Finite Group Databasebeta

← #17966643

Subgroups — #17966643

Order 4352.

730 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14545677C211yesmaximal, characteristic, complemented
2#14545631C211yesmaximal, characteristic
3SD16⋊#3624C214yesmaximal, complemented
4SD16×#3624C214yesmaximal, complemented
5#14544331C211yesmaximal, characteristic, complemented
6#14544086C211yesmaximal, characteristic, complemented
7#14544083C211yesmaximal, characteristic
8#14543987C211yesmaximal, characteristic, complemented
9C2×SD16×Dic17C211yesmaximal, characteristic
10Q8×#3624—22no—
11Q8×#3624C2214yescomplemented
12D4×#3624—22no—
13D4×#3624C2214yes—
14#11763825—22no—
15#11763825C2211yescharacteristic
16#11763825C2211yescharacteristic
17#11763825C2211yescharacteristic
18#11763825C2211yescharacteristic
19C2×C22⋊#3624—22no—
20#11763672C2212yescomplemented
21#11763670C2212yescomplemented
22Q8⋊#3624C2214yescomplemented
23D4⋊#3624C2214yes—
24#11763650C2212yescomplemented
25C8×#3624C2212yescomplemented
26C2×Q8×Dic17C2211yescharacteristic
27C2×D4×Dic17C2211yescharacteristic
28SD16×Dic17C412yes—
29SD16×Dic17C412yes—
30SD16×Dic17C2212yes—
31SD16×Dic17C2212yes—
32C2×#10587488C411yescharacteristic
33C2×#10587506C411yescharacteristic
34C2×#10587503C411yescharacteristic
35C2×C8×Dic17C2211yescharacteristic
36C22×SD16×C17C411yescharacteristic
37C22×Q8×C17C2×C411yescharacteristic
38C22×D4×C17C2×C411yescharacteristic
39C23×Dic17—21no—
40C22×#3624—24no—
41C2×SD16×C17C818yescomplemented
42C2×SD16×C17C2×C412yes—
43C2×SD16×C17C2×C412yes—
44C22×C8×C17C2×C411yescharacteristic
45C2×C22⋊Dic17—21no—
46Q8×Dic17—21no—
47Q8×Dic17—21no—
48Q8×Dic17C2×C412yes—
49Q8×Dic17C2312yes—
50D4×Dic17—21no—
51D4×Dic17—21no—
52D4×Dic17C2×C412yes—
53D4×Dic17C2312yes—
54C2×C4⋊Dic17—21no—
55C2×C4⋊Dic17C2×C411yescharacteristic
56C2×C4⋊Dic17C2×C411yescharacteristic
57C2×C4×Dic17—21no—
58C2×C4×Dic17C2311yescharacteristic
59C22×#3622C2×C411yescharacteristic
60C22⋊#3624—21no—
61C22⋊#3624—21no—
62C22⋊#3624—21no—
63C22⋊#3624—24no—
64C22⋊#3624D412yes—
65C4⋊#3624—21no—
66C4⋊#3624—24no—
67C4⋊#3624D412yes—
68C4⋊#3624C2312yes—
69C4⋊#3624C2312yes—
70C4×#3624—21no—
71C4×#3624—21no—
72C4×#3624—24no—
73C4×#3624C2312yes—
74C4×#3624C2312yes—
75#10587506C2×C412yes—
76#10587506C2×C412yes—
77#10587503C2×C412yes—
78#10587503C2×C412yes—
79#10587488C2×C412yes—
80#10587488C2×C412yes—
81C8×Dic17C2×C411yescharacteristic
82C8×Dic17C2×C411yescharacteristic
83C8×Dic17C2311yescharacteristic
84C8×Dic17C2311yescharacteristic
85C24×C17—21no—
86C2×Q8×C17—21no—
87C2×Q8×C17—22no—
88C2×Q8×C17C2×C814yescomplemented
89C2×Q8×C17C22×C412yes—
90C2×D4×C17—21no—
91C2×D4×C17—22no—
92C2×D4×C17C2×C814yes—
93C2×D4×C17C22×C412yes—
94C22×C4×C17—21no—
95C22×C4×C17C22×C411yescharacteristic
96C22×Dic17—21no—
97C22×Dic17—21no—
98C22×Dic17—22no—
99C22×Dic17—22no—
100C22×Dic17—82no—
101C22×Dic17C2×D411yescharacteristic
102C22×Dic17#4111yescharacteristic
103C2×#3624—22no—
104C2×#3624—22no—
105C2×#3624—22no—
106C2×#3624—84no—
107C2×#3624C2×D414yes—
108SD16×C17C2×C818yescomplemented
109SD16×C17C2×C818yescomplemented
110C2×C8×C17C2×C814yescomplemented
111C2×C8×C17C22×C411yescharacteristic
112C2×C8×C17C22×C411yescharacteristic
113C22⋊Dic17—22no—
