Finite Group Databasebeta

← #29725138

Subgroups — #29725138

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×#16840032C211yesmaximal, characteristic
2#21454672C211yesmaximal, characteristic, complemented
3#21447282C212yesmaximal, complemented
4#21447092C212yesmaximal, complemented
5#21446729C211yesmaximal, characteristic, complemented
6#21446609C211yesmaximal, characteristic, complemented
7#21446309C212yesmaximal, complemented
8#21445943C211yesmaximal, characteristic, complemented
9#21445929C212yesmaximal, complemented
10#21445873C211yesmaximal, characteristic, complemented
11C61×#1757C211yesmaximal, characteristic, complemented
12C2×#14519802C2212yes—
13C22×D488C2211yescharacteristic
14#16840236C2212yes—
15#16840032—22no—
16#16840032—22no—
17#16839875—22no—
18C4×D488—22no—
19#16839810C2212yes—
20C2×#14519713C2211yescharacteristic
21C2×#14519713C2211yescharacteristic
22#16839800C2211yescharacteristic
23#16839779C2212yes—
24#16839768C2212yes—
25#16839766C2212yes—
26C2×#14519693C2212yes—
27C2×#14519693C2212yes—
28#16839534C2212yes—
29#16839533C2212yes—
30#16839532C2212yes—
31#16839518C2212yes—
32C2×#131×C61C2211yescharacteristic
33C61×#413C2211yescharacteristic
34C61×#410C2211yescharacteristic
35C61×#405C2211yescharacteristic
36C61×#405C2211yescharacteristic
37C61×#344C2211yescharacteristic
38C61×#344C2211yescharacteristic
39C22×D244C2312yes—
40#14519802—24no—
41#14519802—42no—
42C2×#11758201—22no—
43C2×D488—21no—
44C2×D488—21no—
45C2×D488—21no—
46C2×D488—21no—
47C2×D488D412yes—
48C2×D488D412yes—
49C2×#11758199—22no—
50C2×#11758194C2312yes—
51C2×#11758194C2312yes—
52C4×D244—24no—
53#14519739—42no—
54#14519736C2312yes—
55C22⋊C4×D61—24no—
56#14519713—21no—
57#14519713—21no—
58#14519713—21no—
59#14519713—21no—
60#14519712—42no—
61#14519711—21no—
62#14519709—21no—
63#14519704—42no—
64#14519699—24no—
65#14519697—42no—
66#14519693—24no—
67#14519693—42no—
68C2×#41×C61C2311yescharacteristic
69#131×C61—21no—
70#131×C61—21no—
71#131×C61—21no—
72#131×C61—21no—
73C2×M16×C61—21no—
74C2×M16×C61C2311yescharacteristic
75C2×M16×C61C2311yescharacteristic
76C2×M16×C61C2311yescharacteristic
77C22×C8×C61—21no—
78C22×C8×C61C2311yescharacteristic
79#117×C61C2311yescharacteristic
80#117×C61C2311yescharacteristic
81#101×C61—21no—
82#100×C61—21no—
83#99×C61—22no—
84#97×C61—21no—
85#97×C61—21no—
86C4×C8×C61—21no—
87C4×C8×C61—21no—
88C23×D61—22no—
89#41×C61—21no—
90#41×C61—21no—
91#41×C61—21no—
92#41×C61—21no—
93C2×Q8×C61#4111yescharacteristic
94C2×D4×C61—21no—
95C2×D4×C61#4111yescharacteristic
96C22×C4×C61—21no—
97C22×C4×C61C2411yescharacteristic
98C2×D244—22no—
99C2×D244—22no—
100C2×D244—22no—
101C2×D244—22no—
102C2×D244—22no—
103C2×D244C2×D414yes—
104C2×C4×D61—44no—
105M16×C61—21no—
106M16×C61—21no—
107M16×C61—21no—
108M16×C61—21no—
109M16×C61—21no—
110M16×C61—21no—
111M16×C61—41no—
112M16×C61—41no—
113C2×C8×C61—21no—
114C2×C8×C61—21no—
115C2×C8×C61—21no—
116C2×C8×C61—41no—
117C2×C8×C61—41no—
118C2×C8×C61C2×D411yescharacteristic
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61#4111yescharacteristic
121C2×C8×C61#4111yescharacteristic
122C2×C8×C61#4111yescharacteristic
123C2×C8×C61#4111yescharacteristic
124C4⋊C4×C61—21no—
125C4⋊C4×C61—21no—
126C22⋊C4×C61—21no—
127C22⋊C4×C61#4112yes—
