Finite Group Databasebeta

← #29725152

Subgroups — #29725152

Order 15616.

620 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840038C211yesmaximal, characteristic
2#21454686C211yesmaximal, characteristic, complemented
3#21447288C212yesmaximal, complemented
4#21447102C212yesmaximal
5#21446734C211yesmaximal, characteristic, complemented
6#21446607C211yesmaximal, characteristic, complemented
7#21446314C212yesmaximal
8#21445932C212yesmaximal, complemented
9#21445870C211yesmaximal, characteristic
10#21445620C211yesmaximal, characteristic
11C61×#1833C211yesmaximal, characteristic, complemented
12C2×#14519810C2212yes—
13C22×Dic244C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15#16840239C2212yes—
16#16840038—22no—
17#16840038—22no—
18#16839868—22no—
19#16839856—22no—
20#16839810C2212yes—
21C2×#14519713C2211yescharacteristic
22#16839780C2212yes—
23#16839770C2212yes—
24#16839770C2212yes—
25#16839744C2211yescharacteristic
26C2×#14519695C2212yes—
27C2×#14519695C2212yes—
28#16839735C2211yescharacteristic
29#16839717C2211yescharacteristic
30#16839534C2211yescharacteristic
31#16839533C2212yes—
32#16839532C2212yes—
33#16839532C2211yescharacteristic
34C2×#108×C61C2211yescharacteristic
35C61×#427C2211yescharacteristic
36C61×#405C2211yescharacteristic
37C61×#344C2212yes—
38C61×H(3,ℤ/4)C2212yes—
39C2×#11758222C2311yescharacteristic
40C22×Dic122C2312yes—
41#14519810—24no—
42#14519810—42no—
43C2×Dic244—21no—
44C2×Dic244—21no—
45C2×Dic244—21no—
46C2×Dic244—21no—
47C2×Dic244—22no—
48C2×Dic244D412yes—
49C2×Dic244D412yes—
50C2×#11758190—22no—
51#14519783—21no—
52#14519783—21no—
53#14519783—21no—
54#14519783—21no—
55C2×#11758194C2311yescharacteristic
56C2×#11758194C2311yescharacteristic
57C2×#11758194C2312yes—
58C2×#11758189—21no—
59C2×C8×D61—21no—
60C4×Dic122—42no—
61#14519742—24no—
62#14519736C2311yescharacteristic
63#14519733—24no—
64#14519713—21no—
65#14519713—21no—
66#14519712—42no—
67#14519705—42no—
68#14519700—24no—
69#14519700—42no—
70#14519695—24no—
71#14519695—21no—
72#14519695—42no—
73#14519693—21no—
74#14519691—22no—
75#14519687—21no—
76C8×Dic61—21no—
77C2×M16×C61C2311yescharacteristic
78C2×M16×C61C2312yes—
79C22×C8×C61C2311yescharacteristic
80#117×C61C2312yes—
81#117×C61C2311yescharacteristic
82#108×C61—21no—
83#108×C61—21no—
84#107×C61—21no—
85#106×C61—21no—
86#97×C61—21no—
87C4×C8×C61—21no—
88C22×C4×C61C2411yescharacteristic
89C2×C22⋊D61—21no—
90C22×Dic61—22no—
91#11758222—21no—
92#11758222—21no—
93#11758222—21no—
94#11758222—21no—
95C2×D244#4111yescharacteristic
96C2×C4×D61—21no—
97C2×Dic122—22no—
98C2×Dic122—22no—
99C2×Dic122—22no—
100C2×Dic122—22no—
101C2×Dic122—22no—
102C2×Dic122C2×D414yes—
103C2×Dic122#4111yescharacteristic
104M16×C61—22no—
105M16×C61—21no—
106M16×C61—22no—
107M16×C61—21no—
108C2×C8×C61—21no—
109C2×C8×C61—22no—
110C2×C8×C61C2×D411yescharacteristic
111C2×C8×C61C2×D411yescharacteristic
112C2×C8×C61#4111yescharacteristic
113C2×C8×C61#4111yescharacteristic
114C2×C8×C61#4111yescharacteristic
115C2×C8×C61#4111yescharacteristic
116C4⋊C4×C61—22no—
117C4⋊C4×C61—21no—
118C22⋊C4×C61—21no—
119C22⋊C4×C61—22no—
120C42×C61—21no—
121C42×C61—22no—
122C22⋊Dic61#4112yes—
123C4⋊Dic61—21no—
