Finite Group Databasebeta

← #30825317

Subgroups — #30825317

Order 17152.

714 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22569319C212yesmaximal, complemented
2C4×#14524066C211yesmaximal, characteristic
3#22559034C211yesmaximal, characteristic, complemented
4#22559032C211yesmaximal, characteristic, complemented
5#22559028C211yesmaximal, characteristic, complemented
6C8×#11761615C212yesmaximal, complemented
7#22559006C212yesmaximal, complemented
8#334×D67C211yesmaximal, characteristic, complemented
9#22558595C212yesmaximal, complemented
10#22558494C211yesmaximal, characteristic, complemented
11C67×#1579C211yesmaximal, characteristic, complemented
12C2×#14524066C411yescharacteristic
13C4×#11761629C2212yes—
14Q8×C67⋊C8C2212yes—
15Q16×Dic67C411yescharacteristic
16C4×#11761615C2212yes—
17#16863481C412yes—
18C4×D536C2211yescharacteristic
19#16863478C411yescharacteristic
20C8×D268—22no—
21C8×D268C2212yes—
22#16863461—22no—
23#16863461C2212yes—
24#16863447—22no—
25#16863447C2211yescharacteristic
26#16863447C2211yescharacteristic
27#16863433C411yescharacteristic
28C4⋊C8×D67—22no—
29C4⋊C8×D67C2212yes—
30#16863391C412yes—
31C4×C8×D67C2211yescharacteristic
32#16863147C2211yescharacteristic
33#16863146C2212yes—
34Q8⋊(C67⋊C8)C2212yes—
35#16863125C2212yes—
36#16863123C2212yes—
37#16863117C2211yescharacteristic
38C8×C67⋊C8C2211yescharacteristic
39C8×C67⋊C8C2211yescharacteristic
40C8×Q8×C67C2212yes—
41C4×Q16×C67C2211yescharacteristic
42#334×C67C2211yescharacteristic
43Q8⋊C8×C67C2212yes—
44#322×C67C2211yescharacteristic
45C2×#11761629C2×C412yes—
46#14524066C818yescomplemented
47C2×#11761615C2×C412yes—
48C2×D536C2×C411yescharacteristic
49C2×C8×D67—22no—
50C2×C8×D67—22no—
51C2×C8×D67—22no—
52C2×C8×D67C2×C411yescharacteristic
53C2×C8×D67C2×C411yescharacteristic
54Q8×Dic67C2×C412yes—
55C4×D268—22no—
56C4×D268C2312yes—
57#14524005—22no—
58#14524005C2×C412yes—
59#14524001—22no—
60#14524001C2×C412yes—
61C42×D67—22no—
62C42×D67C2311yescharacteristic
63#14523977C2×C411yescharacteristic
64#14523974C2×C412yes—
65C4⋊(C67⋊C8)—22no—
66C4⋊(C67⋊C8)C2312yes—
67Dic67⋊C8—22no—
68Dic67⋊C8—21no—
69Dic67⋊C8D412yes—
70#14523967C2×C412yes—
71#14523965C2×C412yes—
72#14523963C2×C412yes—
73(C2×D67)⋊C8—21no—
74(C2×D67)⋊C8—22no—
75(C2×D67)⋊C8—22no—
76(C2×D67)⋊C8D412yes—
77C4×C67⋊C8—22no—
78C4×C67⋊C8C2311yescharacteristic
79C4×C67⋊C8C2311yescharacteristic
80C4×C67⋊C8C2311yescharacteristic
81C8×Dic67C2×C411yescharacteristic
82C8×Dic67C2×C411yescharacteristic
83C2×Q16×C67C2×C411yescharacteristic
84C4×Q8×C67C2312yes—
85#107×C67C2×C411yescharacteristic
86C4⋊C8×C67—22no—
87C4⋊C8×C67C2312yes—
88#103×C67C2×C412yes—
89C4×C8×C67—22no—
90C4×C8×C67C2311yescharacteristic
91C4×C8×C67C2311yescharacteristic
92C4×C8×C67C2311yescharacteristic
93#11761629—24no—
94#11761629C2×C818yescomplemented
95C2×D268—22no—
96C2×D268C22×C412yes—
97C2×C4×D67—22no—
98C2×C4×D67—22no—
99C2×C4×D67—22no—
100C2×C4×D67C22×C411yescharacteristic
101C2×C4×D67C2×D411yescharacteristic
102C2×C4×D67#4111yescharacteristic
103#11761615C2×C818yescomplemented
104D536C2×C814yescomplemented
105C2×C67⋊C8—22no—
106C2×C67⋊C8—21no—
107C2×C67⋊C8C22×C411yescharacteristic
