Finite Group Databasebeta

← #34083197

Subgroups — #34083197

Order 19328.

992 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23752938C211yesmaximal, characteristic, complemented
2#23752937C213yesmaximal, complemented
3#23752933C213yesmaximal, complemented
4#23752904C213yesmaximal, complemented
5#23752902C213yesmaximal, complemented
6Q8×C22⋊D151C211yesmaximal, characteristic, complemented
7Q8×C22⋊D151C211yesmaximal, characteristic, complemented
8#23752823C213yesmaximal, complemented
9#23752822C213yesmaximal, complemented
10#23752764C213yesmaximal, complemented
11#23752742C213yesmaximal
12#23752690C211yesmaximal, characteristic, complemented
13#23752688C211yesmaximal, characteristic, complemented
14C2×Q8×Dic151C211yesmaximal, characteristic, complemented
15D4×Q8×C151C211yesmaximal, characteristic, complemented
16C2×Q8×D151C2211yescharacteristic
17#18012738C2213yes—
18#18012737C2213yes—
19#18012737C2213yes—
20#18012734C2213yes—
21#18012734C2216yes—
22#18012732C2213yes—
23#18012732C2216yes—
24#18012731C2213yes—
25#18012730C2216yes—
26#18012730C2213yes—
27#18012729C2213yes—
28#18012729C2216yes—
29#18012729C2216yes—
30#18012725C2213yes—
31#18012723C2213yes—
32#18012720C2213yes—
33#18012718C2213yes—
34#18012718C2213yes—
35#18012716C2213yes—
36#18012715C2213yes—
37#18012715C2213yes—
38#18012711C2211yescharacteristic
39Q8×Dic151C2213yes—
40Q8×Dic151C2213yes—
41Q8×Dic151C2211yescharacteristic
42Q8×Dic151C2211yescharacteristic
43Q8×Dic151C2213yes—
44Q8×Dic151C2213yes—
45Q8×Dic151C2211yescharacteristic
46C4×Dic302C2213yes—
47C4×Dic302C2213yes—
48#18012704C2213yes—
49#18012704C2216yes—
50C4×C22⋊D151C2213yes—
51C4×C22⋊D151C2213yes—
52#18012698C2213yes—
53#18012698C2213yes—
54#18012697C2213yes—
55#18012696C2213yes—
56#18012695C2216yes—
57C2×C4⋊Dic151C2213yes—
58C2×C4×Dic151C2213yes—
59C22×Q8×C151C2211yescharacteristic
60C22×Q8×C151C2211yescharacteristic
61C4⋊Q8×C151C2213yes—
62C22⋊Q8×C151C2213yes—
63C22⋊Q8×C151C2213yes—
64C4×Q8×C151C2211yescharacteristic
65C4×D4×C151C2213yes—
66C2×Dic302C2313yes—
67C2×Dic302C2313yes—
68Q8×D151—24no—
69C2×C22⋊D151C2311yescharacteristic
70C2×C22⋊D151C2311yescharacteristic
71C22×Dic151C2311yescharacteristic
72C22×Dic151C2313yes—
73C2×C4×D151C2313yes—
74C4⋊Dic151C2313yes—
75C4⋊Dic151C2313yes—
76C4⋊Dic151C2316yes—
77C4⋊Dic151C2313yes—
78C4⋊Dic151C2313yes—
79C4⋊Dic151C2313yes—
80C4⋊Dic151C2313yes—
81Dic151⋊C4C2316yes—
82Dic151⋊C4C2316yes—
83Dic151⋊C4C2313yes—
84Dic151⋊C4C2313yes—
85Dic151⋊C4C2313yes—
86Dic151⋊C4C2313yes—
87Dic151⋊C4C2316yes—
88Dic151⋊C4C2311yescharacteristic
89Dic151⋊C4C2313yes—
90C22⋊Dic151C2313yes—
91C22⋊Dic151C2311yescharacteristic
92C22⋊Dic151C2313yes—
93C22⋊Dic151C2313yes—
94(C2×D151)⋊C4C2311yescharacteristic
95(C2×D151)⋊C4C2313yes—
96(C2×D151)⋊C4C2313yes—
97(C2×D151)⋊C4C2313yes—
98C4×Dic151C2313yes—
99C4×Dic151C2313yes—
100C4×Dic151C2313yes—
101C4×Dic151C2313yes—
102C4×Dic151C2316yes—