114C22⋊Dic17—22no—
115C4⋊Dic17—21no—
116C4⋊Dic17—21no—
117C4⋊Dic17—22no—
118C4⋊Dic17—22no—
119C4⋊Dic17C22×C412yes—
120C4⋊Dic17C22×C412yes—
121C4×Dic17—22no—
122C4×Dic17—22no—
123C4×Dic17C22×C411yescharacteristic
124C4×Dic17C2×D411yescharacteristic
125C4×Dic17#4111yescharacteristic
126C4×Dic17C2411yescharacteristic
127C2×#3622—22no—
128C2×#3622C22×C411yescharacteristic
129C2×#3622C22×C411yescharacteristic
130#17855—171nomaximal, Sylow
131C23×C17—22no—
132C23×C17—24no—
133C23×C17—81no—
134C23×C17—81no—
135C23×C17C4×D411yescharacteristic
136Q8×C17—22no—
137Q8×C17—24no—
138Q8×C17C22×C814yes—
139D4×C17—22no—
140D4×C17—24no—
141D4×C17C22×C814yes—
142C2×C4×C17—22no—
143C2×C4×C17—24no—
144C2×C4×C17C4×D411yescharacteristic
145C2×C4×C17C22×C812yes—
146C2×C4×C17#13112yes—
147C2×C4×C17C23×C411yescharacteristic
148C2×Dic17—22no—
149C2×Dic17—22no—
150C2×Dic17—22no—
151C2×Dic17—41no—
152C2×Dic17—44no—
153C2×Dic17—82no—
154C2×Dic17—81no—
155C2×Dic17—81no—
156C2×Dic17C4×D411yescharacteristic
157C2×Dic17C4×D411yescharacteristic
158C2×Dic17C22×D411yescharacteristic
159C2×Dic17C2×#4111yescharacteristic
160#3624—24no—
161#3624—42no—
162#3624—42no—
163#3624—42no—
164C8×C17C22×C812yes—
165C8×C17C22×C812yes—
166#3622—21no—
167#3622—21no—
168#2982—171no—
169#2972—171no—
170C2×C4×SD16—171no—
171SD16⋊C8—174no—
172C8×SD16—174no—
173#1564—171no—
174#1479—171no—
175#1478—171no—
176#1450—171no—
177C22×C17—21no—
178C22×C17—21no—
179C22×C17—24no—
180C22×C17—42no—
181C22×C17—42no—
182C22×C17—84no—
183C22×C17C8×D412yes—
184C22×C17C2×C4×D411yescharacteristic
185C4×C17—24no—
186C4×C17C8×D412yes—
187C4×C17C23×C811yescharacteristic
188C4×C17C2×#13111yescharacteristic
189Dic17—21no—
190Dic17—21no—
191Dic17—21no—
192Dic17—42no—
193Dic17—42no—
194Dic17#57712yes—
195C22×SD16#59511yescharacteristic
196C2×C4×Q8—171no—
197C2×C4×D4—171no—
198C8×Q8—174no—
199C8×Q8—342no—
200C4×SD16—172no—
201C4×SD16—172no—
202C4×SD16—172no—
203C4×SD16—172no—
204C8×D4—174no—
205C8×D4—342no—
206#431—171no—
207#431—171no—
208#431—171no—
209#431—171no—
210#431—342no—
211C2×#106—171no—
212C2×#103—171no—
213C2×#102—171no—
214C2×C22⋊C8—342no—
215C2×C4×C8—171no—
216#335—172no—
217#334—172no—
218Q8⋊C8—174no—
219D4⋊C8—174no—
220#322—172no—
221C82—172no—
222C2×C17—22no—
223C2×C17—44no—
224C2×C17C2×C8×D411yescharacteristic
225C2×C17#294911yescharacteristic
226C2×C17#295711yescharacteristic
227C22×Q8C2×#59511yescharacteristic
228C22×D4C2×#59511yescharacteristic
229C23×C4—341no—
230C2×SD16#362418yescomplemented
231C2×SD16C2×#59512yes—
232C2×SD16C2×#59512yes—
233C22×C8—171no—
234C22×C8—344no—
235C22×C8C2×#59511yescharacteristic
236C4×Q8—172no—
237C4×Q8—172no—
238C4×Q8—341no—
239C4×Q8—341no—
240C4×D4—172no—
241C4×D4—172no—
242C4×D4—341no—
243C4×D4—341no—
244C2×C4⋊C4—171no—
245C2×C4⋊C4—171no—
246C2×C4⋊C4—341no—
247C2×C22⋊C4—341no—
248C2×C42—171no—
249C2×C42—341no—
250#106—172no—
251#106—172no—
252C4⋊C8—172no—
253C4⋊C8—172no—
254C4⋊C8—172no—
255C4⋊C8—341no—
256C4⋊C8—344no—
257#103—172no—
258#103—172no—
259#102—172no—
260#102—172no—
261C22⋊C8—172no—
262C22⋊C8—344no—
263C22⋊C8—341no—
264C22⋊C8—341no—
265C22⋊C8—341no—
266C4×C8—172no—
267C4×C8—171no—
268C4×C8—171no—
269C4×C8—171no—
270C4×C8—171no—
