Finite Group Databasebeta

← #34056436

Subgroups — #34056436

Order 19072.

472 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14558726C211yesmaximal, characteristic, complemented
2C2×Q16×D149C211yesmaximal, characteristic, complemented
3Q8×C22⋊D149C211yesmaximal, characteristic
4#23750288C211yesmaximal, characteristic
5#23750003C211yesmaximal, characteristic, complemented
6#23749865C211yesmaximal, characteristic, complemented
7#23749864C211yesmaximal, characteristic, complemented
8#23749834C211yesmaximal, characteristic, complemented
9#23749833C211yesmaximal, characteristic, complemented
10#23749783C211yesmaximal, characteristic, complemented
11#23749782C211yesmaximal, characteristic, complemented
12#23749779C211yesmaximal, characteristic, complemented
13#23749717C211yesmaximal, characteristic, complemented
14#23749688C211yesmaximal, characteristic
15#451×C149C211yesmaximal, characteristic, complemented
16C22×Dic298C2211yescharacteristic
17C2×Q8×D149C2211yescharacteristic
18C2×Q8×D149C2211yescharacteristic
19C2×Dic596C2211yescharacteristic
20C2×Dic596C2211yescharacteristic
21C2×#14558726—24no—
22C2×#14558726C2211yescharacteristic
23C2×#14558726C2211yescharacteristic
24C2×#14558726C2211yescharacteristic
25C2×#14558726C2211yescharacteristic
26C2×#14558726C2211yescharacteristic
27Q16×D149—24no—
28C22×#11924746C2211yescharacteristic
29C2×C8×D149C2211yescharacteristic
30#18011910—21no—
31#18011909—21no—
32#18011909C2211yescharacteristic
33#18011909C2211yescharacteristic
34#18011891C2211yescharacteristic
35#18011890—21no—
36#18011890C2211yescharacteristic
37#18011889—21no—
38#18011888—21no—
39#18011887—21no—
40#18011887C2211yescharacteristic
41Q8×Dic149C2211yescharacteristic
42#18011876C2211yescharacteristic
43C4×C22⋊D149—21no—
44C4×C22⋊D149C2211yescharacteristic
45#18011871—21no—
46#18011844C2211yescharacteristic
47#18011840C2211yescharacteristic
48#18011836C2211yescharacteristic
49#18011836C2211yescharacteristic
50#18011835C2211yescharacteristic
51#18011834C2211yescharacteristic
52#18011823C2211yescharacteristic
53C8×Dic149C2211yescharacteristic
54C22×Q8×C149C2211yescharacteristic
55C2×Q16×C149C2211yescharacteristic
56C2×Q16×C149C2211yescharacteristic
57C22⋊Q8×C149C2211yescharacteristic
58#103×C149C2211yescharacteristic
59#103×C149C2211yescharacteristic
60C22⋊C8×C149C2211yescharacteristic
61C2×Dic298—21no—
62C2×Dic298—21no—
63C2×Dic298—22no—
64C2×Dic298—21no—
65C2×Dic298—41no—
66C2×Dic298C2311yescharacteristic
67C2×Dic298C2311yescharacteristic
68C2×Dic298C2311yescharacteristic
69Q8×D149—22no—
70Q8×D149—22no—
71Q8×D149—41no—
72Q8×D149—41no—
73C2×C22⋊D149D411yescharacteristic
74C22×Dic149—21no—
75C2×C4×D149—21no—
76C2×C4×D149—21no—
77C2×C4×D149C2311yescharacteristic
78Dic596—22no—
79Dic596—41no—
80#14558726—22no—
81#14558726—22no—
82#14558726—22no—
83#14558726—22no—
84#14558726D414yescomplemented
85C2×#11924746—22no—
86C2×#11924746C2311yescharacteristic
87C2×#11924746C2311yescharacteristic
88C8×D149—22no—
89C4⋊Dic149—21no—
90C4⋊Dic149—21no—
91C4⋊Dic149C2311yescharacteristic
92#14558715—21no—
93#14558715—21no—
94#14558715—21no—
95#14558715—21no—
96#14558715—21no—
97#14558715—21no—
98#14558715—21no—
99#14558715D411yescharacteristic
100C22⋊Dic149—21no—
101C22⋊Dic149D411yescharacteristic
102#14558713—21no—
103#14558713—21no—
104#14558713D411yescharacteristic