128C42×C61—21no—
129C42×C61—21no—
130C22⋊Dic61—22no—
131#11758201—44no—
132#11758199—44no—
133#11758198—44no—
134C4⋊Dic61—22no—
135C4×Dic61—22no—
136#11758194—22no—
137#11758194—22no—
138#11758194—22no—
139#11758194—22no—
140C2×#92418—22no—
141C2×#92418—22no—
142D488—22no—
143D488—22no—
144D488—22no—
145D488—22no—
146C23×C61—21no—
147C23×C61C2×#4111yescharacteristic
148C22×D61—24no—
149C22×D61—42no—
150C22×D61—42no—
151C22×D61—162no—
152Q8×C61—21no—
153Q8×C61—21no—
154D4×C61—21no—
155D4×C61—21no—
156D4×C61—41no—
157D4×C61—41no—
158C2×C4×C61—21no—
159C2×C4×C61—22no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61C22×D411yescharacteristic
164C2×C4×C61C2×#4111yescharacteristic
165C2×C4×C61C2×#4111yescharacteristic
166C2×C4×C61C2×#4111yescharacteristic
167C2×C4×C61H(5,𝔽2)11yescharacteristic
168C2×C4×C61H(5,𝔽2)11yescharacteristic
169C2×Dic61—42no—
170D244—24no—
171D244—24no—
172D244—42no—
173D244—42no—
174C4×D61—164no—
175C8×C61—41no—
176C8×C61—41no—
177C8×C61—41no—
178C8×C61#12311yescharacteristic
179C8×C61#12311yescharacteristic
180C8×C61#12311yescharacteristic
181C8×C61#12311yescharacteristic
182#92418—42no—
183#92418—42no—
184#24988—611nomaximal, Sylow
185C22×C61—41no—
186C22×C61—41no—
187C22×C61—81no—
188C22×C61#52911yescharacteristic
189C22×C61#54611yescharacteristic
190C22×C61#55511yescharacteristic
191C2×D61—44no—
192C2×D61—84no—
193C2×D61—162no—
194C2×D61—162no—
195C4×C61—42no—
196C4×C61—41no—
197C4×C61—81no—
198C4×C61C2×#12311yescharacteristic
199C4×C61#54011yescharacteristic
200C4×C61#54611yescharacteristic
201C4×C61#55311yescharacteristic
202Dic61—82no—
203C2×#472—611no—
204#2948—611no—
205#2081—612no—
206#2048—612no—
207#1978—611no—
208#1953—611no—
209#1904—612no—
210#1833—611no—
211#1829—612no—
212#1814—611no—
213#1757D6111yescharacteristic, complemented
214C2×C61—21no—
215C2×C61—21no—
216C2×C61—21no—
217C2×C61—81no—
218C2×C61#242011yescharacteristic
219D61—84no—
220C2×AGL(1,ℤ/8)—612no—
221C22×D8—611no—
222C2×#131C2×D6111yescharacteristic
223#518—612no—
224#472—1222no—
225#472—1222no—
226#442—1222no—
227C4×D8—1222no—
228#430—612no—
229C2×#108—611no—
230C2×#108—611no—
231#427—611no—
232#421—612no—
233#418—612no—
234#418—612no—
235#413C2×D6111yescharacteristic
236C2×#100—612no—
237C2×#100—612no—
238#410C2×D6111yescharacteristic
239#405C2×D6111yescharacteristic
240#405C2×D6111yescharacteristic
241#344—612no—
242#344—612no—
243#344—612no—
244#344C2×D6111yescharacteristic
245#344C2×D6111yescharacteristic
246H(3,ℤ/4)—612no—
247C61#2498811yescharacteristic, complemented, Sylow
248C2×#41C22×D6111yescharacteristic
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—21no—
261#131—21no—
262#131—21no—
263#131—21no—
264C2×M16—21no—
265C2×M16—612no—
266C2×M16—612no—
267C2×M16C22×D6111yescharacteristic
268C2×M16C22×D6111yescharacteristic
269C2×M16C22×D6111yescharacteristic
270C22×C8—21no—
271C22×C8C22×D6111yescharacteristic
272C4×D4—1224no—
273C4×D4—2442no—
274#117—612no—
275#117C22×D6111yescharacteristic
276#117C22×D6111yescharacteristic
277C2×C22⋊C4—1224no—
278#108—1221no—
279#108—1221no—
280#108—1221no—
281#108—1221no—
282#108—2442no—
283#107—1221no—
284#106—1221no—
285#104—2442no—
286#102—1224no—
287#102—2442no—
288#101—21no—
289#100—21no—