108C2×C67⋊C8C22×C411yescharacteristic
109C2×C67⋊C8C2×D412yes—
110C8×D67—22no—
111C8×D67—82no—
112C8×D67—82no—
113C8×D67—82no—
114C8×D67C2×C814yescomplemented
115C4⋊Dic67—22no—
116C4⋊Dic67C22×C412yes—
117(C2×D67)⋊C4—22no—
118(C2×D67)⋊C4—22no—
119(C2×D67)⋊C4—22no—
120C4×Dic67—22no—
121C4×Dic67—22no—
122C4×Dic67C22×C411yescharacteristic
123C4×Dic67C2×D411yescharacteristic
124C4×Dic67#4111yescharacteristic
125C2×Q8×C67C22×C412yes—
126Q16×C67C2×C814yescomplemented
127C2×C8×C67—22no—
128C2×C8×C67—21no—
129C2×C8×C67C22×C411yescharacteristic
130C2×C8×C67C22×C411yescharacteristic
131C2×C8×C67C2×D412yes—
132C4⋊C4×C67—22no—
133C4⋊C4×C67C22×C412yes—
134C42×C67—22no—
135C42×C67C2411yescharacteristic
136C22×D67—22no—
137C22×D67C4×D411yescharacteristic
138Q8×C67—22no—
139Q8×C67C22×C814yes—
140C2×C4×C67—22no—
141C2×C4×C67—22no—
142C2×C4×C67C23×C411yescharacteristic
143C2×C4×C67C22×D411yescharacteristic
144C2×C4×C67C2×#4111yescharacteristic
145C2×Dic67—22no—
146C2×Dic67C4×D411yescharacteristic
147C2×Dic67C4×D411yescharacteristic
148C2×Dic67C4×D411yescharacteristic
149D268—28no—
150D268—22no—
151D268C22×C814yes—
152C4×D67—22no—
153C4×D67—28no—
154C4×D67—41no—
155C4×D67—42no—
156C4×D67—42no—
157C4×D67—82no—
158C4×D67C22×C812yes—
159C4×D67#13112yes—
160C8×C67—21no—
161C8×C67—22no—
162C8×C67—41no—
163C8×C67—42no—
164C8×C67C22×C812yes—
165C67⋊C8—21no—
166C67⋊C8—22no—
167C67⋊C8—41no—
168C67⋊C8—42no—
169C67⋊C8C22×C812yes—
170C22×C67C2×C4×D411yescharacteristic
171C2×D67—21no—
172C2×D67—21no—
173C2×D67—24no—
174C2×D67—42no—
175C2×D67—42no—
176C2×D67C8×D412yes—
177C4×C67—24no—
178C4×C67—21no—
179C4×C67—42no—
180C4×C67C23×C811yescharacteristic
181C4×C67C2×#13111yescharacteristic
182C4×C67#57712yes—
183Dic67—21no—
184Dic67—21no—
185Dic67—42no—
186Dic67C8×D412yes—
187#17858—671nomaximal, Sylow
188C2×C67C2×C8×D411yescharacteristic
189C2×C67#295211yescharacteristic
190C2×C67#295711yescharacteristic
191D67—22no—
192D67—44no—
193#2988—672no—
194C4×#135—671no—
195#1580—671no—
196#1580—671no—
197#1579—671no—
198#1579D6711yescharacteristic, complemented
199C8×SD16—672no—
200C8×SD16—672no—
201C2×#334—671no—
202#1477—672no—
203#1450—671no—
204C67#1785811yescharacteristic, complemented, Sylow
205C2×#135—671no—
206C4×#41—672no—
207C8×Q8—672no—
208C8×Q8C2×D6712yes—
209C4×Q16—671no—
210C4×Q16C2×D6711yescharacteristic
211C4×SD16—672no—
212C4×SD16—672no—
213C4×D8—671no—
214C4×D8—671no—
215C8×D4—672no—
216C8×D4—672no—
217C8×D4—1342no—
218C8×D4—1342no—
219#431—671no—
220#431—671no—
221#431—1342no—
222#427—671no—
223C2×C4⋊C8—672no—
224C2×C4⋊C8—1342no—
225#416—672no—
226C2×C4×C8—671no—
227#334—672no—
228#334—671no—
229#334C2×D6711yescharacteristic
230Q8⋊C8—672no—
231Q8⋊C8C2×D6712yes—
232D4⋊C8—672no—
233D4⋊C8—672no—
234#322—671no—
235#322C2×D6711yescharacteristic
236C82—671no—
237C82—671no—
238C2×#41—672no—
239#135—678no—
240C2×Q16C4×D6711yescharacteristic
241C2×SD16—672no—
242C2×D8—671no—
243C22×C8—671no—
244C22×C8—671no—
245C22×C8—1342no—
246C22×C8—1342no—
247C22×C8—1342no—
248C4×Q8—672no—
249C4×Q8C22×D6712yes—
250C4×D4—672no—
251C4×D4—672no—
252C4×D4—1342no—
253C4×D4—1342no—
254#117—672no—
255#117—1342no—