124#11758196—44no—
125C4×Dic61—21no—
126C4×Dic61—44no—
127#11758194—21no—
128#11758194—21no—
129#11758194—22no—
130#11758194—21no—
131#11758194—22no—
132#11758194—21no—
133C2×#92418—21no—
134C2×#92418—21no—
135C2×#92418—22no—
136Dic244—22no—
137Dic244—22no—
138Dic244—22no—
139Dic244—22no—
140Dic244—44no—
141#11758190—44no—
142#11758189—41no—
143#11758189—41no—
144C8×D61—41no—
145C8×D61—41no—
146C23×C61C2×#4111yescharacteristic
147C22×D61—21no—
148C2×C4×C61—42no—
149C2×C4×C61—41no—
150C2×C4×C61C22×D411yescharacteristic
151C2×C4×C61C2×#4111yescharacteristic
152C2×C4×C61C2×#4111yescharacteristic
153C2×C4×C61C2×#4111yescharacteristic
154C2×C4×C61H(5,𝔽2)11yescharacteristic
155C2×C4×C61H(5,𝔽2)11yescharacteristic
156C22⋊D61—41no—
157C22⋊D61—41no—
158C2×Dic61—21no—
159C2×Dic61—22no—
160C2×Dic61—24no—
161C2×Dic61—42no—
162C2×Dic61—42no—
163D244—21no—
164D244—21no—
165C4×D61—41no—
166C4×D61—41no—
167Dic122—24no—
168Dic122—24no—
169Dic122—21no—
170Dic122—21no—
171Dic122—42no—
172Dic122—42no—
173C8×C61—42no—
174C8×C61—41no—
175C8×C61#12311yescharacteristic
176C8×C61#12311yescharacteristic
177C8×C61#12311yescharacteristic
178C8×C61#12311yescharacteristic
179#92418—41no—
180#92418—42no—
181#92418—41no—
182#24989—611nomaximal, Sylow
183C22×C61—41no—
184C22×C61#52911yescharacteristic
185C22×C61#54611yescharacteristic
186C22×C61#55511yescharacteristic
187C2×D61—41no—
188C2×D61—81no—
189C4×C61—82no—
190C4×C61—81no—
191C4×C61C2×#12311yescharacteristic
192C4×C61#54011yescharacteristic
193C4×C61#54611yescharacteristic
194C4×C61#55311yescharacteristic
195Dic61—42no—
196Dic61—44no—
197Dic61—41no—
198C2×#473—611no—
199#2950—611no—
200#2082—612no—
201#2049—612no—
202#1979—611no—
203#1953—611no—
204#1905—612no—
205#1833D6111yescharacteristic, complemented
206#1830—612no—
207#1814—611no—
208#1757—611no—
209C2×C61—21no—
210C2×C61—21no—
211C2×C61—21no—
212C2×C61#242011yescharacteristic
213D61—81no—
214C2×#137—612no—
215C22×Q16—611no—
216C2×#131—611no—
217#519—612no—
218#473—1222no—
219#473—1222no—
220#441—1222no—
221C4×Q16—1222no—
222#430—612no—
223C2×#108—611no—
224C2×#108C2×D6111yescharacteristic
225#427C2×D6111yescharacteristic
226#421—612no—
227#419—612no—
228#419—612no—
229#413—611no—
230C2×#101—612no—
231C2×#101—612no—
232#410—611no—
233#405—611no—
234#405C2×D6111yescharacteristic
235#344—611no—
236#344—612no—
237#344—612no—
238#344—611no—
239#344C2×D6112yes—
240H(3,ℤ/4)C2×D6112yes—
241C61#2498911yescharacteristic, complemented, Sylow
242C2×#41—611no—
243C22×Q8—612no—
244#137—1224no—
245#137—2442no—
246C2×Q16—612no—
247C2×Q16—612no—
248C2×Q16—1222no—
249C2×Q16—1221no—
250C2×Q16—1221no—
251C2×Q16—1221no—
252C2×Q16—1221no—
253C2×SD16—1222no—
254#131—1221no—
255#131—1221no—
256#131—1221no—
257#131—1221no—
258C2×M16—611no—
259C2×M16—612no—
260C2×M16—611no—
261C2×M16—1221no—
262C2×M16C22×D6112yes—
263C2×M16C22×D6111yescharacteristic
264C22×C8—1221no—
265C22×C8C22×D6111yescharacteristic
266C4×Q8—1224no—
267C4×Q8—2442no—
268#117—611no—
269#117—1224no—
270#117C22×D6112yes—
271#117C22×D6111yescharacteristic
272#108—21no—
273#108—21no—
274#108—1221no—
275#108—1221no—
276#108—2442no—
277#107—21no—
278#106—21no—
279#104—2442no—
280#103—1224no—
