Finite Group Databasebeta

← #30855846

Subgroups — #30855846

Order 17152.

1326 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic134C211yesmaximal, characteristic
2C2×#16863449C211yesmaximal, characteristic
3#22560293C212yesmaximal
4#22560221C214yesmaximal, complemented
5#22560143C212yesmaximal
6C4×#14523966C214yesmaximal, complemented
7C67×#2283C211yesmaximal, characteristic
8C23×Dic134C2211yescharacteristic
9C2×C4×Dic134—24no—
10C2×C4×Dic134—21no—
11C2×C4×Dic134—21no—
12C2×C4×Dic134C418yescomplemented
13C2×C4×Dic134C2212yes—
14C2×C4×Dic134C2212yes—
15C22×C4⋊Dic67C2211yescharacteristic
16C22×Dic67⋊C4—21no—
17C22×C4×Dic67—21no—
18#16863449—24no—
19C2×C4⋊(C67⋊C8)C2212yes—
20C2×#14523966C2218yes—
21C2×C22⋊(C67⋊C8)C2212yes—
22C2×C4×C67⋊C8C2212yes—
23#16863158C2218yes—
24C22×C42×C67C2211yescharacteristic
25#383×C67—21no—
26#382×C67C2212yes—
27#381×C67—21no—
28#380×C67C2212yes—
29C4×C4⋊C4×C67C2212yes—
30C4×C22⋊C4×C67—21no—
31#376×C67—21no—
32C22×Dic134—28no—
33C22×Dic134—21no—
34C22×Dic134—22no—
35C22×Dic134C2×C414yes—
36C22×Dic134C2312yes—
37C23×Dic67—21no—
38C23×Dic67—21no—
39C22×C67⋊C8C2314yes—
40C4×Dic134—28no—
41C4×Dic134—28no—
42C4×Dic134—24no—
43C4×Dic134—24no—
44C4×Dic134C2×C418yes—
45C4×Dic134C2×C418yes—
46C2×C4⋊Dic67—21no—
47C2×C4⋊Dic67—22no—
48C2×C4⋊Dic67C2×C414yes—
49C2×C4⋊Dic67C2312yes—
50C2×Dic67⋊C4—28no—
51C2×Dic67⋊C4—22no—
52C2×Dic67⋊C4—22no—
53C2×C4×Dic67—28no—
54C2×C4×Dic67—22no—
55C2×C4×Dic67—22no—
56C4⋊(C67⋊C8)—24no—
57#14523966—216no—
58C22⋊(C67⋊C8)—24no—
59C4×C67⋊C8—24no—
60C23×C4×C67D411yescharacteristic
61C23×C4×C67D411yescharacteristic
62C23×C4×C67C2311yescharacteristic
63C2×C4⋊C4×C67C2314yes—
64C2×C22⋊C4×C67—22no—
65C2×C42×C67—22no—
66C2×C42×C67C2×C414yes—
67C2×C42×C67D414yes—
68C2×C42×C67D411yescharacteristic
69C2×C42×C67D411yescharacteristic
70C2×C42×C67C2311yescharacteristic
71C2×C42×C67C2311yescharacteristic
72#95×C67—22no—
73#95×C67—22no—
74#95×C67—22no—
75#95×C67—22no—
76C2×Dic134—28no—
77C2×Dic134—216no—
78C2×Dic134—24no—
79C2×Dic134—22no—
80C2×Dic134—24no—
81C2×Dic134—216no—
82C2×Dic134C22×C414yes—
83C2×Dic134#4118yes—
84C22×Dic67—28no—
85C22×Dic67—24no—
86C22×Dic67—22no—
87C22×Dic67—24no—
88C22×Dic67—22no—
89C22×Dic67—44no—
90C2×C67⋊C8—24no—
91C2×C67⋊C8—28no—
92C4⋊Dic67—24no—
93C4⋊Dic67—22no—
94C4⋊Dic67C22×C414yes—
95Dic67⋊C4—28no—
96Dic67⋊C4—28no—
97C4×Dic67—28no—
98C4×Dic67—28no—
99C24×C67C2×D411yescharacteristic
100C22×C4×C67—24no—
101C22×C4×C67—24no—
102C22×C4×C67—24no—
103C22×C4×C67—24no—
104C22×C4×C67C22⋊C414yes—
105C22×C4×C67C22⋊C414yes—
106C22×C4×C67C22×C412yes—
107C22×C4×C67C2×D411yescharacteristic
108C22×C4×C67C2×D411yescharacteristic
109C22×C4×C67C2×D411yescharacteristic
110C22×C4×C67C2×D411yescharacteristic
111C22×C4×C67C2×D411yescharacteristic
112C22×C4×C67C2×D412yes—
113C22×C4×C67C2411yescharacteristic
114C4⋊C4×C67—28no—
115C22⋊C4×C67—44no—