105C4×Dic149—21no—
106C4×Dic149—21no—
107C4×Dic149C2311yescharacteristic
108C2×Q8×C149—21no—
109C2×Q8×C149—22no—
110C2×Q8×C149—41no—
111C2×Q8×C149C2311yescharacteristic
112C2×Q8×C149C2311yescharacteristic
113C2×Q8×C149C2311yescharacteristic
114C22×C4×C149—21no—
115C22×C4×C149C2311yescharacteristic
116Q16×C149—22no—
117Q16×C149—41no—
118C2×C8×C149C2311yescharacteristic
119C2×C8×C149C2311yescharacteristic
120C4⋊C4×C149—21no—
121C4⋊C4×C149C2311yescharacteristic
122C22⋊C4×C149—21no—
123C22×D149C2×D411yescharacteristic
124Dic298—21no—
125Dic298—21no—
126Dic298—21no—
127Dic298—21no—
128Dic298—21no—
129Dic298—42no—
130Dic298—42no—
131Dic298—42no—
132Dic298C2×D412yes—
133C22⋊D149Q1614yescomplemented
134C2×Dic149—21no—
135C2×Dic149—21no—
136C2×Dic149—21no—
137C2×Dic149—21no—
138C2×Dic149—42no—
139C2×Dic149C2×D411yescharacteristic
140C2×Dic149C2×D411yescharacteristic
141C2×Dic149C2×D411yescharacteristic
142C4×D149—21no—
143C4×D149—21no—
144C4×D149—42no—
145C4×D149—42no—
146#11924746—21no—
147#11924746C2×D412yes—
148C23×C149C2×D411yescharacteristic
149Q8×C149—21no—
150Q8×C149—21no—
151Q8×C149—21no—
152Q8×C149—21no—
153Q8×C149—21no—
154Q8×C149—42no—
155Q8×C149C2×D412yes—
156C2×C4×C149—21no—
157C2×C4×C149—21no—
158C2×C4×C149—21no—
159C2×C4×C149—21no—
160C2×C4×C149—21no—
161C2×C4×C149—21no—
162C2×C4×C149—42no—
163C2×C4×C149C2×D411yescharacteristic
164C2×C4×C149C2411yescharacteristic
165C8×C149—21no—
166C8×C149—21no—
167C22×C149—21no—
168C22×C149—21no—
169C22×C149C2×Q1612yescomplemented
170C22×C149C22×D411yescharacteristic
171C2×D149—21no—
172C2×D149—21no—
173C2×D149C2×Q1612yescomplemented
174C4×C149—21no—
175C4×C149—22no—
176C4×C149—41no—
177C4×C149—41no—
178C4×C149—41no—
179C4×C149C22×D411yescharacteristic
180C4×C149H(5,𝔽2)11yescharacteristic
181Dic149—21no—
182Dic149—22no—
183Dic149—21no—
184Dic149—41no—
185Dic149—41no—
186Dic149—41no—
187Dic149C2×Q1612yescomplemented
188C2×C149—22no—
189C2×C149D4211yescharacteristic
190C2×C149C22×Q1611yescharacteristic
191C2×C149#57811yescharacteristic
192D149—22no—
193C149D4×Q1611yescharacteristic, complemented, Sylow
194D4×Q16—1491nomaximal, Sylow
195C22×Q16—1491no—
196C22×Q16—1491no—
197D4×Q8—1491no—
198D4×Q8—1491no—
199#494—1491no—
200#467—1491no—
201#467—1491no—
202#462—1491no—
203#462—1491no—
204#451—1491no—
205#451—1491no—
206#451—1491no—
207#451D14911yescharacteristic, complemented
208C4×Q16—1491no—
209C8×D4—1491no—
210C22×Q8—1491no—
211C22×Q8—1491no—
212C22×Q8—1491no—
213C22×Q8C2×D14911yescharacteristic
214C2×Q16—2984no—
215C2×Q16—2984no—
216C2×Q16—1491no—
217C2×Q16—1491no—
218C2×Q16—1491no—
219C2×Q16—1491no—
220C2×Q16—1491no—
221C2×Q16—1491no—
222C2×Q16—1491no—
223C2×Q16C2×D14911yescharacteristic
224C2×Q16C2×D14911yescharacteristic
225C22×C8—1491no—
226C22×C8—1491no—
227C4⋊Q8—2981no—
228C4⋊Q8—2981no—
229C4⋊Q8—1491no—
230C4⋊Q8—1491no—
231C22⋊Q8—2981no—
232C22⋊Q8—2981no—
233C22⋊Q8—2981no—
234C22⋊Q8—2981no—
235C22⋊Q8—1491no—
236C22⋊Q8—1491no—
237C22⋊Q8—1491no—
238C22⋊Q8C2×D14911yescharacteristic
239C4×Q8—1491no—
240C4×Q8—1491no—
241C4×D4—2981no—
242C4×D4—2981no—
243C4×D4—1491no—
244#107—1491no—
245C4⋊C8—1491no—
246#103—1491no—
247#103—1491no—
248#103—1491no—
249#103—1491no—
250#103C2×D14911yescharacteristic
251#103C2×D14911yescharacteristic
252C22⋊C8—1491no—
253C22⋊C8C2×D14911yescharacteristic
