Finite Group Databasebeta

← #34083121

Subgroups — #34083121

Order 19328.

1010 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×Dic302C211yesmaximal, characteristic, complemented
2#23752935C213yesmaximal, complemented
3#23752934C213yesmaximal, complemented
4#23752930C213yesmaximal, complemented
5#23752925C213yesmaximal, complemented
6Q8×C22⋊D151C211yesmaximal, characteristic, complemented
7#23752874C211yesmaximal, characteristic, complemented
8#23752873C211yesmaximal, characteristic, complemented
9#23752834C213yesmaximal, complemented
10#23752833C213yesmaximal, complemented
11#23752766C213yesmaximal, complemented
12#23752740C213yesmaximal, complemented
13#23752688C211yesmaximal, characteristic, complemented
14C4×#14559005C211yesmaximal, characteristic, complemented
15#519×C151C211yesmaximal, characteristic, complemented
16C2×#14559005C2211yescharacteristic
17C2×Q8×D151C2211yescharacteristic
18#18012739C2213yes—
19#18012738C2213yes—
20#18012738C2213yes—
21#18012737C2213yes—
22#18012734C2213yes—
23#18012734C2216yes—
24#18012733C2216yes—
25#18012732C2213yes—
26#18012729C2213yes—
27#18012728C2213yes—
28#18012727C2213yes—
29#18012727C2216yes—
30#18012725C2211yescharacteristic
31#18012725C2216yes—
32#18012724C2213yes—
33#18012721C2213yes—
34#18012719C2213yes—
35#18012718C2213yes—
36#18012717C2213yes—
37#18012717C2216yes—
38#18012715C2213yes—
39#18012715C2213yes—
40#18012714C2213yes—
41#18012713C2213yes—
42#18012712C2213yes—
43Q8×Dic151C2211yescharacteristic
44Q8×Dic151C2211yescharacteristic
45C4×Dic302C2213yes—
46C4×Dic302C2213yes—
47C4×Dic302C2216yes—
48#18012704C2213yes—
49C4×D604C2213yes—
50C4×D604C2213yes—
51#18012701C2213yes—
52C4×C22⋊D151C2213yes—
53C4×C22⋊D151C2213yes—
54#18012697C2213yes—
55#18012697C2213yes—
56#18012696C2213yes—
57#18012696C2216yes—
58C42×D151C2213yes—
59C22×Q8×C151C2211yescharacteristic
60C4×Q8×C151C2213yes—
61C4×Q8×C151C2211yescharacteristic
62C4×Q8×C151C2213yes—
63C4×Q8×C151C2211yescharacteristic
64#117×C151C2213yes—
65#117×C151C2213yes—
66#14559005—24no—
67C2×Dic302C2313yes—
68C2×Dic302C2311yescharacteristic
69C2×Dic302C2316yes—
70Q8×D151—24no—
71C2×D604C2313yes—
72C2×D604C2313yes—
73C2×C22⋊D151C2311yescharacteristic
74C2×C22⋊D151C2313yes—
75C2×C4×D151C2313yes—
76C2×C4×D151C2313yes—
77C2×C4×D151C2313yes—
78C2×C4×D151C2313yes—
79C4⋊Dic151C2313yes—
80C4⋊Dic151C2313yes—
81C4⋊Dic151C2313yes—
82C4⋊Dic151C2313yes—
83Dic151⋊C4C2313yes—
84Dic151⋊C4C2313yes—
85Dic151⋊C4C2313yes—
86Dic151⋊C4C2313yes—
87Dic151⋊C4C2313yes—
88Dic151⋊C4C2316yes—
89Dic151⋊C4C2313yes—
90C22⋊Dic151C2311yescharacteristic
91C22⋊Dic151C2313yes—
92(C2×D151)⋊C4C2311yescharacteristic
93(C2×D151)⋊C4C2313yes—
94(C2×D151)⋊C4C2316yes—
95(C2×D151)⋊C4C2316yes—
96(C2×D151)⋊C4C2316yes—
97(C2×D151)⋊C4C2313yes—
98(C2×D151)⋊C4C2313yes—
99(C2×D151)⋊C4C2311yescharacteristic
100(C2×D151)⋊C4C2313yes—
101C4×Dic151C2311yescharacteristic
102C4×Dic151C2313yes—
