758222—21no—
94#11758222—21no—
95C2×D244—22no—
96C2×D244—22no—
97C2×D244—22no—
98C2×D244—22no—
99C2×D244—22no—
100C2×D244C2×D414yes—
101C2×D244#4111yescharacteristic
102C2×C4×D61—21no—
103C2×C4×D61—44no—
104C2×Dic122#4111yescharacteristic
105M16×C61—22no—
106M16×C61—22no—
107M16×C61—21no—
108M16×C61—21no—
109M16×C61—22no—
110M16×C61—22no—
111C2×C8×C61—21no—
112C2×C8×C61—22no—
113C2×C8×C61—22no—
114C2×C8×C61C2×D411yescharacteristic
115C2×C8×C61C2×D411yescharacteristic
116C2×C8×C61#4111yescharacteristic
117C2×C8×C61#4111yescharacteristic
118C2×C8×C61#4111yescharacteristic
119C2×C8×C61#4111yescharacteristic
120C4⋊C4×C61—21no—
121C22⋊C4×C61#4112yes—
122C42×C61—21no—
123C22⋊Dic61—21no—
124C22⋊Dic61—22no—
125#11758198—44no—
126C4⋊Dic61—22no—
127C4⋊Dic61—21no—
128C4×Dic61—21no—
129C4×Dic61—22no—
130#11758194—21no—
131#11758194—21no—
132#11758194—21no—
133#11758194—21no—
134C2×#92418—21no—
135C2×#92418—21no—
136D488—22no—
137D488—22no—
138D488—22no—
139D488—22no—
140D488—44no—
141#11758190—44no—
142#11758189—41no—
143#11758189—41no—
144C8×D61—41no—
145C8×D61—41no—
146C23×C61C2×#4111yescharacteristic
147C22×D61—21no—
148C22×D61—24no—
149C22×D61—42no—
150C22×D61—42no—
151C22×D61—162no—
152C2×C4×C61—22no—
153C2×C4×C61C22×D411yescharacteristic
154C2×C4×C61C2×#4111yescharacteristic
155C2×C4×C61C2×#4111yescharacteristic
156C2×C4×C61C2×#4111yescharacteristic
157C2×C4×C61H(5,𝔽2)11yescharacteristic
158C2×C4×C61H(5,𝔽2)11yescharacteristic
159C22⋊D61—41no—
160C22⋊D61—41no—
161C2×Dic61—21no—
162C2×Dic61—42no—
163C2×Dic61—41no—
164D244—24no—
165D244—24no—
166D244—21no—
167D244—21no—
168D244—42no—
169D244—42no—
170C4×D61—41no—
171C4×D61—41no—
172C4×D61—164no—
173Dic122—21no—
174Dic122—21no—
175C8×C61—42no—
176C8×C61—42no—
177C8×C61—41no—
178C8×C61#12311yescharacteristic
179C8×C61#12311yescharacteristic
180C8×C61#12311yescharacteristic
181C8×C61#12311yescharacteristic
182#92418—41no—
183#92418—41no—
184#24988—611nomaximal, Sylow
185C22×C61—41no—
186C22×C61#52911yescharacteristic
187C22×C61#54611yescharacteristic
188C22×C61#55511yescharacteristic
189C2×D61—41no—
190C2×D61—44no—
191C2×D61—84no—
192C2×D61—81no—
193C2×D61—162no—
194C2×D61—162no—
195C4×C61—42no—
196C4×C61C2×#12311yescharacteristic
197C4×C61#54011yescharacteristic
198C4×C61#54611yescharacteristic
199C4×C61#55311yescharacteristic
200Dic61—41no—
201Dic61—82no—
202Dic61—81no—
203C2×#472—611no—
204#2948—611no—
205#2081—612no—
206#2048—612no—
207#1978—611no—
208#1953D6111yescharacteristic, complemented
209#1904—612no—
210#1833—611no—
211#1829—612no—
212#1814—611no—
213#1757—611no—
214C2×C61—21no—
215C2×C61—21no—
216C2×C61—21no—
217C2×C61#242011yescharacteristic
218D61—81no—
219D61—84no—
220C2×AGL(1,ℤ/8)—612no—
221C22×D8—611no—
222C2×#131—611no—
223#518—612no—
224#472—1222no—
225#472—1222no—
226#442—1222no—
227C4×D8—1222no—
228#430C2×D6112yes—
229C2×#108C2×D6111yescharacteristic
230C2×#108C2×D6111yescharacteristic
231#427—611no—
232#421—612no—
233#418—612no—
234#418—612no—
235#413—611no—
236C2×#100—612no—
237C2×#100—612no—
238#410—611no—
239#405—611no—
240#405C2×D6111yescharacteristic
241#344—612no—
242#344—612no—
243#344—611no—
244#344—611no—
245#344C2×D6112yes—
246H(3,ℤ/4)—612no—
247C61#2498811yescharacteristic, complemented, Sylow
248C2×#41—611no—
249C22×D4—612no—
250AGL(1,ℤ/8)—1224no—
251AGL(1,ℤ/8)—2442no—
252C2×SD16—1222no—
253C2×D8—612no—
254C2×D8—612no—
255C2×D8—1222no—
256C2×D8—1221no—
257C2×D8—1221no—
258C2×D8—1221no—
259C2×D8—1221no—
260#131—1221no—
261#131—1221no—
262#131—1221no—
263#131—1221no—
264C2×M16—611no—
265C2×M16—611no—
266C2×M16—1221no—
267C2×M16C22×D6112yes—
268C2×M16C22×D6112yes—
269C2×M16C22×D6111yescharacteristic