271C4×C8—172no—
272C4×C8—341no—
273C4×C8—341no—
274C4×C8—344no—
275C17#1785511yescharacteristic, complemented, Sylow
276C24—21no—
277C2×Q8—22no—
278C2×Q8—21no—
279C2×Q8C2×#362414yescomplemented
280C2×Q8C22×#59512yes—
281C2×D4—22no—
282C2×D4—21no—
283C2×D4C2×#362414yes—
284C2×D4C22×#59512yes—
285C22×C4—21no—
286C22×C4—171no—
287C22×C4—171no—
288C22×C4—342no—
289C22×C4—342no—
290C22×C4—341no—
291C22×C4—341no—
292C22×C4—1362no—
293C22×C4C22×#59511yescharacteristic
294SD16C2×#362418yescomplemented
295SD16C2×#362418yescomplemented
296C2×C8—174no—
297C2×C8—171no—
298C2×C8—171no—
299C2×C8—342no—
300C2×C8—342no—
301C2×C8—342no—
302C2×C8—342no—
303C2×C8—1364no—
304C2×C8C2×#362414yescomplemented
305C2×C8C22×#59511yescharacteristic
306C2×C8C22×#59511yescharacteristic
307C4⋊C4—172no—
308C4⋊C4—172no—
309C4⋊C4—342no—
310C4⋊C4—342no—
311C4⋊C4—341no—
312C4⋊C4—341no—
313C22⋊C4—342no—
314C22⋊C4—342no—
315C42—171no—
316C42—171no—
317C42—171no—
318C42—171no—
319C42—342no—
320C42—342no—
321C23—22no—
322C23—24no—
323C23—81no—
324C23—81no—
325C23D4×#59511yescharacteristic
326Q8—24no—
327Q8—22no—
328Q8C22×#362414yes—
329D4—24no—
330D4—22no—
331D4C22×#362414yes—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—171no—
335C2×C4—171no—
336C2×C4—171no—
337C2×C4—171no—
338C2×C4—342no—
339C2×C4—342no—
340C2×C4—342no—
341C2×C4—684no—
342C2×C4—681no—
343C2×C4—1362no—
344C2×C4—1361no—
345C2×C4—1361no—
346C2×C4#1058767812yes—
347C2×C4D4×#59511yescharacteristic
348C2×C4C22×#362412yes—
349C2×C4C23×#59511yescharacteristic
350C8—344no—
351C8—341no—
352C8—341no—
353C8—682no—
354C8—682no—
355C8—682no—
356C8C22×#362412yes—
357C8C22×#362412yes—
358C22—21no—
359C22—21no—
360C22—24no—
361C22—42no—
362C22—42no—
363C22—84no—
364C22D4×#362412yes—
365C22C2×D4×#59511yescharacteristic
366C4—24no—
367C4—172no—
368C4—341no—
369C4—341no—
370C4—341no—
371C4—682no—
372C4—682no—
373C4D4×#362412yes—
374C4C23×#362411yescharacteristic
375C4C2×#1058767811yescharacteristic
376C2—22no—
377C2—44no—
378C2C2×D4×#362411yescharacteristic
379C2#1454554411yescharacteristic
380C2#1454556211yescharacteristic
subgroupG/H
centerC22C2×D4×#595
commutatorC4×C17C23×C8
FrattiniC2×C4C23×#595
FittingC22×SD16×C17C4
socleC22×C17C2×C4×D4
prime radicalC22×D4×C17C2×C4
cosocleC4×Dic17C24
hypercenterC22×SD16#595
hypocenterC17#17855
LCSG ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. #14545677C2semidirect
3. SD16⋊#3624C2semidirect
4. SD16×#3624C2semidirect
5. #14544331C2semidirect
6. #14544086C2semidirect
8. #14543987C2semidirect
11. Q8×#3624C22semidirect
20. #11763672C22semidirect
21. #11763670C22semidirect
22. Q8⋊#3624C22semidirect
24. #11763650C22semidirect
25. C8×#3624C22semidirect
41. C2×SD16×C17C8semidirect
88. C2×Q8×C17C2×C8semidirect
108. SD16×C17C2×C8semidirect
109. SD16×C17C2×C8semidirect
110. C2×C8×C17C2×C8semidirect
230. C2×SD16#3624semidirect
275. C17#17855semidirect
279. C2×Q8C2×#3624semidirect
294. SD16C2×#3624semidirect
295. SD16C2×#3624semidirect
304. C2×C8C2×#3624semidirect

Maximal subgroups

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

subgroupindex
#145456772
#145456312
SD16⋊#36242
SD16×#36242
#145443312
#145440862
#145440832
#145439872
C2×SD16×Dic172
#1785517