Finite Group Databasebeta

← #46822630

Subgroups — #46822630

Order 34432.

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×#16864392C211yesmaximal, characteristic, complemented
2C2×Q16×D269C211yesmaximal, characteristic, complemented
3Q8×C22⋊D269C211yesmaximal, characteristic
4#31771268C211yesmaximal, characteristic
5#31770983C211yesmaximal, characteristic, complemented
6#31770845C211yesmaximal, characteristic, complemented
7#31770844C211yesmaximal, characteristic, complemented
8#31770814C211yesmaximal, characteristic, complemented
9#31770813C211yesmaximal, characteristic, complemented
10#31770763C211yesmaximal, characteristic, complemented
11#31770762C211yesmaximal, characteristic, complemented
12#31770759C211yesmaximal, characteristic, complemented
13#31770697C211yesmaximal, characteristic, complemented
14#31770668C211yesmaximal, characteristic
15#451×C269C211yesmaximal, characteristic, complemented
16C22×Dic538C2211yescharacteristic
17C2×Q8×D269C2211yescharacteristic
18C2×Q8×D269C2211yescharacteristic
19C2×Dic1076C2211yescharacteristic
20C2×Dic1076C2211yescharacteristic
21C2×#16864392—24no—
22C2×#16864392C2211yescharacteristic
23C2×#16864392C2211yescharacteristic
24C2×#16864392C2211yescharacteristic
25C2×#16864392C2211yescharacteristic
26C2×#16864392C2211yescharacteristic
27Q16×D269—24no—
28C22×#14524130C2211yescharacteristic
29C2×C8×D269C2211yescharacteristic
30#22575452—21no—
31#22575451—21no—
32#22575451C2211yescharacteristic
33#22575451C2211yescharacteristic
34#22575433C2211yescharacteristic
35#22575432—21no—
36#22575432C2211yescharacteristic
37#22575431—21no—
38#22575430—21no—
39#22575429—21no—
40#22575429C2211yescharacteristic
41Q8×Dic269C2211yescharacteristic
42#22575418C2211yescharacteristic
43C4×C22⋊D269—21no—
44C4×C22⋊D269C2211yescharacteristic
45#22575413—21no—
46#22575386C2211yescharacteristic
47#22575382C2211yescharacteristic
48#22575378C2211yescharacteristic
49#22575378C2211yescharacteristic
50#22575377C2211yescharacteristic
51#22575376C2211yescharacteristic
52#22575365C2211yescharacteristic
53C8×Dic269C2211yescharacteristic
54C22×Q8×C269C2211yescharacteristic
55C2×Q16×C269C2211yescharacteristic
56C2×Q16×C269C2211yescharacteristic
57C22⋊Q8×C269C2211yescharacteristic
58#103×C269C2211yescharacteristic
59#103×C269C2211yescharacteristic
60C22⋊C8×C269C2211yescharacteristic
61C2×Dic538—21no—
62C2×Dic538—21no—
63C2×Dic538—22no—
64C2×Dic538—21no—
65C2×Dic538—41no—
66C2×Dic538C2311yescharacteristic
67C2×Dic538C2311yescharacteristic
68C2×Dic538C2311yescharacteristic
69Q8×D269—22no—
70Q8×D269—22no—
71Q8×D269—41no—
72Q8×D269—41no—
73C2×C22⋊D269D411yescharacteristic
74C22×Dic269—21no—
75C2×C4×D269—21no—
76C2×C4×D269—21no—
77C2×C4×D269C2311yescharacteristic
78Dic1076—22no—
79Dic1076—41no—
80#16864392—22no—
81#16864392—22no—
82#16864392—22no—
83#16864392—22no—
84#16864392D414yescomplemented
85C2×#14524130—22no—
86C2×#14524130C2311yescharacteristic
87C2×#14524130C2311yescharacteristic
88C8×D269—22no—
89C4⋊Dic269—21no—
90C4⋊Dic269—21no—
91C4⋊Dic269C2311yescharacteristic
92#16864381—21no—
93#16864381—21no—
94#16864381—21no—
95#16864381—21no—
96#16864381—21no—
97#16864381—21no—
98#16864381—21no—
99#16864381D411yescharacteristic
100C22⋊Dic269—21no—
101C22⋊Dic269D411yescharacteristic
102#16864379—21no—
103#16864379—21no—
104#16864379D411yescharacteristic
