Finite Group Databasebeta

← #35218233

Subgroups — #35218233

Order 20864.

608 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104734C211yesmaximal, characteristic, complemented
2C2×#19104715C211yesmaximal, characteristic
3C2×#19104713C211yesmaximal, characteristic
4C2×C4⋊C4×D163C211yesmaximal, characteristic
5#23781597C211yesmaximal, characteristic, complemented
6#23781586C211yesmaximal, characteristic, complemented
7#23781585C211yesmaximal, characteristic, complemented
8#23781562C211yesmaximal, characteristic, complemented
9#23781560C211yesmaximal, characteristic, complemented
10#23781539C211yesmaximal, characteristic
11#23781536C211yesmaximal, characteristic
12#23781534C211yesmaximal, characteristic, complemented
13#23781524C211yesmaximal, characteristic
14#23781512C211yesmaximal, characteristic
15#398×C163C211yesmaximal, characteristic, complemented
16C22×Dic326C2211yescharacteristic
17C22×Dic326C2211yescharacteristic
18C22×C4×D163C2211yescharacteristic
19C22×C4×D163C2211yescharacteristic
20C22×C4×D163C2211yescharacteristic
21#19104734—24no—
22#19104715—24no—
23#19104713—24no—
24C2×C4⋊Dic163C2211yescharacteristic
25C2×C4⋊Dic163C2211yescharacteristic
26C2×C4⋊Dic163C2211yescharacteristic
27C2×C4⋊Dic163C2211yescharacteristic
28C2×Dic163⋊C4C2211yescharacteristic
29C2×Dic163⋊C4C2211yescharacteristic
30C2×Dic163⋊C4C2211yescharacteristic
31C2×Dic163⋊C4C2211yescharacteristic
32C2×Dic163⋊C4C2211yescharacteristic
33C2×Dic163⋊C4C2211yescharacteristic
34C2×Dic163⋊C4C2211yescharacteristic
35C2×Dic163⋊C4C2211yescharacteristic
36C4⋊C4×D163—24no—
37C2×(C2×D163)⋊C4C2211yescharacteristic
38C2×(C2×D163)⋊C4C2211yescharacteristic
39C2×(C2×D163)⋊C4C2211yescharacteristic
40C2×(C2×D163)⋊C4C2211yescharacteristic
41C2×C4×Dic163C2211yescharacteristic
42C2×C4×Dic163C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C163⋊#95C2211yescharacteristic
45C163⋊#95C2211yescharacteristic
46C163⋊#95C2211yescharacteristic
47C163⋊#95C2211yescharacteristic
48C2×C4⋊C4×C163C2211yescharacteristic
49C2×C4⋊C4×C163C2211yescharacteristic
50C2×C4⋊C4×C163C2211yescharacteristic
51C2×C4⋊C4×C163C2211yescharacteristic
52#95×C163C2211yescharacteristic
53#95×C163C2211yescharacteristic
54#95×C163C2211yescharacteristic
55C23×D163C2311yescharacteristic
56C2×Dic326—22no—
57C2×Dic326—22no—
58C2×Dic326—22no—
59C2×Dic326—22no—
60C2×Dic326—41no—
61C2×Dic326—41no—
62C22×Dic163C2311yescharacteristic
63C22×Dic163C2311yescharacteristic
64C22×Dic163C2311yescharacteristic
65C22×Dic163C2311yescharacteristic
66C22×Dic163C2311yescharacteristic
67C22×Dic163C2311yescharacteristic
68C22×Dic163C2311yescharacteristic
69C2×C4×D163—22no—
70C2×C4×D163—22no—
71C2×C4×D163—22no—
72C2×C4×D163—22no—
73C2×C4×D163—22no—
74C2×C4×D163—22no—
75C2×C4×D163—41no—
76C2×C4×D163—41no—
77C2×C4×D163Q814yescomplemented
78C4⋊Dic163—22no—
79C4⋊Dic163—22no—
80C4⋊Dic163—22no—
81C4⋊Dic163—22no—
82Dic163⋊C4—22no—
83Dic163⋊C4—22no—
84Dic163⋊C4—22no—
85Dic163⋊C4—22no—
86Dic163⋊C4—22no—
87Dic163⋊C4—41no—
88Dic163⋊C4—41no—
89Dic163⋊C4D414yes—
90(C2×D163)⋊C4—22no—
91(C2×D163)⋊C4—22no—
92(C2×D163)⋊C4D414yes—
93(C2×D163)⋊C4Q814yescomplemented
94C4×Dic163—22no—
95C4×Dic163—22no—
96C22×C4×C163C2311yescharacteristic
97C22×C4×C163C2311yescharacteristic
