Finite Group Databasebeta

← #34056560

Subgroups — #34056560

Order 19072.

388 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
1#23750333C211yesmaximal, characteristic
2Q8×C22⋊D149C211yesmaximal, characteristic
3#23749974C211yesmaximal, characteristic, complemented
4#23749961C211yesmaximal, characteristic, complemented
5#23749959C211yesmaximal, characteristic, complemented
6#23749918C211yesmaximal, characteristic, complemented
7#23749864C211yesmaximal, characteristic, complemented
8#23749863C211yesmaximal, characteristic, complemented
9#23749833C211yesmaximal, characteristic, complemented
10#23749779C211yesmaximal, characteristic, complemented
11Q16×Dic149C211yesmaximal, characteristic, complemented
12#23749688C211yesmaximal, characteristic
13C2×#18011835C211yesmaximal, characteristic, complemented
14#103×D149C211yesmaximal, characteristic, complemented
15#451×C149C211yesmaximal, characteristic, complemented
16C2×Q8×D149C2211yescharacteristic
17C2×#14558726C2211yescharacteristic
18C22×#11924746C2211yescharacteristic
19C2×C8×D149C2211yescharacteristic
20#18011910C2211yescharacteristic
21#18011909—21no—
22#18011909C2211yescharacteristic
23#18011901—21no—
24#18011891—21no—
25#18011890—21no—
26#18011890C2211yescharacteristic
27#18011888—21no—
28#18011887—21no—
29#18011887C2211yescharacteristic
30#18011887C2211yescharacteristic
31Q8×Dic149C2211yescharacteristic
32Q8×Dic149C2211yescharacteristic
33C4×C22⋊D149—21no—
34C4×C22⋊D149C2211yescharacteristic
35#18011871—21no—
36C2×C4⋊Dic149C2211yescharacteristic
37C4⋊C4×D149C2211yescharacteristic
38#18011845C2211yescharacteristic
39#18011845C2211yescharacteristic
40#18011844C2211yescharacteristic
41#18011840C2211yescharacteristic
42#18011836C2211yescharacteristic
43#18011836C2211yescharacteristic
44#18011835C2211yescharacteristic
45#18011835C2211yescharacteristic
46#18011835C2211yescharacteristic
47#18011835C2211yescharacteristic
48#18011835C2211yescharacteristic
49#18011834C2211yescharacteristic
50#18011823C2211yescharacteristic
51C8×Dic149C2211yescharacteristic
52C22×Q8×C149C2211yescharacteristic
53C2×Q16×C149C2211yescharacteristic
54C2×Q16×C149C2211yescharacteristic
55C22⋊Q8×C149C2211yescharacteristic
56#103×C149C2211yescharacteristic
57#103×C149C2211yescharacteristic
58C22⋊C8×C149C2211yescharacteristic
59C2×Dic298—21no—
60C2×Dic298—21no—
61C2×Dic298C2311yescharacteristic
62Q8×D149—22no—
63Q8×D149—41no—
64C2×C22⋊D149D411yescharacteristic
65C22×Dic149—21no—
66C2×C4×D149—21no—
67C2×C4×D149—21no—
68C2×C4×D149C2311yescharacteristic
69#14558726—41no—
70C2×#11924746—41no—
71C2×#11924746C2311yescharacteristic
72C2×#11924746C2311yescharacteristic
73C8×D149—41no—
74C4⋊Dic149—21no—
75C4⋊Dic149—21no—
76C4⋊Dic149—21no—
77C4⋊Dic149C2311yescharacteristic
78C4⋊Dic149C2311yescharacteristic
79C4⋊Dic149C2311yescharacteristic
80#14558715—21no—
81#14558715—21no—
82#14558715—21no—
83#14558715—21no—
84#14558715—21no—
85#14558715—21no—
86#14558715—21no—
87#14558715D411yescharacteristic
88C22⋊Dic149—21no—
89C22⋊Dic149D411yescharacteristic
90#14558713—21no—
