Finite Group Databasebeta

← #29834931

Subgroups — #29834931

Order 15616.

546 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455482C211yesmaximal, characteristic
2#21454427C211yesmaximal, characteristic, complemented
3#21448040C214yesmaximal, complemented
4#21448035C214yesmaximal, complemented
5C2×#16839584C211yesmaximal, characteristic, complemented
6#21447569C211yesmaximal, characteristic, complemented
7#21447566C211yesmaximal, characteristic, complemented
8#21447439C211yesmaximal, characteristic, complemented
9C2×C61×#365C211yesmaximal, characteristic, complemented
10C2×#14519799C2211yescharacteristic
11#16840528—21no—
12C2×#14519783C2211yescharacteristic
13#16840343—21no—
14C2×#14519717C2211yescharacteristic
15#16840021C2212yes—
16#16840016C2212yes—
17#16839992—22no—
18#16839810—21no—
19#16839809C2214yes—
20C2×#14519713C2211yescharacteristic
21#108×D61—22no—
22#16839761—21no—
23#16839584C2214yes—
24#16839579C2212yes—
25#16839577C2212yes—
26#16839576C2212yes—
27#16839574C2212yes—
28#16839543C2214yes—
29C2×M32×C61C2211yescharacteristic
30C2×#108×C61C2211yescharacteristic
31C2×#106×C61C2211yescharacteristic
32C61×#365C2214yes—
33C2×#11758222—21no—
34C2×#11758222D411yescharacteristic
35#14519799—21no—
36#14519799—21no—
37#14519799—21no—
38#14519799—21no—
39C2×Dic244C2311yescharacteristic
40C2×#11758190—21no—
41C2×D488C2311yescharacteristic
42#14519785—42no—
43#14519783—21no—
44#14519783—21no—
45#14519783—21no—
46#14519783—21no—
47#14519783—42no—
48C2×#11758194—21no—
49C2×#11758194D411yescharacteristic
50C2×#11758189C2312yes—
51M16×D61—24no—
52C2×C8×D61D412yes—
53#14519757—22no—
54#14519757—41no—
55#14519752—22no—
56#14519752—41no—
57#14519751—42no—
58#14519717—21no—
59#14519717C2312yes—
60C2×#11758185—21no—
61#14519713—21no—
62#14519713C2312yes—
63#14519712—24no—
64#14519700—22no—
65#14519697—22no—
66C2×M16×C61—21no—
67C22×C8×C61C2311yescharacteristic
68C2×C4⋊C4×C61—21no—
69M32×C61—21no—
70M32×C61C2312yes—
71C2×C16×C61—21no—
72#108×C61—21no—
73#108×C61C2312yes—
74#106×C61—21no—
75#106×C61C2312yes—
76C22×C4×C61—41no—
77C22×C4×C61C2×D411yescharacteristic
78C2×C22⋊D61—41no—
79C2×C22⋊D61SD1612yes—
80#11758222—21no—
81#11758222—21no—
82#11758222—21no—
83#11758222—21no—
84#11758222—41no—
85#11758222—41no—
86#11758222—41no—
87#11758222—41no—
88C2×D244—21no—
89C2×D244—42no—
90C2×D244SD1611yescharacteristic
91C2×C4×D61—41no—
92C2×C4×D61C2×D412yes—
93C2×Dic122—21no—
94C2×Dic122SD1611yescharacteristic
95M16×C61—22no—
96M16×C61—41no—
97C2×C8×C61—41no—
98C2×C8×C61C2×D411yescharacteristic
99C2×C8×C61#4111yescharacteristic
100C2×C8×C61#4111yescharacteristic
101C2×C8×C61#4111yescharacteristic
102C2×C8×C61#4111yescharacteristic
103C2×C8×C61C2411yescharacteristic
104C4⋊C4×C61—22no—
105C4⋊C4×C61—41no—
106#11758198—42no—
107#11758198—42no—
108C4⋊Dic61—42no—
109#11758194—21no—
110#11758194—21no—
111#11758194—22no—
112#11758194—41no—
113C2×#92418—41no—
114C2×#92418C2×D412yes—
115Dic244—21no—
116Dic244—21no—
117D488—21no—
118D488—21no—