98C22×C4×C163C2311yescharacteristic
99C22×C4×C163C2311yescharacteristic
100C22×C4×C163C2311yescharacteristic
101C22×C4×C163C2311yescharacteristic
102C22×C4×C163C2311yescharacteristic
103C4⋊C4×C163—22no—
104C4⋊C4×C163—22no—
105C4⋊C4×C163—22no—
106C4⋊C4×C163—22no—
107C22×D163—21no—
108C22×D163—21no—
109C22×D163—21no—
110C22×D163—21no—
111C22×D163C2×D412yes—
112C22×D163C2×Q812yes—
113C22×D163C2×Q812yes—
114Dic326—44no—
115Dic326—44no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C2×Dic163—21no—
128C2×Dic163—21no—
129C2×Dic163—21no—
130C2×Dic163—21no—
131C2×Dic163C2×D412yes—
132C2×Dic163C2×D412yes—
133C2×Dic163C2×D412yes—
134C2×Dic163C2×Q812yes—
135C2×Dic163C2×Q812yes—
136C2×Dic163#4112yes—
137C4×D163—28no—
138C4×D163—44no—
139C4×D163—44no—
140C23×C163C2411yescharacteristic
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—21no—
150C2×C4×C163—21no—
151C2×C4×C163—21no—
152C2×C4×C163—21no—
153C2×C4×C163—21no—
154C2×C4×C163—21no—
155C2×C4×C163—21no—
156C2×C4×C163—21no—
157C2×C4×C163—21no—
158C2×C4×C163C2×D412yes—
159C2×C4×C163C2×D412yes—
160C2×C4×C163C2×Q812yes—
161C2×C4×C163C2×Q812yes—
162C22×C163C22×D411yescharacteristic
163C22×C163C22×D411yescharacteristic
164C22×C163C22×Q811yescharacteristic
165C22×C163C22×Q811yescharacteristic
166C22×C163C2×#4111yescharacteristic
167C22×C163H(5,𝔽2)11yescharacteristic
168C22×C1632−1+411yescharacteristic
169C2×D163—22no—
170C2×D163—22no—
171C2×D163—22no—
172C2×D163—22no—
173C2×D163—22no—
174C2×D163—22no—
175C2×D163C4⋊Q814yescomplemented
176C4×C163—22no—
177C4×C163—22no—
178C4×C163—22no—
179C4×C163—22no—
180C4×C163—41no—
181C4×C163—41no—
182C4×C163—41no—
183Dic163—22no—
184Dic163—22no—
185Dic163—22no—
186Dic163—41no—
187Dic163—41no—
188Dic163—41no—
189Dic163C22⋊Q814yescomplemented
190C2×C163C2×C22⋊Q811yescharacteristic
191C2×C163C2×C4⋊Q811yescharacteristic
192C2×C163#54411yescharacteristic
193C2×C163D4211yescharacteristic
194C2×C163#54711yescharacteristic
195C2×C163D4×Q811yescharacteristic
196C2×C163#55411yescharacteristic
197D163—24no—
198C163#244011yescharacteristic, complemented, Sylow
199#2440—1631nomaximal, Sylow
200C2×C4⋊Q8—1631no—
201C2×C22⋊Q8—1631no—
202C2×C22⋊Q8—1631no—
203C22×C4⋊C4—1631no—
204#398—1631no—
205#398D16311yescharacteristic, complemented
206#395—1631no—
207#395—1631no—
208#389—1631no—
209#389—1631no—
210#385—1631no—
211#385—1631no—
212#384—1631no—
213#380—1631no—
214C4×C22⋊C4—1631no—
215C22×Q8—1631no—
216C22×Q8—1631no—
217C23×C4—1631no—
218C23×C4—1631no—
219C23×C4—1631no—
220C4⋊Q8—3264no—
221C22⋊Q8—3264no—
222C22⋊Q8—3264no—
223C2×C4⋊C4—3264no—
224C2×C4⋊C4—1631no—
225C2×C4⋊C4—1631no—
226C2×C4⋊C4—1631no—
227C2×C4⋊C4—1631no—
228C2×C4⋊C4—1631no—
229C2×C4⋊C4—1631no—
230C2×C4⋊C4—1631no—
231C2×C4⋊C4—1631no—
232C2×C4⋊C4—1631no—
233C2×C4⋊C4—1631no—
234C2×C4⋊C4—1631no—
235C2×C4⋊C4—1631no—
236C2×C4⋊C4C2×D16311yescharacteristic
237C2×C4⋊C4C2×D16311yescharacteristic
238C2×C4⋊C4C2×D16311yescharacteristic
239C2×C4⋊C4C2×D16311yescharacteristic
240C2×C22⋊C4—1631no—
241C2×C22⋊C4—1631no—
242C2×C22⋊C4—1631no—
243C2×C22⋊C4—1631no—
244C2×C42—1631no—
245C2×C42—1631no—
246#95—1631no—
247#95—1631no—
248#95—1631no—
249#95—1631no—
250#95—1631no—
251#95C2×D16311yescharacteristic
252#95C2×D16311yescharacteristic
253#95C2×D16311yescharacteristic
