Finite Group Databasebeta

← #34083186

Subgroups — #34083186

Order 19328.

1104 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23752944C213yesmaximal, complemented
2Q8×Dic302C211yesmaximal, characteristic, complemented
3#23752920C213yesmaximal, complemented
4#23752881C211yesmaximal, characteristic, complemented
5#23752873C211yesmaximal, characteristic, complemented
6#23752866C213yesmaximal, complemented
7#23752862C213yesmaximal, complemented
8#23752835C213yesmaximal, complemented
9#23752831C213yesmaximal, complemented
10#23752757C213yesmaximal, complemented
11#23752742C213yesmaximal, complemented
12#23752695C211yesmaximal, characteristic, complemented
13#23752694C211yesmaximal, characteristic, complemented
14C2×Q8×Dic151C211yesmaximal, characteristic, complemented
15#548×C151C211yesmaximal, characteristic, complemented
16C2×#14559005C2211yescharacteristic
17C2×#14559003C2216yes—
18#18012739C2213yes—
19#18012738C2213yes—
20#18012738C2213yes—
21#18012735C2213yes—
22#18012734C2213yes—
23#18012732C2213yes—
24#18012729C2213yes—
25#18012728C2213yes—
26#18012725C2213yes—
27#18012725C2216yes—
28#18012723C2213yes—
29#18012723C2216yes—
30#18012721C2216yes—
31#18012720C2213yes—
32#18012719C2216yes—
33#18012716C2211yescharacteristic
34#18012716C2216yes—
35#18012715C2213yes—
36#18012713C2213yes—
37#18012713C2213yes—
38#18012711C2213yes—
39#18012710C2213yes—
40Q8×Dic151C2213yes—
41Q8×Dic151C2211yescharacteristic
42Q8×Dic151C2211yescharacteristic
43Q8×Dic151C2213yes—
44Q8×Dic151C2211yescharacteristic
45C4×Dic302C2213yes—
46C4×D604C2213yes—
47D4×Dic151C2213yes—
48D4×Dic151C2213yes—
49#18012701C2213yes—
50C4×C22⋊D151C2213yes—
51C4×C22⋊D151C2213yes—
52(C22⋊D151)⋊C4C2216yes—
53#18012698C2213yes—
54#18012696C2213yes—
55C2×C4⋊Dic151C2213yes—
56C2×C4⋊Dic151C2213yes—
57C4⋊C4×D151C2213yes—
58C2×C4×Dic151C2213yes—
59C2×#41×C151C2211yescharacteristic
60C22×Q8×C151C2211yescharacteristic
61#124×C151C2213yes—
62C22⋊Q8×C151C2213yes—
63#121×C151C2213yes—
64C4×Q8×C151C2211yescharacteristic
65C4×D4×C151C2213yes—
66#14559005—24no—
67#14559003—412no—
68C2×Dic302C2311yescharacteristic
69C2×Dic302C2316yes—
70C2×D604C2313yes—
71C2×C22⋊D151C2313yes—
72C2×C22⋊D151C2313yes—
73C2×C22⋊D151C2316yes—
74C22×Dic151C2311yescharacteristic
75C22×Dic151C2313yes—
76C22×Dic151C2316yes—
77C2×C4×D151C2313yes—
78C2×C4×D151C2316yes—
79C4⋊Dic151C2313yes—
80C4⋊Dic151C2313yes—
81C4⋊Dic151C2313yes—
82C4⋊Dic151C2313yes—
83C4⋊Dic151C2316yes—
84C4⋊Dic151C2311yescharacteristic
85C4⋊Dic151C2313yes—
86Dic151⋊C4C2313yes—
87Dic151⋊C4C2311yescharacteristic
88Dic151⋊C4C2313yes—
89Dic151⋊C4C2311yescharacteristic
90Dic151⋊C4C2313yes—
91Dic151⋊C4C2313yes—
92C22⋊Dic151C2313yes—
93C22⋊Dic151C2313yes—
94C22⋊Dic151C2311yescharacteristic
95C22⋊Dic151C2313yes—
96(C2×D151)⋊C4C2313yes—
97(C2×D151)⋊C4C2313yes—
98(C2×D151)⋊C4C2316yes—
99(C2×D151)⋊C4C2313yes—
100(C2×D151)⋊C4C2313yes—
101C4×Dic151C2313yes—
102C4×Dic151C2313yes—
103C4×Dic151C2313yes—
104C4×Dic151C2313yes—
105C4×Dic151C2313yes—