116C42×C67—44no—
117C42×C67C22⋊C414yes—
118C42×C67C22⋊C414yes—
119C42×C67C22×C412yes—
120C42×C67C22×C412yes—
121C42×C67C2×D412yes—
122C42×C67C2×D412yes—
123C23×C67—22no—
124C23×C67—22no—
125C23×C67C2×C22⋊C412yes—
126C23×C67#13514yes—
127C23×C67C22×D411yescharacteristic
128C2×C4×C67—22no—
129C2×C4×C67—22no—
130C2×C4×C67—22no—
131C2×C4×C67—22no—
132C2×C4×C67—28no—
133C2×C4×C67—24no—
134C2×C4×C67—24no—
135C2×C4×C67—28no—
136C2×C4×C67—44no—
137C2×C4×C67—44no—
138C2×C4×C67—44no—
139C2×C4×C67C2×C22⋊C412yes—
140C2×C4×C67C2×C22⋊C412yes—
141C2×C4×C67C2×C22⋊C412yes—
142C2×C4×C67C2×C22⋊C412yes—
143C2×C4×C67C2×C22⋊C412yes—
144C2×C4×C67#13714yes—
145C2×C4×C67C23×C411yescharacteristic
146C2×C4×C67C22×D411yescharacteristic
147C2×C4×C67C2×#4114yes—
148C2×Dic67—216no—
149C2×Dic67—24no—
150C2×Dic67—28no—
151C2×Dic67—24no—
152C2×Dic67—44no—
153C2×Dic67—48no—
154Dic134—216no—
155Dic134—24no—
156Dic134—28no—
157Dic134#11718yes—
158C67⋊C8—44no—
159C22×C67—28no—
160C22×C67—22no—
161C22×C67—22no—
162C22×C67#41614yes—
163C22×C67C22×C22⋊C411yescharacteristic
164C22×C67#52512yes—
165C22×C67C2×#13512yes—
166C22×C67C2×#13712yes—
167C4×C67—22no—
168C4×C67—22no—
169C4×C67—22no—
170C4×C67—22no—
171C4×C67—84no—
172C4×C67#41914yes—
173C4×C67C2×#11712yes—
174C4×C67#52512yes—
175Dic67—28no—
176Dic67—44no—
177#20735—671nomaximal, Sylow
178C2×C67—24no—
179C2×C67#228011yescharacteristic
180C2×C67C2×#41611yescharacteristic
181C2×C67C2×#41911yescharacteristic
182C2×C67#310814yes—
183C22×C4×Q8—671no—
184C2×#431—671no—
185#2283D6711yescharacteristic
186#1823—672no—
187#1805—674no—
188#1787—672no—
189C4×#103—674no—
190C67#2073511yescharacteristic, complemented, Sylow
191C23×Q8—671no—
192C2×C4×Q8—678no—
193C2×C4×Q8—672no—
194C2×C4×Q8—672no—
195C2×C4×Q8—1344no—
196C2×C4×Q8—1341no—
197C2×C4×Q8—1341no—
198C22×C4⋊C4—671no—
199C22×C4⋊C4—1341no—
200C22×C42—1341no—
201C22×C42C2×D6711yescharacteristic
202#431—1344no—
203C2×C4⋊C8—672no—
204C2×#103—678no—
205C2×C22⋊C8—672no—
206C2×C4×C8—672no—
207#383—21no—
208#382C2×D6712yes—
209#381—21no—
210#380C2×D6712yes—
211C4×C4⋊C4C2×D6712yes—
212C4×C22⋊C4—21no—
213#376—21no—
214#340—678no—
215C22×Q8—674no—
216C22×Q8—672no—
217C22×Q8—1348no—
218C22×Q8—1341no—
219C22×Q8—1342no—
220C23×C4—1341no—
221C23×C4—1341no—
222C23×C4C22⋊D6711yescharacteristic
223C23×C4C22⋊D6711yescharacteristic
224C23×C4C22×D6711yescharacteristic
225C22×C8—674no—
226C4×Q8—678no—
227C4×Q8—678no—
228C4×Q8—1348no—
229C4×Q8—1348no—
230C4×Q8—1344no—
231C4×Q8—1344no—
232C2×C4⋊C4—674no—
233C2×C4⋊C4—672no—
234C2×C4⋊C4—1348no—
235C2×C4⋊C4—1341no—
236C2×C4⋊C4—1342no—
237C2×C4⋊C4—1342no—
238C2×C4⋊C4—1342no—
239C2×C4⋊C4C22×D6714yes—
240C2×C22⋊C4—22no—
241C2×C42—22no—
242C2×C42—1348no—
243C2×C42—1342no—
244C2×C42—1342no—
245C2×C42C4×D6714yes—
246C2×C42D26814yes—
247C2×C42C22⋊D6711yescharacteristic
248C2×C42C22⋊D6711yescharacteristic
249C2×C42C22×D6711yescharacteristic