254C4×C8—1491no—
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—2981no—
259C2×Q8—2981no—
260C2×Q8—2982no—
261C2×Q8—2981no—
262C2×Q8—2982no—
263C2×Q8—2982no—
264C2×Q8—5961no—
265C2×Q8—5961no—
266C2×Q8—5961no—
267C2×Q8—1491no—
268C2×Q8—1491no—
269C2×Q8—1491no—
270C2×Q8C22×D14911yescharacteristic
271C2×Q8C22×D14911yescharacteristic
272C2×Q8C22×D14911yescharacteristic
273C2×D4—1491no—
274C22×C4—21no—
275C22×C4—2981no—
276C22×C4—2981no—
277C22×C4—2981no—
278C22×C4—1491no—
279C22×C4C22×D14911yescharacteristic
280Q16—22no—
281Q16—41no—
282Q16—2982no—
283Q16—2982no—
284Q16—2982no—
285Q16—2982no—
286Q16—2982no—
287Q16—5961no—
288Q16—1494no—
289C2×C8—2982no—
290C2×C8—2982no—
291C2×C8—1491no—
292C2×C8—1491no—
293C2×C8C22×D14911yescharacteristic
294C2×C8C22×D14911yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—2981no—
297C4⋊C4—2981no—
298C4⋊C4—2981no—
299C4⋊C4—2981no—
300C4⋊C4—2981no—
301C4⋊C4—2981no—
302C4⋊C4—2981no—
303C4⋊C4—2981no—
304C4⋊C4—2981no—
305C4⋊C4—1491no—
306C4⋊C4—1491no—
307C4⋊C4C22×D14911yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—2981no—
310C22⋊C4—2981no—
311C22⋊C4—2981no—
312C22⋊C4—1491no—
313C22⋊C4—1491no—
314C42—2981no—
315C42—2981no—
316C42—1491no—
317C23—1491no—
318C23D4×D14911yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—42no—
325Q8—2981no—
326Q8—2981no—
327Q8—2981no—
328Q8—2981no—
329Q8—2981no—
330Q8—5962no—
331Q8—5962no—
332Q8—5962no—
333Q8—1492no—
334Q8D4×D14912yes—
335D4—1494no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—2981no—
344C2×C4—2981no—
345C2×C4—2981no—
346C2×C4—2981no—
347C2×C4—2981no—
348C2×C4—2981no—
349C2×C4—5962no—
350C2×C4—5962no—
351C2×C4—5962no—
352C2×C4—1491no—
353C2×C4—1491no—
354C2×C4—1491no—
355C2×C4D4×D14911yescharacteristic
356C2×C4C23×D14911yescharacteristic
357C8—21no—
358C8—21no—
359C8—2981no—
360C8—1492no—
361C22—21no—
362C22—21no—
363C22—2981no—
364C22—2981no—
365C22—1492no—
366C22Q16×D14912yescomplemented
367C22C2×D4×D14911yescharacteristic
368C4—21no—
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—2981no—
374C4—2981no—
375C4—2982no—
376C4—5961no—
377C4—5961no—
378C4—5961no—
379C4—1492no—
380C4C2×D4×D14911yescharacteristic
381C4#1801195611yescharacteristic
382C2—22no—
383C2—2982no—
384C2#2375025711yescharacteristic
385C2C2×Q16×D14911yescharacteristic
386C2#2375044111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#451×C149C2
socleC22×C149C22×D4
prime radicalC2×C22⋊D149D4
hypercenter#451D149
hypocenterC149D4×Q16
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
1. C22×#14558726C2semidirect
2. C2×Q16×D149C2semidirect
5. #23750003C2semidirect
6. #23749865C2semidirect
7. #23749864C2semidirect
8. #23749834C2semidirect
9. #23749833C2semidirect
10. #23749783C2semidirect
11. #23749782C2semidirect
12. #23749779C2semidirect
13. #23749717C2semidirect
15. #451×C149C2semidirect
84. #14558726D4semidirect
133. C22⋊D149Q16semidirect
169. C22×C149C2×Q16semidirect
173. C2×D149C2×Q16semidirect
187. Dic149C2×Q16semidirect
193. C149D4×Q16semidirect
207. #451D149semidirect
366. C22Q16×D149semidirect

Maximal subgroups

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

subgroupindex
C22×#145587262
C2×Q16×D1492
Q8×C22⋊D1492
#237502882
#237500032
#237498652
#237498642
#237498342
#237498332
#237497832
#237497822
#237497792
#237497172
#237496882
#451×C1492
D4×Q16149