106#41×C151—24no—
107C2×Q8×C151—24no—
108C2×Q8×C151C2313yes—
109C2×Q8×C151C2311yescharacteristic
110C2×D4×C151C2313yes—
111C2×D4×C151C2313yes—
112C22×C4×C151C2313yes—
113C22×C4×C151C2313yes—
114C4⋊C4×C151C2313yes—
115C4⋊C4×C151C2313yes—
116C22⋊C4×C151C2316yes—
117C22⋊C4×C151C2311yescharacteristic
118C22⋊C4×C151C2313yes—
119C42×C151C2313yes—
120C22×D151C2413yes—
121Dic302—22no—
122Dic302—46no—
123D604—26no—
124C22⋊D151—26no—
125C22⋊D151—26no—
126C22⋊D151—46no—
127C2×Dic151—26no—
128C2×Dic151—22no—
129C2×Dic151—46no—
130C2×Dic151C2411yescharacteristic
131C2×Dic151C2411yescharacteristic
132C2×Dic151C2413yes—
133C2×Dic151C2413yes—
134C2×Dic151C2413yes—
135C2×Dic151C2411yescharacteristic
136C2×Dic151C2411yescharacteristic
137C4×D151—26no—
138C4×D151—46no—
139C23×C151C2413yes—
140C23×C151C2411yescharacteristic
141Q8×C151—26no—
142Q8×C151#4114yescomplemented
143D4×C151—26no—
144D4×C151—26no—
145C2×C4×C151—26no—
146C2×C4×C151—26no—
147C2×C4×C151C2413yes—
148C2×C4×C151C2411yescharacteristic
149C2×C4×C151C2411yescharacteristic
150C2×C4×C151C2413yes—
151C2×C4×C151C2413yes—
152C22×C151—21no—
153C22×C151—21no—
154C22×C151—23no—
155C22×C151—23no—
156C22×C151—23no—
157C22×C151H(5,𝔽2)12yes—
158C22×C151C2511yescharacteristic
159C2×D151—23no—
160C2×D151—23no—
161C2×D151—23no—
162C4×C151—21no—
163C4×C151—23no—
164C4×C151—23no—
165C4×C151—21no—
166C4×C151C2×#4116yes—
167Dic151—21no—
168Dic151—23no—
169Dic151—23no—
170Dic151—23no—
171Dic151—21no—
172Dic151—21no—
173Dic1512−1+412yes—
174C2×C151—22no—
175C2×C151—43no—
176C2×C151C22×#4111yescharacteristic
177C2×C151C2×H(5,𝔽2)11yescharacteristic
178C2×C151C2×2−1+411yescharacteristic
179D151—43no—
180C151#352211yescharacteristic, complemented, Sylow
181#3522—1511nomaximal, Sylow
182#559—1513no—
183Q82—1511no—
184#554—1513no—
185#550—1511no—
186#548—1511no—
187#548D15111yescharacteristic, complemented
188#547—1513no—
189#547—1513no—
190#542—1513no—
191#542—1513no—
192#532—1513no—
193#529—1513no—
194#520—1511no—
195#520—1511no—
196C2×C4×Q8—1511no—
197C2×#41—1511no—
198C2×#41—1516no—
199C2×#41C2×D15111yescharacteristic
200C22×Q8C2×D15111yescharacteristic
201C4⋊Q8—1513no—
202C4⋊Q8—1513no—
203C4⋊Q8—1513no—
204#127—1513no—
205#126—1513no—
206#126—1513no—
207#125—1513no—
208#124—1513no—
209#124—1513no—
210#124—1516no—
211#124C2×D15113yes—
212#123—1513no—
213#123—1513no—
214#123—1516no—
215#123—1516no—
216C22⋊Q8—1513no—
217C22⋊Q8—1516no—
218C22⋊Q8—1511no—
219C22⋊Q8—1516no—
220C22⋊Q8C2×D15113yes—
221#121—1513no—
222#121—1513no—
223#121—1513no—
224#121—1513no—
225#121C2×D15113yes—
226C4×Q8—1511no—
227C4×Q8—1513no—
228C4×Q8—1511no—
229C4×Q8—1513no—
230C4×Q8—1511no—
231C4×Q8—1513no—
232C4×Q8C2×D15111yescharacteristic
233C4×D4—1513no—
234C4×D4—1513no—
235C4×D4—1513no—
236C4×D4—1513no—
237C4×D4—1513no—
238C4×D4—1516no—
239C4×D4—1513no—
240C4×D4C2×D15113yes—
241#117—1513no—
242#117—1513no—
243C2×C4⋊C4—1513no—
244C2×C4⋊C4—1513no—
245C2×C4⋊C4—1513no—
246C2×C42—1513no—
247#41—24no—
248#41—3024no—
249#41—60412no—
250C2×Q8—24no—