281#103—2442no—
282#101—1224no—
283#101—1221no—
284#101—2442no—
285#100—1221no—
286#99—1222no—
287#97—21no—
288#97—1221no—
289C4×C8—21no—
290C4×C8—1221no—
291#41—1221no—
292#41—1221no—
293#41—1221no—
294#41—1221no—
295C2×Q8—614no—
296C2×Q8—611no—
297C2×Q8—1222no—
298C2×Q8—1222no—
299C2×Q8—1222no—
300C2×Q8—1222no—
301C2×Q8—1222no—
302C2×D4—611no—
303C2×D4—1221no—
304C22×C4—1222no—
305C22×C4—1221no—
306C22×C4C23×D6111yescharacteristic
307Q16—1222no—
308Q16—1222no—
309Q16—1222no—
310Q16—1222no—
311Q16—2444no—
312SD16—2444no—
313M16—22no—
314M16—21no—
315M16—21no—
316M16—22no—
317M16—1221no—
318M16—1221no—
319M16—1221no—
320M16—1222no—
321M16—1222no—
322M16—1221no—
323M16—2441no—
324M16—2441no—
325C2×C8—22no—
326C2×C8—21no—
327C2×C8—1221no—
328C2×C8—1221no—
329C2×C8—1222no—
330C2×C8—2441no—
331C2×C8—2441no—
332C2×C8#1175822211yescharacteristic
333C2×C8#1175822211yescharacteristic
334C2×C8D4×D6111yescharacteristic
335C2×C8D4×D6111yescharacteristic
336C2×C8#1175822611yescharacteristic
337C2×C8#1175822611yescharacteristic
338C4⋊C4—22no—
339C4⋊C4—21no—
340C4⋊C4—1221no—
341C4⋊C4—2444no—
342C22⋊C4—21no—
343C22⋊C4—22no—
344C22⋊C4—612no—
345C42—21no—
346C42—22no—
347C42—1221no—
348C42—2444no—
349C23—1221no—
350C23#41×D6111yescharacteristic
351Q8—1221no—
352Q8—1224no—
353Q8—1224no—
354Q8—1221no—
355Q8—2442no—
356Q8—2442no—
357D4—1221no—
358D4—1221no—
359D4—2441no—
360D4—2441no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—1221no—
364C2×C4—1222no—
365C2×C4—1224no—
366C2×C4—2442no—
367C2×C4—2442no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4C2×D4×D6111yescharacteristic
371C2×C4#41×D6111yescharacteristic
372C2×C4C2×#1175822211yescharacteristic
373C2×C4C2×#1175822611yescharacteristic
374C2×C4#1451982311yescharacteristic
375C2×C4#1451982311yescharacteristic
376C8—41no—
377C8—42no—
378C8—2441no—
379C8—2442no—
380C8—2441no—
381C8#1451976211yescharacteristic
382C8#1451976211yescharacteristic
383C8#1451976211yescharacteristic
384C8#1451976211yescharacteristic
385C22—41no—
386C22—4881no—
387C22—2441no—
388C22#1684029711yescharacteristic
389C22#1684040511yescharacteristic
390C22#1684047511yescharacteristic
391C4—82no—
392C4—81no—
393C4—2442no—
394C4—2441no—
395C4—2444no—
396C4C2×#1451976211yescharacteristic
397C4#1684036811yescharacteristic
398C4#1684040511yescharacteristic
399C4#1684045611yescharacteristic
400C2—21no—
401C2—21no—
402C2—21no—
403C2—4881no—
404C2#2144931411yescharacteristic
subgroupG/H
centerC2#21449314
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1833C2
socleC2×C61#2420
prime radicalC2×#11758222C23
hypercenter#1833D61
hypocenterC61#24989
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1833

Complements

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

NH ≅ G/Ntype
2. #21454686C2semidirect
3. #21447288C2semidirect
5. #21446734C2semidirect
6. #21446607C2semidirect
8. #21445932C2semidirect
11. C61×#1833C2semidirect
205. #1833D61semidirect
241. C61#24989semidirect

Maximal subgroups

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

subgroupindex
C2×#168400382
#214546862
#214472882
#214471022
#214467342
#214466072
#214463142
#214459322
#214458702
#214456202
C61×#18332
#2498961