103C4×Dic151C2313yes—
104C4×Dic151C2316yes—
105C4×Dic151C2313yes—
106C2×Q8×C151—24no—
107C2×Q8×C151C2313yes—
108C2×Q8×C151C2311yescharacteristic
109C22×C4×C151C2313yes—
110C4⋊C4×C151C2313yes—
111C4⋊C4×C151C2313yes—
112C4⋊C4×C151C2313yes—
113C4⋊C4×C151C2313yes—
114C22⋊C4×C151C2313yes—
115C22⋊C4×C151C2311yescharacteristic
116C42×C151C2313yes—
117C42×C151C2316yes—
118C42×C151C2313yes—
119C22×D151C2411yescharacteristic
120C22×D151C2413yes—
121Dic302—26no—
122Dic302—22no—
123Dic302—46no—
124D604—26no—
125D604—43no—
126C22⋊D151—22no—
127C22⋊D151—43no—
128C2×Dic151C2411yescharacteristic
129C2×Dic151C2411yescharacteristic
130C2×Dic151C2413yes—
131C2×Dic151C2413yes—
132C2×Dic151C2413yes—
133C2×Dic151C2411yescharacteristic
134C4×D151—26no—
135C4×D151—26no—
136C4×D151—26no—
137C4×D151—43no—
138C23×C151C2411yescharacteristic
139Q8×C151—26no—
140Q8×C151#4114yes—
141C2×C4×C151—26no—
142C2×C4×C151C2413yes—
143C2×C4×C151C2413yes—
144C2×C4×C151C2411yescharacteristic
145C2×C4×C151C2411yescharacteristic
146C2×C4×C151C2413yes—
147C2×C4×C151C2413yes—
148C22×C151—21no—
149C22×C151—21no—
150C22×C151—21no—
151C22×C151C2511yescharacteristic
152C2×D151—21no—
153C2×D151—23no—
154C2×D151—23no—
155C2×D151—21no—
156C2×D151—23no—
157C2×D151—21no—
158C4×C151—23no—
159C4×C151—23no—
160C4×C151—21no—
161C4×C151—21no—
162C4×C151—23no—
163C4×C151C2×#4116yes—
164Dic151—21no—
165Dic151—23no—
166Dic151—21no—
167Dic151—23no—
168Dic151—23no—
169Dic1512−1+412yes—
170C2×C151—41no—
171C2×C151C22×#4111yescharacteristic
172C2×C151C2×2−1+411yescharacteristic
173C2×C151#58511yescharacteristic
174D151—41no—
175D151—43no—
176C151#351811yescharacteristic, complemented, Sylow
177#3518—1511nomaximal, Sylow
178Q82—1511no—
179#556—1513no—
180#556—1513no—
181#556—1513no—
182#555—1513no—
183D4×Q8—1511no—
184#548—1511no—
185#548—1511no—
186#542—1513no—
187#542—1513no—
188#533—1513no—
189#529—1513no—
190#519—1511no—
191#519D15111yescharacteristic, complemented
192C4×#41—1511no—
193C2×#41—1511no—
194C22×Q8—1511no—
195C22×Q8C2×D15111yescharacteristic
196C4⋊Q8—1513no—
197C4⋊Q8—1513no—
198C4⋊Q8—1513no—
199C4⋊Q8—1513no—
200#126—1513no—
201#126—1516no—
202#126—1513no—
203#126—1516no—
204#125—1513no—
205#124—1513no—
206#124—1511no—
207#124—1516no—
208#124—1513no—
209#124—1516no—
210#123—1513no—
211#123—1513no—
212C22⋊Q8—1513no—
213C22⋊Q8—1513no—
214C22⋊Q8—1513no—
215C22⋊Q8—1513no—
216C22⋊Q8—1516no—
217C22⋊Q8—1513no—
218#121—1513no—
219#121—1513no—
220#121—1513no—
221C4×Q8—1511no—
222C4×Q8—1511no—
223C4×Q8—1513no—
224C4×Q8—1513no—
225C4×Q8—1513no—
226C4×Q8—1516no—
227C4×Q8C2×D15111yescharacteristic
228C4×Q8C2×D15113yes—
229C4×Q8C2×D15111yescharacteristic
230C4×Q8C2×D15113yes—
231C4×D4—1513no—
232C4×D4—1513no—
233C4×D4—1513no—
234C4×D4—1513no—
235C4×D4—1513no—
236#117—1513no—
237#117—1513no—
238#117—1513no—
239#117—1516no—
240#117C2×D15113yes—
241#117C2×D15113yes—
242C2×C42—1513no—
243#41—3024no—