254C24—1631no—
255C2×Q8—3262no—
256C2×Q8—3262no—
257C2×Q8—3262no—
258C2×Q8—3262no—
259C2×Q8—6521no—
260C2×Q8—6521no—
261C22×C4—3262no—
262C22×C4—3262no—
263C22×C4—3262no—
264C22×C4—3262no—
265C22×C4—3262no—
266C22×C4—3262no—
267C22×C4—6521no—
268C22×C4—6521no—
269C22×C4—1631no—
270C22×C4—1631no—
271C22×C4—1631no—
272C22×C4—1631no—
273C22×C4—1631no—
274C22×C4—1631no—
275C22×C4—1634no—
276C22×C4—1631no—
277C22×C4C22×D16311yescharacteristic
278C22×C4C22×D16311yescharacteristic
279C22×C4C22×D16311yescharacteristic
280C22×C4C22×D16311yescharacteristic
281C22×C4C22×D16311yescharacteristic
282C22×C4C22×D16311yescharacteristic
283C22×C4C22×D16311yescharacteristic
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—3262no—
289C4⋊C4—3262no—
290C4⋊C4—3262no—
291C4⋊C4—3262no—
292C4⋊C4—3262no—
293C4⋊C4—3262no—
294C4⋊C4—3262no—
295C4⋊C4—3262no—
296C4⋊C4—3262no—
297C4⋊C4—6521no—
298C4⋊C4—6521no—
299C4⋊C4—1634no—
300C22⋊C4—3262no—
301C22⋊C4—3262no—
302C22⋊C4—1634no—
303C22⋊C4—1634no—
304C42—3262no—
305C42—3262no—
306C23—3261no—
307C23—3261no—
308C23—3261no—
309C23—3261no—
310C23—1632no—
311C23—1632no—
312C23—1632no—
313C23C23×D16311yescharacteristic
314Q8—6524no—
315Q8—6524no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—3268no—
334C2×C4—3261no—
335C2×C4—3261no—
336C2×C4—3261no—
337C2×C4—3261no—
338C2×C4—3261no—
339C2×C4—3261no—
340C2×C4—3261no—
341C2×C4—3261no—
342C2×C4—3261no—
343C2×C4—3261no—
344C2×C4—3261no—
345C2×C4—3261no—
346C2×C4—3261no—
347C2×C4—3261no—
348C2×C4—3261no—
349C2×C4—6524no—
350C2×C4—6524no—
351C2×C4—1632no—
352C2×C4—1632no—
353C2×C4—1632no—
354C2×C4—1632no—
355C2×C4—1632no—
356C2×C4—1632no—
357C2×C4D4×D16312yes—
358C2×C4D4×D16312yes—
359C2×C4Q8×D16312yes—
360C2×C4Q8×D16312yes—
361C22—3262no—
362C22—3262no—
363C22—3262no—
364C22—3262no—
365C22—3262no—
366C22—3262no—
367C22—1634no—
368C22C2×D4×D16311yescharacteristic
369C22C2×D4×D16311yescharacteristic
370C22C2×Q8×D16311yescharacteristic
371C22C2×Q8×D16311yescharacteristic
372C22#41×D16311yescharacteristic
373C22#1910478111yescharacteristic
374C22#1910478211yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—3262no—
383C4—3262no—
384C4—3262no—
385C4—6521no—
386C4—6521no—
387C4—6521no—
388C4—1634no—
389C2—3264no—
390C2C22⋊Q8×D16311yescharacteristic
391C2C4⋊Q8×D16311yescharacteristic
392C2#2378229511yescharacteristic
393C2#2378229911yescharacteristic
394C2#2378231311yescharacteristic
395C2#2378232811yescharacteristic
396C2#2378237111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#398×C163C2
prime radicalC23×D163C23
hypercenter#398D163
hypocenterC163#2440
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#19104734C2semidirect
5. #23781597C2semidirect
6. #23781586C2semidirect
7. #23781585C2semidirect
8. #23781562C2semidirect
9. #23781560C2semidirect
12. #23781534C2semidirect
15. #398×C163C2semidirect
77. C2×C4×D163Q8semidirect
93. (C2×D163)⋊C4Q8semidirect
175. C2×D163C4⋊Q8semidirect
189. Dic163C22⋊Q8semidirect
198. C163#2440semidirect
205. #398D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047342
C2×#191047152
C2×#191047132
C2×C4⋊C4×D1632
#237815972
#237815862
#237815852
#237815622
#237815602
#237815392
#237815362
#237815342
#237815242
#237815122
#398×C1632
#2440163