290#100—1224no—
291#100—2442no—
292#99—22no—
293#97—21no—
294#97—21no—
295C4×C8—21no—
296C4×C8—21no—
297C24—1222no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302C2×Q8#1175822211yescharacteristic
303C2×D4—21no—
304C2×D4—614no—
305C2×D4—1222no—
306C2×D4—1222no—
307C2×D4—1222no—
308C2×D4—1222no—
309C2×D4—1222no—
310C2×D4#1175822211yescharacteristic
311C22×C4—21no—
312C22×C4—2444no—
313C22×C4C23×D6111yescharacteristic
314SD16—2444no—
315D8—1222no—
316D8—1222no—
317D8—1222no—
318D8—1222no—
319D8—2444no—
320M16—21no—
321M16—21no—
322M16—21no—
323M16—21no—
324M16—21no—
325M16—21no—
326M16—41no—
327M16—41no—
328M16—1222no—
329M16—1222no—
330M16—1222no—
331M16—1222no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—1222no—
338C2×C8—1222no—
339C2×C8#1175822211yescharacteristic
340C2×C8#1175822211yescharacteristic
341C2×C8D4×D6111yescharacteristic
342C2×C8D4×D6111yescharacteristic
343C2×C8#1175822411yescharacteristic
344C2×C8#1175822411yescharacteristic
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—1222no—
348C22⋊C4—21no—
349C22⋊C4—1222no—
350C22⋊C4—2444no—
351C22⋊C4#1175822212yes—
352C42—21no—
353C42—21no—
354C42—1222no—
355C23—21no—
356C23—1224no—
357C23—9762no—
358C23—2442no—
359C23—2442no—
360C23C2×#1175822211yescharacteristic
361Q8—21no—
362Q8—21no—
363D4—21no—
364D4—21no—
365D4—41no—
366D4—41no—
367D4—1224no—
368D4—1224no—
369D4—2442no—
370D4—2442no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—9764no—
377C2×C4—2442no—
378C2×C4C2×D4×D6111yescharacteristic
379C2×C4C2×#1175822411yescharacteristic
380C2×C4C2×#1175822211yescharacteristic
381C2×C4C2×#1175822211yescharacteristic
382C2×C4#1451982311yescharacteristic
383C2×C4#1451982311yescharacteristic
384C8—41no—
385C8—41no—
386C8—41no—
387C8—2442no—
388C8—2442no—
389C8#1451975911yescharacteristic
390C8#1451975911yescharacteristic
391C8#1451975911yescharacteristic
392C8#1451975911yescharacteristic
393C22—41no—
394C22—41no—
395C22—81no—
396C22—9762no—
397C22—9762no—
398C22—4884no—
399C22—2444no—
400C22#1684029611yescharacteristic
401C22#1684040811yescharacteristic
402C22#1684047711yescharacteristic
403C4—42no—
404C4—41no—
405C4—81no—
406C4—4882no—
407C4C2×#1451975911yescharacteristic
408C4#1684036711yescharacteristic
409C4#1684040811yescharacteristic
410C4#1684045811yescharacteristic
411C2—21no—
412C2—21no—
413C2—21no—
414C2—81no—
415C2—4884no—
416C2#2144931111yescharacteristic
subgroupG/H
centerC2#21449311
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1757C2
socleC2×C61#2420
prime radicalC2×#16840032C2
hypercenter#1757D61
hypocenterC61#24988
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1757

Complements

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

NH ≅ G/Ntype
2. #21454672C2semidirect
3. #21447282C2semidirect
4. #21447092C2semidirect
5. #21446729C2semidirect
6. #21446609C2semidirect
7. #21446309C2semidirect
8. #21445943C2semidirect
9. #21445929C2semidirect
10. #21445873C2semidirect
11. C61×#1757C2semidirect
213. #1757D61semidirect
247. C61#24988semidirect

Maximal subgroups

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

subgroupindex
C2×#168400322
#214546722
#214472822
#214470922
#214467292
#214466092
#214463092
#214459432
#214459292
#214458732
C61×#17572
#2498861