119#11758190—41no—
120#11758190—41no—
121#11758189—22no—
122#11758189—22no—
123#11758189—44no—
124C8×D61—22no—
125C8×D61—22no—
126C8×D61—44no—
127C16×C61—22no—
128#11758185—22no—
129C23×C61C2×SD1611yescharacteristic
130C22×D61—41no—
131C22×D61C2×SD1612yes—
132C2×C4×C61—42no—
133C2×C4×C61—161no—
134C2×C4×C61#12311yescharacteristic
135C2×C4×C61#12311yescharacteristic
136C2×C4×C61C2×SD1611yescharacteristic
137C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C61C22×D411yescharacteristic
140C22⋊D61—22no—
141C22⋊D61—22no—
142C22⋊D61—81no—
143C22⋊D61—81no—
144C2×Dic61—41no—
145C2×Dic61C2×SD1612yes—
146D244—21no—
147D244—21no—
148D244—41no—
149D244—41no—
150D244—162no—
151C4×D61—22no—
152C4×D61—22no—
153C4×D61—81no—
154C4×D61—81no—
155Dic122—21no—
156Dic122—21no—
157Dic122—41no—
158Dic122—41no—
159C8×C61—42no—
160C8×C61#12311yescharacteristic
161C8×C61#12311yescharacteristic
162C8×C61C2×#4111yescharacteristic
163C8×C61C2×#4111yescharacteristic
164#92418—22no—
165#92418—42no—
166#33659—611nomaximal, Sylow
167C22×C61—41no—
168C22×C61#48111yescharacteristic
169C22×C61#48111yescharacteristic
170C22×C61C22×SD1611yescharacteristic
171C2×D61—22no—
172C2×D61—42no—
173C2×D61—81no—
174C2×D61—81no—
175C2×D61—81no—
176C4×C61—82no—
177C4×C61#48111yescharacteristic
178C4×C61#48111yescharacteristic
179C4×C61C2×#12311yescharacteristic
180C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
181Dic61—22no—
182Dic61—81no—
183#3060—611no—
184#2916—611no—
185#2239—614no—
186#2238—614no—
187C2×#365—611no—
188C2×#365D6111yescharacteristic, complemented
189#2148—611no—
190#2148—611no—
191#2115—611no—
192C2×C61—21no—
193C2×C61—21no—
194C2×C61—21no—
195C2×C61C2×#48111yescharacteristic
196D61—42no—
197D61—161no—
198C2×#135—611no—
199#568—1221no—
200C2×#131—611no—
201#537—1221no—
202C2×M32—611no—
203C2×M32C2×D6111yescharacteristic
204#470—612no—
205#469—612no—
206#465—1222no—
207#430—614no—
208#430—1221no—
209C2×#108—611no—
210C2×#108—1222no—
211C2×#108C2×D6111yescharacteristic
212C2×#106C2×D6111yescharacteristic
213#417—1221no—
214#365—614no—
215#365C2×D6114yes—
216#362—612no—
217#362—612no—
218#361—612no—
219#361—612no—
220#349—614no—
221C61#3365911yescharacteristic, complemented, Sylow
222C2×#41—611no—
223C2×#41—1221no—
224#135—1221no—
225#135—1221no—
226#135—1221no—
227#135—1221no—
228C2×Q16—611no—
229C2×SD16—1221no—
230C2×D8—611no—
231#131—1221no—
232#131—1221no—
233#131—1221no—
234#131—1221no—
235#131—2442no—
236#131—2442no—
237C2×M16—21no—
238C2×M16—611no—
239C2×M16—612no—
240C2×M16—1224no—
241C2×M16—1221no—
242C22×C8—612no—
243C22×C8C22×D6111yescharacteristic
244C22⋊Q8—1222no—
245C22⋊Q8—2441no—
246#121—1222no—
247#121—2441no—
248#121—2442no—
249C2×C4⋊C4—21no—
250M32—21no—
251M32—612no—
252M32—1221no—
253M32C22×D6112yes—
254C2×C16—21no—
255C2×C16—1221no—
256#108—21no—
257#108—612no—
258#108—1224no—
259#108—1221no—
260#108C22×D6112yes—
261#106—21no—
262#106C22×D6112yes—
263#103—1222no—
264#102—1222no—
265#41—1221no—
266#41—1221no—
267#41—1221no—