250C2×C42C22×D6711yescharacteristic
251C4⋊C8—1344no—
252#103—13416no—
253C22⋊C8—1344no—
254C4×C8—1344no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259C24C2×C22⋊D6711yescharacteristic
260C2×Q8—674no—
261C2×Q8—678no—
262C2×Q8—13416no—
263C2×Q8—1348no—
264C2×Q8—13416no—
265C2×Q8—1344no—
266C2×Q8—1342no—
267C2×Q8—1344no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—2684no—
273C22×C4—1344no—
274C22×C4—1342no—
275C22×C4—1348no—
276C22×C4—1344no—
277C22×C4—1342no—
278C22×C4(C2×D67)⋊C414yes—
279C22×C4(C2×D67)⋊C414yes—
280C22×C4C2×C4×D6712yes—
281C22×C4C2×C22⋊D6711yescharacteristic
282C22×C4C2×C22⋊D6711yescharacteristic
283C22×C4C2×C22⋊D6711yescharacteristic
284C22×C4C2×C22⋊D6711yescharacteristic
285C22×C4C2×C22⋊D6711yescharacteristic
286C22×C4C2×D26812yes—
287C22×C4C23×D6711yescharacteristic
288C2×C8—1344no—
289C2×C8—1348no—
290C4⋊C4—28no—
291C4⋊C4—674no—
292C4⋊C4—1344no—
293C4⋊C4—1348no—
294C4⋊C4—1348no—
295C4⋊C4—1342no—
296C22⋊C4—44no—
297C42—44no—
298C42—1348no—
299C42—1348no—
300C42(C2×D67)⋊C414yes—
301C42(C2×D67)⋊C414yes—
302C42C2×C4×D6712yes—
303C42C2×C4×D6712yes—
304C42C2×D26812yes—
305C42C2×D26812yes—
306C23—22no—
307C23—22no—
308C23C2×(C2×D67)⋊C412yes—
309C23#1452406314yes—
310C23C22×C22⋊D6711yescharacteristic
311Q8—678no—
312Q8—13416no—
313Q8—1344no—
314Q8—1348no—
315C2×C4—28no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—28no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—2688no—
327C2×C4—2684no—
328C2×C4—1344no—
329C2×C4—13416no—
330C2×C4—1344no—
331C2×C4—1348no—
332C2×C4C2×(C2×D67)⋊C412yes—
333C2×C4C2×(C2×D67)⋊C412yes—
334C2×C4C2×(C2×D67)⋊C412yes—
335C2×C4C2×(C2×D67)⋊C412yes—
336C2×C4C2×(C2×D67)⋊C412yes—
337C2×C4#1452407314yes—
338C2×C4C22×C4×D6711yescharacteristic
339C2×C4C22×D26811yescharacteristic
340C2×C4#41×D6714yes—
341C8—2684no—
342C22—22no—
343C22—22no—
344C22—28no—
345C22#1686339014yes—
346C22C22×(C2×D67)⋊C411yescharacteristic
347C22#1686390312yes—
348C22C2×#1452406312yes—
349C22C2×#1452407312yes—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—84no—
355C4—2684no—
356C4—1348no—
357C4#1686340414yes—
358C4#117×D6712yes—
359C4#1686390812yes—
360C2—24no—
361C2#2256258811yescharacteristic
362C2C2×#1686339011yescharacteristic
363C2C2×#1686340411yescharacteristic
364C2#2257028514yes—
subgroupG/H
centerC22×C4C2×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#2283C2
socleC23×C67C22×D4
prime radicalC24×C67C2×D4
cosocleC22×C4×C67C24
hypercenter#2283D67
hypocenterC67#20735
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ #2283

Complements

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

NH ≅ G/Ntype
4. #22560221C2semidirect
6. C4×#14523966C2semidirect
12. C2×C4×Dic134C4semidirect
190. C67#20735semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1342
C2×#168634492
#225602932
#225602212
#225601432
C4×#145239662
C67×#22832
#2073567