91#14558713—21no—
92#14558713D411yescharacteristic
93C4×Dic149—21no—
94C4×Dic149—21no—
95C4×Dic149C2311yescharacteristic
96C2×Q8×C149—21no—
97C2×Q8×C149—22no—
98C2×Q8×C149—41no—
99C2×Q8×C149C2311yescharacteristic
100C2×Q8×C149C2311yescharacteristic
101C2×Q8×C149C2311yescharacteristic
102C22×C4×C149—21no—
103C22×C4×C149C2311yescharacteristic
104Q16×C149—22no—
105Q16×C149—41no—
106C2×C8×C149C2311yescharacteristic
107C2×C8×C149C2311yescharacteristic
108C4⋊C4×C149—21no—
109C4⋊C4×C149C2311yescharacteristic
110C22⋊C4×C149—21no—
111C22×D149C2×D411yescharacteristic
112Dic298—21no—
113Dic298—21no—
114Dic298—42no—
115Dic298—81no—
116C22⋊D149Q1614yescomplemented
117C2×Dic149—21no—
118C2×Dic149—21no—
119C2×Dic149—21no—
120C2×Dic149—21no—
121C2×Dic149—81no—
122C2×Dic149C2×D411yescharacteristic
123C2×Dic149C2×D411yescharacteristic
124C2×Dic149C2×D411yescharacteristic
125C4×D149—21no—
126C4×D149—21no—
127C4×D149—42no—
128C4×D149—81no—
129#11924746—21no—
130#11924746—21no—
131C23×C149C2×D411yescharacteristic
132Q8×C149—21no—
133Q8×C149—21no—
134Q8×C149—21no—
135Q8×C149—21no—
136Q8×C149—21no—
137Q8×C149—42no—
138Q8×C149#4112yes—
139C2×C4×C149—21no—
140C2×C4×C149—21no—
141C2×C4×C149—21no—
142C2×C4×C149—21no—
143C2×C4×C149—21no—
144C2×C4×C149—21no—
145C2×C4×C149—42no—
146C2×C4×C149C2×D411yescharacteristic
147C2×C4×C149C2411yescharacteristic
148C8×C149—21no—
149C8×C149—21no—
150C22×C149—21no—
151C22×C149—21no—
152C22×C149C2×Q1612yes—
153C22×C149C22×D411yescharacteristic
154C2×D149—21no—
155C2×D149—21no—
156C2×D149C2×Q1612yes—
157C4×C149—21no—
158C4×C149—22no—
159C4×C149—41no—
160C4×C149—41no—
161C4×C149—41no—
162C4×C149C2×#4111yescharacteristic
163C4×C149H(5,𝔽2)11yescharacteristic
164Dic149—21no—
165Dic149—21no—
166Dic149—41no—
167Dic149—41no—
168Dic149—41no—
169Dic149—41no—
170Dic149C2×Q1612yes—
171C2×C149—22no—
172C2×C149#54611yescharacteristic
173C2×C149C22×Q1611yescharacteristic
174C2×C149#57711yescharacteristic
175D149—22no—
176C149#329911yescharacteristic, complemented, Sylow
177#3299—1491nomaximal, Sylow
178#554—1491no—
179D4×Q8—1491no—
180#487—1491no—
181#484—1491no—
182#484—1491no—
183#477—1491no—
184#467—1491no—
185#467—1491no—
186#462—1491no—
187#451—1491no—
188#451D14911yescharacteristic, complemented
189C4×Q16—1491no—
190C8×D4—1491no—
191C2×#103—1491no—
192C2×#103—1491no—
193C22×Q8—1491no—
194C22×Q8C2×D14911yescharacteristic
195C2×Q16—1491no—
196C2×Q16C2×D14911yescharacteristic
197C2×Q16C2×D14911yescharacteristic
198C22×C8—1491no—
199C22×C8—1491no—
200C4⋊Q8—2981no—
201C4⋊Q8—1491no—
202C4⋊Q8—1491no—
203#125—2981no—
204C22⋊Q8—2981no—
205C22⋊Q8—2981no—
206C22⋊Q8—2981no—
207C22⋊Q8—2981no—
208C22⋊Q8—1491no—
209C22⋊Q8—1491no—
210C22⋊Q8—1491no—
211C22⋊Q8C2×D14911yescharacteristic
212C4×Q8—1491no—
213C4×Q8—1491no—
214C4×D4—2981no—
215C4×D4—2981no—
216C4×D4—1491no—
217C2×C4⋊C4—1491no—
218C2×C4⋊C4—1491no—
219#107—1491no—
220#107—1491no—
221#107—1491no—
222C4⋊C8—1491no—
223#103—1491no—
224#103—1491no—
225#103—1491no—
226#103—1491no—
227#103—1491no—
228#103—1491no—
229#103—1491no—