251C2×Q8—1511no—
252C2×Q8—1516no—
253C2×Q8C22×D15113yes—
254C2×Q8C22×D15111yescharacteristic
255C2×D4—1513no—
256C2×D4—1513no—
257C2×D4—1516no—
258C2×D4—1513no—
259C2×D4C22×D15113yes—
260C2×D4C22×D15113yes—
261C22×C4—1511no—
262C22×C4—1513no—
263C22×C4—1513no—
264C22×C4—1516no—
265C22×C4—1516no—
266C22×C4C22×D15113yes—
267C22×C4C22×D15113yes—
268C4⋊C4—1513no—
269C4⋊C4—1511no—
270C4⋊C4—1511no—
271C4⋊C4—1513no—
272C4⋊C4—1513no—
273C4⋊C4—1513no—
274C4⋊C4—1513no—
275C4⋊C4—1513no—
276C4⋊C4—1513no—
277C4⋊C4—1513no—
278C4⋊C4—1516no—
279C4⋊C4—1511no—
280C4⋊C4—1513no—
281C4⋊C4C22×D15113yes—
282C4⋊C4C22×D15113yes—
283C22⋊C4—1513no—
284C22⋊C4—1513no—
285C22⋊C4—1511no—
286C22⋊C4—1513no—
287C22⋊C4—1513no—
288C22⋊C4—1513no—
289C22⋊C4—1513no—
290C22⋊C4—1516no—
291C22⋊C4—1513no—
292C22⋊C4C22×D15116yes—
293C22⋊C4C22×D15111yescharacteristic
294C22⋊C4C22×D15113yes—
295C42—1513no—
296C42—1513no—
297C42—1513no—
298C42—1513no—
299C42—1513no—
300C42C22×D15113yes—
301C23—1513no—
302C23C23×D15113yes—
303C23C23×D15111yescharacteristic
304Q8—26no—
305Q8—3022no—
306Q8—6046no—
307Q8#1455900314yescomplemented
308D4—26no—
309D4—26no—
310D4—3026no—
311D4—3026no—
312D4—3026no—
313D4—6046no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—3026no—
317C2×C4—3022no—
318C2×C4—3026no—
319C2×C4—6046no—
320C2×C4—6046no—
321C2×C4—1511no—
322C2×C4—1511no—
323C2×C4—1513no—
324C2×C4—1513no—
325C2×C4—1513no—
326C2×C4—1511no—
327C2×C4—1511no—
328C2×C4C23×D15113yes—
329C2×C4C23×D15111yescharacteristic
330C2×C4C23×D15113yes—
331C2×C4C23×D15111yescharacteristic
332C2×C4C23×D15113yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—3023no—
339C22—3023no—
340C22—3023no—
341C22#1801278512yes—
342C22C24×D15111yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—3021no—
348C4—3023no—
349C4—3023no—
350C4—3023no—
351C4—3021no—
352C4—3021no—
353C4—1512no—
354C4C2×#1455900316yes—
355C2—22no—
356C2—43no—
357C2—6043no—
358C2C22×#1455900311yescharacteristic
359C2C2×#1801278511yescharacteristic
360C22−1+4×D15111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D151
commutator, socle, cosocleC22×C151C25
Fitting#548×C151C2
hypercenter#548D151
hypocenterC151#3522
LCSG ▷ C22×C151 ▷ C151
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #23752944C2semidirect
2. Q8×Dic302C2semidirect
3. #23752920C2semidirect
4. #23752881C2semidirect
5. #23752873C2semidirect
6. #23752866C2semidirect
7. #23752862C2semidirect
8. #23752835C2semidirect
9. #23752831C2semidirect
10. #23752757C2semidirect
11. #23752742C2semidirect
12. #23752695C2semidirect
13. #23752694C2semidirect
14. C2×Q8×Dic151C2semidirect
15. #548×C151C2semidirect
142. Q8×C151#41semidirect
180. C151#3522semidirect
187. #548D151semidirect
307. Q8#14559003semidirect

Maximal subgroups

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

subgroupindex
#237529442
Q8×Dic3022
#237529202
#237528812
#237528732
#237528662
#237528622
#237528352
#237528312
#237527572
#237527422
#237526952
#237526942
C2×Q8×Dic1512
#548×C1512
#3522151