105C4×Dic269—21no—
106C4×Dic269—21no—
107C4×Dic269C2311yescharacteristic
108C2×Q8×C269—21no—
109C2×Q8×C269—22no—
110C2×Q8×C269—41no—
111C2×Q8×C269C2311yescharacteristic
112C2×Q8×C269C2311yescharacteristic
113C2×Q8×C269C2311yescharacteristic
114C22×C4×C269—21no—
115C22×C4×C269C2311yescharacteristic
116Q16×C269—22no—
117Q16×C269—41no—
118C2×C8×C269C2311yescharacteristic
119C2×C8×C269C2311yescharacteristic
120C4⋊C4×C269—21no—
121C4⋊C4×C269C2311yescharacteristic
122C22⋊C4×C269—21no—
123C22×D269C2×D411yescharacteristic
124Dic538—21no—
125Dic538—21no—
126Dic538—21no—
127Dic538—21no—
128Dic538—21no—
129Dic538—42no—
130Dic538—42no—
131Dic538—42no—
132Dic538C2×D412yes—
133C22⋊D269Q1614yescomplemented
134C2×Dic269—21no—
135C2×Dic269—21no—
136C2×Dic269—21no—
137C2×Dic269—21no—
138C2×Dic269—42no—
139C2×Dic269C2×D411yescharacteristic
140C2×Dic269C2×D411yescharacteristic
141C2×Dic269C2×D411yescharacteristic
142C4×D269—21no—
143C4×D269—21no—
144C4×D269—42no—
145C4×D269—42no—
146#14524130—21no—
147#14524130C2×D412yes—
148C23×C269C2×D411yescharacteristic
149Q8×C269—21no—
150Q8×C269—21no—
151Q8×C269—21no—
152Q8×C269—21no—
153Q8×C269—21no—
154Q8×C269—42no—
155Q8×C269C2×D412yes—
156C2×C4×C269—21no—
157C2×C4×C269—21no—
158C2×C4×C269—21no—
159C2×C4×C269—21no—
160C2×C4×C269—21no—
161C2×C4×C269—21no—
162C2×C4×C269—42no—
163C2×C4×C269C2×D411yescharacteristic
164C2×C4×C269C2411yescharacteristic
165C8×C269—21no—
166C8×C269—21no—
167C22×C269—21no—
168C22×C269—21no—
169C22×C269C2×Q1612yescomplemented
170C22×C269C22×D411yescharacteristic
171C2×D269—21no—
172C2×D269—21no—
173C2×D269C2×Q1612yescomplemented
174C4×C269—21no—
175C4×C269—22no—
176C4×C269—41no—
177C4×C269—41no—
178C4×C269—41no—
179C4×C269C22×D411yescharacteristic
180C4×C269H(5,𝔽2)11yescharacteristic
181Dic269—21no—
182Dic269—22no—
183Dic269—21no—
184Dic269—41no—
185Dic269—41no—
186Dic269—41no—
187Dic269C2×Q1612yescomplemented
188C2×C269—22no—
189C2×C269D4211yescharacteristic
190C2×C269C22×Q1611yescharacteristic
191C2×C269#57811yescharacteristic
192D269—22no—
193C269D4×Q1611yescharacteristic, complemented, Sylow
194D4×Q16—2691nomaximal, Sylow
195C22×Q16—2691no—
196C22×Q16—2691no—
197D4×Q8—2691no—
198D4×Q8—2691no—
199#494—2691no—
200#467—2691no—
201#467—2691no—
202#462—2691no—
203#462—2691no—
204#451—2691no—
205#451—2691no—
206#451—2691no—
207#451D26911yescharacteristic, complemented
208C4×Q16—2691no—
209C8×D4—2691no—
210C22×Q8—2691no—
211C22×Q8—2691no—
212C22×Q8—2691no—
213C22×Q8C2×D26911yescharacteristic
214C2×Q16—2691no—
215C2×Q16—2691no—
216C2×Q16—2691no—
217C2×Q16—2691no—
218C2×Q16—2691no—
219C2×Q16—2691no—
220C2×Q16—2691no—
221C2×Q16—5384no—
222C2×Q16—5384no—
223C2×Q16C2×D26911yescharacteristic
224C2×Q16C2×D26911yescharacteristic
225C22×C8—2691no—
226C22×C8—2691no—
227C4⋊Q8—2691no—
228C4⋊Q8—2691no—
229C4⋊Q8—5381no—
230C4⋊Q8—5381no—
231C22⋊Q8—2691no—
232C22⋊Q8—2691no—
233C22⋊Q8—2691no—
234C22⋊Q8—5381no—
235C22⋊Q8—5381no—
236C22⋊Q8—5381no—
237C22⋊Q8—5381no—
238C22⋊Q8C2×D26911yescharacteristic
239C4×Q8—2691no—
240C4×Q8—2691no—
241C4×D4—2691no—
242C4×D4—5381no—
243C4×D4—5381no—
244#107—2691no—
245C4⋊C8—2691no—
246#103—2691no—
247#103—2691no—
248#103—2691no—
249#103—2691no—
250#103C2×D26911yescharacteristic
251#103C2×D26911yescharacteristic
252C22⋊C8—2691no—
253C22⋊C8C2×D26911yescharacteristic