103C4×Dic151C2316yes—
104C4×Dic151C2313yes—
105C2×Q8×C151—24no—
106C2×Q8×C151—24no—
107C2×Q8×C151C2313yes—
108C2×Q8×C151C2313yes—
109C2×Q8×C151C2311yescharacteristic
110C2×D4×C151C2311yescharacteristic
111C22×C4×C151C2313yes—
112C22×C4×C151C2313yes—
113C4⋊C4×C151C2313yes—
114C4⋊C4×C151C2316yes—
115C4⋊C4×C151C2313yes—
116C22⋊C4×C151C2313yes—
117C22⋊C4×C151C2313yes—
118C42×C151C2313yes—
119C22×D151C2411yescharacteristic
120Dic302—26no—
121Dic302—26no—
122C22⋊D151—22no—
123C22⋊D151—22no—
124C2×Dic151—26no—
125C2×Dic151—22no—
126C2×Dic151C2411yescharacteristic
127C2×Dic151C2413yes—
128C2×Dic151C2413yes—
129C2×Dic151C2411yescharacteristic
130C2×Dic151C2413yes—
131C2×Dic151C2413yes—
132C2×Dic151C2411yescharacteristic
133C4×D151—26no—
134C23×C151C2411yescharacteristic
135C23×C151C2411yescharacteristic
136Q8×C151—26no—
137Q8×C151—26no—
138Q8×C151#4114yes—
139D4×C151—22no—
140C2×C4×C151—26no—
141C2×C4×C151—26no—
142C2×C4×C151C2413yes—
143C2×C4×C151C2413yes—
144C2×C4×C151C2413yes—
145C2×C4×C151C2411yescharacteristic
146C2×C4×C151C2413yes—
147C22×C151—21no—
148C22×C151—21no—
149C22×C151—21no—
150C22×C151—21no—
151C22×C151—21no—
152C22×C1512−1+412yes—
153C22×C151C2511yescharacteristic
154C2×D151—21no—
155C2×D151—21no—
156C2×D151—21no—
157C4×C151—23no—
158C4×C151—23no—
159C4×C151—21no—
160C4×C151—23no—
161C4×C151C2×#4116yes—
162Dic151—23no—
163Dic151—21no—
164Dic151—23no—
165Dic151—23no—
166Dic151—21no—
167Dic151—23no—
168Dic1512−1+412yes—
169C2×C151—22no—
170C2×C151—41no—
171C2×C151C22×#4111yescharacteristic
172C2×C151C2×2−1+411yescharacteristic
173C2×C151C2×2−1+411yescharacteristic
174D151—41no—
175C151#352311yescharacteristic, complemented, Sylow
176#3523—1511nomaximal, Sylow
177#557—1513no—
178#557—1511no—
179#556—1513no—
180#552—1513no—
181#552—1513no—
182D4×Q8—1511no—
183D4×Q8—1511no—
184D4×Q8D15111yescharacteristic, complemented
185#541—1513no—
186#541—1513no—
187#533—1513no—
188#529—1513no—
189#519—1511no—
190#519—1511no—
191C2×C4×Q8—1511no—
192C22×Q8—1511no—
193C22×Q8C2×D15111yescharacteristic
194C22×Q8C2×D15111yescharacteristic
195C4⋊Q8—1513no—
196C4⋊Q8—1513no—
197C4⋊Q8—1513no—
198C4⋊Q8C2×D15113yes—
199#126—1513no—
200#126—1516no—
201#126—1513no—
202#126—1516no—
203#125—1513no—
204#125—1516no—
205#125—1516no—
206#125—1516no—
207#125—1513no—
208#125—1513no—
209#124—1513no—
210#123—1513no—
211#123—1513no—
212C22⋊Q8—1513no—
213C22⋊Q8—1513no—
214C22⋊Q8—1513no—
215C22⋊Q8—1513no—
216C22⋊Q8—1513no—
217C22⋊Q8C2×D15113yes—
218C22⋊Q8C2×D15113yes—
219#121—1511no—
220C4×Q8—1511no—
221C4×Q8—1513no—
222C4×Q8—1511no—
223C4×Q8—1513no—
224C4×Q8—1513no—
225C4×Q8—1511no—
226C4×Q8—1513no—
227C4×Q8—1513no—
228C4×Q8—1516no—
229C4×Q8—1513no—
230C4×Q8—1513no—
231C4×Q8C2×D15111yescharacteristic
232C4×D4—1513no—
233C4×D4—1513no—
234C4×D4C2×D15113yes—
235#117—1513no—
236#117—1513no—
237#117—1513no—
238#117—1516no—
239#117—1513no—
240C2×C4⋊C4—1513no—