268#41—1221no—
269#41—2441no—
270#41—2441no—
271#41—2441no—
272#41—2441no—
273C2×Q8—611no—
274C2×Q8—1221no—
275C2×D4—612no—
276C2×D4—611no—
277C2×D4—1221no—
278C2×D4—2442no—
279C2×D4—2441no—
280C22×C4—41no—
281C22×C4—612no—
282C22×C4—2441no—
283C22×C4D4×D6111yescharacteristic
284Q16—1221no—
285Q16—1221no—
286SD16—2441no—
287SD16—2441no—
288D8—1221no—
289D8—1221no—
290M16—22no—
291M16—41no—
292M16—1221no—
293M16—1221no—
294M16—1222no—
295M16—1222no—
296M16—1222no—
297M16—2444no—
298M16—2441no—
299C2×C8—41no—
300C2×C8—612no—
301C2×C8—1222no—
302C2×C8—1222no—
303C2×C8—2444no—
304C2×C8—2441no—
305C2×C8#1175822211yescharacteristic
306C2×C8#1175822211yescharacteristic
307C2×C8D4×D6111yescharacteristic
308C2×C8#1175822611yescharacteristic
309C2×C8#1175822611yescharacteristic
310C2×C8C23×D6111yescharacteristic
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—2442no—
314C22⋊C4—2442no—
315C22⋊C4—2442no—
316C16—22no—
317C16—1222no—
318C23—612no—
319C23—2441no—
320C23SD16×D6111yescharacteristic
321Q8—1221no—
322Q8—1221no—
323Q8—2441no—
324Q8—2441no—
325D4—1222no—
326D4—1222no—
327D4—1221no—
328D4—1221no—
329D4—9762no—
330D4—4881no—
331D4—4881no—
332D4—2441no—
333D4—2441no—
334C2×C4—42no—
335C2×C4—161no—
336C2×C4—612no—
337C2×C4—1222no—
338C2×C4—1222no—
339C2×C4—4881no—
340C2×C4—4881no—
341C2×C4—2441no—
342C2×C4#1451976211yescharacteristic
343C2×C4#1451976211yescharacteristic
344C2×C4SD16×D6111yescharacteristic
345C2×C4#1451980411yescharacteristic
346C2×C4#1451980411yescharacteristic
347C2×C4C2×D4×D6111yescharacteristic
348C8—42no—
349C8—1222no—
350C8—2442no—
351C8#1451976211yescharacteristic
352C8#1451976211yescharacteristic
353C8C2×#1175822211yescharacteristic
354C8C2×#1175822611yescharacteristic
355C22—41no—
356C22—1222no—
357C22—4881no—
358C22—4881no—
359C22—4881no—
360C22—2442no—
361C22#1684008211yescharacteristic
362C22#1684008211yescharacteristic
363C22C2×SD16×D6111yescharacteristic
364C4—82no—
365C4—1222no—
366C4—4881no—
367C4#1684008211yescharacteristic
368C4#1684008211yescharacteristic
369C4C2×#1451976211yescharacteristic
370C4C2×#1451980411yescharacteristic
371C2—21no—
372C2—21no—
373C2—21no—
374C2—9761no—
375C2—2442no—
376C2C2×#1684008211yescharacteristic
subgroupG/H
centerC2C2×#16840082
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#365C2
socleC2×C61C2×#481
prime radical#21455482C2
hypercenterC2×#365D61
hypocenterC61#33659
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ C2×#365

Complements

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

NH ≅ G/Ntype
2. #21454427C2semidirect
3. #21448040C2semidirect
4. #21448035C2semidirect
5. C2×#16839584C2semidirect
6. #21447569C2semidirect
7. #21447566C2semidirect
8. #21447439C2semidirect
9. C2×C61×#365C2semidirect
188. C2×#365D61semidirect
221. C61#33659semidirect

Maximal subgroups

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

subgroupindex
#214554822
#214544272
#214480402
#214480352
C2×#168395842
#214475692
#214475662
#214474392
C2×C61×#3652
#3365961