230#103—1491no—
231#103C2×D14911yescharacteristic
232#103C2×D14911yescharacteristic
233C22⋊C8—1491no—
234C22⋊C8C2×D14911yescharacteristic
235C4×C8—1491no—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—2981no—
240C2×Q8—2981no—
241C2×Q8—2982no—
242C2×Q8—5961no—
243C2×Q8—1491no—
244C2×Q8C22×D14911yescharacteristic
245C2×Q8C22×D14911yescharacteristic
246C2×Q8C22×D14911yescharacteristic
247C2×D4—1491no—
248C22×C4—21no—
249C22×C4—2981no—
250C22×C4—2981no—
251C22×C4—2981no—
252C22×C4—1491no—
253C22×C4C22×D14911yescharacteristic
254Q16—22no—
255Q16—41no—
256Q16—5961no—
257C2×C8—5961no—
258C2×C8—5961no—
259C2×C8—1491no—
260C2×C8—1491no—
261C2×C8C22×D14911yescharacteristic
262C2×C8C22×D14911yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—2981no—
265C4⋊C4—2981no—
266C4⋊C4—2981no—
267C4⋊C4—2981no—
268C4⋊C4—2981no—
269C4⋊C4—2981no—
270C4⋊C4—2981no—
271C4⋊C4—2981no—
272C4⋊C4—2981no—
273C4⋊C4—2981no—
274C4⋊C4—1491no—
275C4⋊C4—1491no—
276C4⋊C4—1491no—
277C4⋊C4—1491no—
278C4⋊C4C22×D14911yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—2981no—
281C22⋊C4—2981no—
282C22⋊C4—2981no—
283C22⋊C4—1491no—
284C22⋊C4—1491no—
285C42—2981no—
286C42—2981no—
287C42—1491no—
288C23—1491no—
289C23D4×D14911yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8—2981no—
297Q8—2981no—
298Q8—5962no—
299Q8—11921no—
300Q8#1455873512yes—
301D4—1494no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—2981no—
310C2×C4—2981no—
311C2×C4—2981no—
312C2×C4—2981no—
313C2×C4—2981no—
314C2×C4—2981no—
315C2×C4—5962no—
316C2×C4—1491no—
317C2×C4—1491no—
318C2×C4—1491no—
319C2×C4—11921no—
320C2×C4—11921no—
321C2×C4D4×D14911yescharacteristic
322C2×C4C23×D14911yescharacteristic
323C8—21no—
324C8—21no—
325C8—2981no—
326C8—2981no—
327C22—21no—
328C22—21no—
329C22—2981no—
330C22—2981no—
331C22—1492no—
332C22Q16×D14912yes—
333C22C2×D4×D14911yescharacteristic
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—2981no—
340C4—2981no—
341C4—5961no—
342C4—5961no—
343C4—5961no—
344C4—5961no—
345C4—1492no—
346C4C2×#1455873511yescharacteristic
347C4#1801195611yescharacteristic
348C2—22no—
349C2—2982no—
350C2#2375026211yescharacteristic
351C2C2×Q16×D14911yescharacteristic
352C2#2375043411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#451×C149C2
socleC22×C149C22×D4
prime radicalC2×C22⋊D149D4
hypercenter#451D149
hypocenterC149#3299
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #23749974C2semidirect
4. #23749961C2semidirect
5. #23749959C2semidirect
6. #23749918C2semidirect
7. #23749864C2semidirect
8. #23749863C2semidirect
9. #23749833C2semidirect
10. #23749779C2semidirect
11. Q16×Dic149C2semidirect
13. C2×#18011835C2semidirect
14. #103×D149C2semidirect
15. #451×C149C2semidirect
116. C22⋊D149Q16semidirect
176. C149#3299semidirect
188. #451D149semidirect

Maximal subgroups

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

subgroupindex
#237503332
Q8×C22⋊D1492
#237499742
#237499612
#237499592
#237499182
#237498642
#237498632
#237498332
#237497792
Q16×Dic1492
#237496882
C2×#180118352
#103×D1492
#451×C1492
#3299149