244C2×Q8—24no—
245C2×Q8—3024no—
246C2×Q8—1511no—
247C2×Q8—1513no—
248C2×Q8—1516no—
249C2×Q8C22×D15113yes—
250C2×Q8C22×D15111yescharacteristic
251C2×D4—1511no—
252C2×D4—1513no—
253C2×D4—1513no—
254C2×D4—1513no—
255C22×C4—1513no—
256C22×C4—1513no—
257C22×C4—1513no—
258C22×C4—1513no—
259C22×C4C22×D15113yes—
260C4⋊C4—1513no—
261C4⋊C4—1513no—
262C4⋊C4—1513no—
263C4⋊C4—1513no—
264C4⋊C4—1513no—
265C4⋊C4—1516no—
266C4⋊C4—1513no—
267C4⋊C4—1513no—
268C4⋊C4—1513no—
269C4⋊C4—1513no—
270C4⋊C4—1513no—
271C4⋊C4C22×D15113yes—
272C4⋊C4C22×D15113yes—
273C4⋊C4C22×D15113yes—
274C4⋊C4C22×D15113yes—
275C22⋊C4—1511no—
276C22⋊C4—1513no—
277C22⋊C4—1513no—
278C22⋊C4—1511no—
279C22⋊C4—1516no—
280C22⋊C4—1516no—
281C22⋊C4—1516no—
282C22⋊C4—1511no—
283C22⋊C4—1513no—
284C22⋊C4—1513no—
285C22⋊C4—1513no—
286C22⋊C4C22×D15113yes—
287C22⋊C4C22×D15111yescharacteristic
288C42—1511no—
289C42—1513no—
290C42—1513no—
291C42—1516no—
292C42—1513no—
293C42C22×D15113yes—
294C42C22×D15116yes—
295C42C22×D15113yes—
296C23—1511no—
297C23—1513no—
298C23C23×D15111yescharacteristic
299Q8—26no—
300Q8—3022no—
301Q8—3026no—
302Q8—6046no—
303Q8#1455900414yes—
304D4—3022no—
305D4—3026no—
306D4—6043no—
307D4—6043no—
308C2×C4—26no—
309C2×C4—3026no—
310C2×C4—3026no—
311C2×C4—3026no—
312C2×C4—6043no—
313C2×C4—1511no—
314C2×C4—1511no—
315C2×C4—1513no—
316C2×C4—1513no—
317C2×C4—1513no—
318C2×C4—1511no—
319C2×C4C23×D15113yes—
320C2×C4C23×D15111yescharacteristic
321C2×C4C23×D15111yescharacteristic
322C2×C4C23×D15113yes—
323C2×C4C23×D15113yes—
324C2×C4C23×D15113yes—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—3021no—
329C22—3023no—
330C22—3023no—
331C22—3021no—
332C22—3023no—
333C22—3021no—
334C22C24×D15111yescharacteristic
335C4—23no—
336C4—23no—
337C4—21no—
338C4—21no—
339C4—23no—
340C4—3021no—
341C4—3023no—
342C4—3021no—
343C4—3023no—
344C4—3023no—
345C4—1512no—
346C4C2×#1455900416yes—
347C2—41no—
348C2—6041no—
349C2—6043no—
350C2C22×#1455900411yescharacteristic
351C22−1+4×D15111yescharacteristic
352C2#2375304911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D151
commutator, socle, cosocleC22×C151C25
Fitting#519×C151C2
hypercenter#519D151
hypocenterC151#3518
LCSG ▷ C22×C151 ▷ C151
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. Q8×Dic302C2semidirect
2. #23752935C2semidirect
3. #23752934C2semidirect
4. #23752930C2semidirect
5. #23752925C2semidirect
6. Q8×C22⋊D151C2semidirect
7. #23752874C2semidirect
8. #23752873C2semidirect
9. #23752834C2semidirect
10. #23752833C2semidirect
11. #23752766C2semidirect
12. #23752740C2semidirect
13. #23752688C2semidirect
14. C4×#14559005C2semidirect
15. #519×C151C2semidirect
176. C151#3518semidirect
191. #519D151semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic3022
#237529352
#237529342
#237529302
#237529252
Q8×C22⋊D1512
#237528742
#237528732
#237528342
#237528332
#237527662
#237527402
#237526882
C4×#145590052
#519×C1512
#3518151