241C2×C42—1513no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—3024no—
245C2×Q8—1513no—
246C2×Q8—1513no—
247C2×Q8C22×D15113yes—
248C2×Q8C22×D15113yes—
249C2×Q8C22×D15111yescharacteristic
250C2×D4—1511no—
251C2×D4—1511no—
252C2×D4C22×D15111yescharacteristic
253C22×C4—1511no—
254C22×C4—1513no—
255C22×C4—1513no—
256C22×C4C22×D15113yes—
257C22×C4C22×D15113yes—
258C4⋊C4—1513no—
259C4⋊C4—1513no—
260C4⋊C4—1516no—
261C4⋊C4—1516no—
262C4⋊C4—1513no—
263C4⋊C4—1513no—
264C4⋊C4—1511no—
265C4⋊C4—1516no—
266C4⋊C4—1513no—
267C4⋊C4—1513no—
268C4⋊C4—1513no—
269C4⋊C4—1516no—
270C4⋊C4—1513no—
271C4⋊C4—1513no—
272C4⋊C4—1513no—
273C4⋊C4—1513no—
274C4⋊C4C22×D15113yes—
275C4⋊C4C22×D15113yes—
276C4⋊C4C22×D15116yes—
277C22⋊C4—1511no—
278C22⋊C4—1513no—
279C22⋊C4—1513no—
280C22⋊C4—1513no—
281C22⋊C4—1511no—
282C22⋊C4—1513no—
283C22⋊C4—1513no—
284C22⋊C4—1513no—
285C22⋊C4C22×D15113yes—
286C22⋊C4C22×D15113yes—
287C42—1513no—
288C42—1513no—
289C42—1513no—
290C42—1513no—
291C42—1516no—
292C42—1516no—
293C42—1513no—
294C42C22×D15113yes—
295C23—1511no—
296C23C23×D15111yescharacteristic
297C23C23×D15111yescharacteristic
298Q8—26no—
299Q8—26no—
300Q8—3026no—
301Q8—3026no—
302Q8#1455900314yes—
303D4—22no—
304D4—3022no—
305D4—3022no—
306C2×C4—26no—
307C2×C4—26no—
308C2×C4—3026no—
309C2×C4—3022no—
310C2×C4—3026no—
311C2×C4—1511no—
312C2×C4—1513no—
313C2×C4—1513no—
314C2×C4—1511no—
315C2×C4—1513no—
316C2×C4—1513no—
317C2×C4—1511no—
318C2×C4C23×D15113yes—
319C2×C4C23×D15113yes—
320C2×C4C23×D15113yes—
321C2×C4C23×D15111yescharacteristic
322C2×C4C23×D15113yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—3021no—
329C22—3021no—
330C22—3021no—
331C22#1801278712yes—
332C22C24×D15111yescharacteristic
333C4—23no—
334C4—23no—
335C4—23no—
336C4—21no—
337C4—3023no—
338C4—3021no—
339C4—3023no—
340C4—3023no—
341C4—3023no—
342C4—3021no—
343C4—1512no—
344C4C2×#1455900316yes—
345C2—22no—
346C2—41no—
347C2—6041no—
348C2C22×#1455900311yescharacteristic
349C22−1+4×D15111yescharacteristic
350C2C2×#1801278711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D151
commutator, socle, cosocleC22×C151C25
FittingD4×Q8×C151C2
prime radical#18012711C22
hypercenterD4×Q8D151
hypocenterC151#3523
LCSG ▷ C22×C151 ▷ C151
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #23752938C2semidirect
2. #23752937C2semidirect
3. #23752933C2semidirect
4. #23752904C2semidirect
5. #23752902C2semidirect
6. Q8×C22⋊D151C2semidirect
7. Q8×C22⋊D151C2semidirect
8. #23752823C2semidirect
9. #23752822C2semidirect
10. #23752764C2semidirect
12. #23752690C2semidirect
13. #23752688C2semidirect
14. C2×Q8×Dic151C2semidirect
15. D4×Q8×C151C2semidirect
175. C151#3523semidirect
184. D4×Q8D151semidirect

Maximal subgroups

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

subgroupindex
#237529382
#237529372
#237529332
#237529042
#237529022
Q8×C22⋊D1512
#237528232
#237528222
#237527642
#237527422
#237526902
#237526882
C2×Q8×Dic1512
D4×Q8×C1512
#3523151