254C4×C8—2691no—
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—2691no—
259C2×Q8—2691no—
260C2×Q8—2691no—
261C2×Q8—5381no—
262C2×Q8—5381no—
263C2×Q8—5382no—
264C2×Q8—5381no—
265C2×Q8—5382no—
266C2×Q8—5382no—
267C2×Q8—10761no—
268C2×Q8—10761no—
269C2×Q8—10761no—
270C2×Q8C22×D26911yescharacteristic
271C2×Q8C22×D26911yescharacteristic
272C2×Q8C22×D26911yescharacteristic
273C2×D4—2691no—
274C22×C4—21no—
275C22×C4—2691no—
276C22×C4—5381no—
277C22×C4—5381no—
278C22×C4—5381no—
279C22×C4C22×D26911yescharacteristic
280Q16—22no—
281Q16—41no—
282Q16—2694no—
283Q16—5382no—
284Q16—5382no—
285Q16—5382no—
286Q16—5382no—
287Q16—5382no—
288Q16—10761no—
289C2×C8—2691no—
290C2×C8—2691no—
291C2×C8—5382no—
292C2×C8—5382no—
293C2×C8C22×D26911yescharacteristic
294C2×C8C22×D26911yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—2691no—
297C4⋊C4—2691no—
298C4⋊C4—5381no—
299C4⋊C4—5381no—
300C4⋊C4—5381no—
301C4⋊C4—5381no—
302C4⋊C4—5381no—
303C4⋊C4—5381no—
304C4⋊C4—5381no—
305C4⋊C4—5381no—
306C4⋊C4—5381no—
307C4⋊C4C22×D26911yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—2691no—
310C22⋊C4—2691no—
311C22⋊C4—5381no—
312C22⋊C4—5381no—
313C22⋊C4—5381no—
314C42—2691no—
315C42—5381no—
316C42—5381no—
317C23—2691no—
318C23D4×D26911yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—42no—
325Q8—2692no—
326Q8—5381no—
327Q8—5381no—
328Q8—5381no—
329Q8—5381no—
330Q8—5381no—
331Q8—10762no—
332Q8—10762no—
333Q8—10762no—
334Q8D4×D26912yes—
335D4—2694no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—2691no—
344C2×C4—2691no—
345C2×C4—2691no—
346C2×C4—5381no—
347C2×C4—5381no—
348C2×C4—5381no—
349C2×C4—5381no—
350C2×C4—5381no—
351C2×C4—5381no—
352C2×C4—10762no—
353C2×C4—10762no—
354C2×C4—10762no—
355C2×C4D4×D26911yescharacteristic
356C2×C4C23×D26911yescharacteristic
357C8—21no—
358C8—21no—
359C8—2692no—
360C8—5381no—
361C22—21no—
362C22—21no—
363C22—2692no—
364C22—5381no—
365C22—5381no—
366C22Q16×D26912yescomplemented
367C22C2×D4×D26911yescharacteristic
368C4—21no—
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—2692no—
374C4—5381no—
375C4—5381no—
376C4—5382no—
377C4—10761no—
378C4—10761no—
379C4—10761no—
380C4C2×D4×D26911yescharacteristic
381C4#2257549811yescharacteristic
382C2—22no—
383C2—5382no—
384C2#3177123711yescharacteristic
385C2C2×Q16×D26911yescharacteristic
386C2#3177142111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#451×C269C2
socleC22×C269C22×D4
prime radicalC2×C22⋊D269D4
hypercenter#451D269
hypocenterC269D4×Q16
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C22×#16864392C2semidirect
2. C2×Q16×D269C2semidirect
5. #31770983C2semidirect
6. #31770845C2semidirect
7. #31770844C2semidirect
8. #31770814C2semidirect
9. #31770813C2semidirect
10. #31770763C2semidirect
11. #31770762C2semidirect
12. #31770759C2semidirect
13. #31770697C2semidirect
15. #451×C269C2semidirect
84. #16864392D4semidirect
133. C22⋊D269Q16semidirect
169. C22×C269C2×Q16semidirect
173. C2×D269C2×Q16semidirect
187. Dic269C2×Q16semidirect
193. C269D4×Q16semidirect
207. #451D269semidirect
366. C22Q16×D269semidirect

Maximal subgroups

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

subgroupindex
C22×#168643922
C2×Q16×D2692
Q8×C22⋊D2692
#317712682
#317709832
#317708452
#317708442
#317708142
#317708132
#317707632
#317707622
#317707592
#317706972
#317706682
#451×C2692
D4×Q16269