270C22×C8—1221no—
271C22×C8C22×D6111yescharacteristic
272C4×D4—1224no—
273C4×D4—2442no—
274#117—612no—
275#117—611no—
276#117C22×D6111yescharacteristic
277C2×C22⋊C4—1224no—
278#108—21no—
279#108—21no—
280#108—21no—
281#108—21no—
282#108—42no—
283#107—1221no—
284#106—1221no—
285#104—2442no—
286#102—1224no—
287#102—2442no—
288#101—1221no—
289#100—1224no—
290#100—1221no—
291#100—2442no—
292#99—1222no—
293#97—21no—
294#97—1221no—
295C4×C8—21no—
296C4×C8—1221no—
297C24—1222no—
298#41—1221no—
299#41—1221no—
300#41—1221no—
301#41—1221no—
302C2×Q8—611no—
303C2×D4—614no—
304C2×D4—611no—
305C2×D4—1221no—
306C2×D4—1222no—
307C2×D4—1222no—
308C2×D4—1222no—
309C2×D4—1222no—
310C2×D4—1222no—
311C22×C4—1221no—
312C22×C4—2444no—
313C22×C4C23×D6111yescharacteristic
314SD16—2444no—
315D8—1222no—
316D8—1222no—
317D8—1222no—
318D8—1222no—
319D8—2444no—
320M16—22no—
321M16—21no—
322M16—22no—
323M16—21no—
324M16—22no—
325M16—22no—
326M16—1221no—
327M16—1221no—
328M16—1221no—
329M16—1221no—
330M16—2441no—
331M16—2441no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—1221no—
336C2×C8—1221no—
337C2×C8—2441no—
338C2×C8—2441no—
339C2×C8#1175822211yescharacteristic
340C2×C8#1175822211yescharacteristic
341C2×C8D4×D6111yescharacteristic
342C2×C8D4×D6111yescharacteristic
343C2×C8#1175822611yescharacteristic
344C2×C8#1175822611yescharacteristic
345C4⋊C4—21no—
346C4⋊C4—1222no—
347C4⋊C4—1221no—
348C22⋊C4—1221no—
349C22⋊C4—1222no—
350C22⋊C4—2444no—
351C22⋊C4#1175822612yes—
352C42—21no—
353C42—1221no—
354C42—1222no—
355C23—1221no—
356C23—1224no—
357C23—9762no—
358C23—2442no—
359C23—2442no—
360C23C2×#1175822611yescharacteristic
361Q8—1221no—
362Q8—1221no—
363D4—1221no—
364D4—1224no—
365D4—1221no—
366D4—1224no—
367D4—2441no—
368D4—2441no—
369D4—2442no—
370D4—2442no—
371C2×C4—22no—
372C2×C4—1221no—
373C2×C4—9764no—
374C2×C4—2441no—
375C2×C4—2442no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4C2×D4×D6111yescharacteristic
379C2×C4#41×D6111yescharacteristic
380C2×C4C2×#1175822211yescharacteristic
381C2×C4C2×#1175822611yescharacteristic
382C2×C4#1451982211yescharacteristic
383C2×C4#1451982311yescharacteristic
384C8—42no—
385C8—41no—
386C8—42no—
387C8—2441no—
388C8—2441no—
389C8#1451976211yescharacteristic
390C8#1451976211yescharacteristic
391C8#1451976211yescharacteristic
392C8#1451976211yescharacteristic
393C22—41no—
394C22—9762no—
395C22—9762no—
396C22—4884no—
397C22—4881no—
398C22—2441no—
399C22—2444no—
400C22#1684029911yescharacteristic
401C22#1684040311yescharacteristic
402C22#1684047811yescharacteristic
403C4—42no—
404C4—4882no—
405C4—4881no—
406C4—2441no—
407C4C2×#1451976211yescharacteristic
408C4#1684036311yescharacteristic
409C4#1684040911yescharacteristic
410C4#1684046511yescharacteristic
411C2—21no—
412C2—21no—
413C2—21no—
414C2—4884no—
415C2—4881no—
416C2#2144931211yescharacteristic
subgroupG/H
centerC2#21449312
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1953C2
socleC2×C61#2420
prime radicalC2×#16840033C2
hypercenter#1953D61
hypocenterC61#24988
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1953

Complements

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

NH ≅ G/Ntype
2. #21454672C2semidirect
3. #21447277C2semidirect
4. #21447094C2semidirect
5. #21446727C2semidirect
6. #21446305C2semidirect
7. #21445944C2semidirect
8. #21445929C2semidirect
9. #21445874C2semidirect
10. #21445620C2semidirect
11. C61×#1953C2semidirect
208. #1953D61semidirect
247. C61#24988semidirect

Maximal subgroups

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

subgroupindex
C2×#168400332
#214546722
#214472772
#214470942
#214467272
#214463052
#214459442
#214459292
#214458742
#214456202
C61×#19532
#2498861