256C2×C42—671no—
257C2×C42—1342no—
258#107—671no—
259#107C4×D6711yescharacteristic
260#106—672no—
261C4⋊C8—22no—
262C4⋊C8—672no—
263C4⋊C8—672no—
264C4⋊C8—1342no—
265C4⋊C8—1342no—
266C4⋊C8—1341no—
267C4⋊C8C22×D6712yes—
268#103—672no—
269#103C4×D6712yes—
270#102—672no—
271#102—672no—
272C22⋊C8—672no—
273C22⋊C8—1342no—
274C22⋊C8—1341no—
275C22⋊C8—1342no—
276C4×C8—22no—
277C4×C8—671no—
278C4×C8—671no—
279C4×C8—671no—
280C4×C8—671no—
281C4×C8—671no—
282C4×C8—1342no—
283C4×C8C22×D6711yescharacteristic
284C4×C8C22×D6711yescharacteristic
285C4×C8C22×D6711yescharacteristic
286#41—678no—
287#41—1344no—
288C2×Q8C2×C4×D6712yes—
289C2×D4—672no—
290C2×D4—1342no—
291C22×C4—671no—
292C22×C4—671no—
293C22×C4—671no—
294C22×C4—1342no—
295C22×C4—1342no—
296C22×C4—1342no—
297Q16C8×D6714yescomplemented
298SD16—678no—
299D8—674no—
300C2×C8—22no—
301C2×C8—21no—
302C2×C8—5362no—
303C2×C8—5362no—
304C2×C8—5362no—
305C2×C8—671no—
306C2×C8—674no—
307C2×C8—671no—
308C2×C8—672no—
309C2×C8—1342no—
310C2×C8—1342no—
311C2×C8—1341no—
312C2×C8C2×C4×D6711yescharacteristic
313C2×C8C2×C4×D6711yescharacteristic
314C2×C8D4×D6712yes—
315C4⋊C4—22no—
316C4⋊C4—672no—
317C4⋊C4—1342no—
318C4⋊C4C2×C4×D6712yes—
319C22⋊C4—1342no—
320C22⋊C4—1342no—
321C22⋊C4—1342no—
322C42—22no—
323C42—671no—
324C42—671no—
325C42—671no—
326C42—1342no—
327C42—1342no—
328C42C23×D6711yescharacteristic
329C23—671no—
330C23—1342no—
331Q8—22no—
332Q8C2×C8×D6714yes—
333D4—674no—
334D4—1348no—
335D4—1342no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—2681no—
339C2×C4—2682no—
340C2×C4—2682no—
341C2×C4—5362no—
342C2×C4—671no—
343C2×C4—672no—
344C2×C4—672no—
345C2×C4—671no—
346C2×C4—671no—
347C2×C4—1348no—
348C2×C4—1342no—
349C2×C4—1342no—
350C2×C4C22×C4×D6711yescharacteristic
351C2×C4C2×D4×D6711yescharacteristic
352C2×C4#41×D6711yescharacteristic
353C8—21no—
354C8—22no—
355C8—41no—
356C8—42no—
357C8—2681no—
358C8—2682no—
359C8—672no—
360C8—1341no—
361C8—1342no—
362C8C2×C8×D6712yes—
363C22—2682no—
364C22—2682no—
365C22—672no—
366C22—1344no—
367C22—1341no—
368C22—1341no—
369C22C4×D4×D6711yescharacteristic
370C4—21no—
371C4—24no—
372C4—42no—
373C4—2682no—
374C4—672no—
375C4—1341no—
376C4—1341no—
377C4C22×C8×D6711yescharacteristic
378C4#131×D6711yescharacteristic
379C4#1686420512yes—
380C2—2684no—
381C2—1342no—
382C2C8×D4×D6711yescharacteristic
383C2#2256908911yescharacteristic
384C2#2256912311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC4×C67C23×C8
FrattiniC42C23×D67
FittingC67×#1579C2
socleC22×C67C2×C4×D4
prime radicalC2×#14524066C4
cosocleC42×C67C24
hypercenter#1579D67
hypocenterC67#17858
LCSG ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1579

Complements

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

NH ≅ G/Ntype
1. #22569319C2semidirect
3. #22559034C2semidirect
4. #22559032C2semidirect
5. #22559028C2semidirect
6. C8×#11761615C2semidirect
7. #22559006C2semidirect
8. #334×D67C2semidirect
9. #22558595C2semidirect
10. #22558494C2semidirect
11. C67×#1579C2semidirect
46. #14524066C8semidirect
94. #11761629C2×C8semidirect
103. #11761615C2×C8semidirect
104. D536C2×C8semidirect
114. C8×D67C2×C8semidirect
126. Q16×C67C2×C8semidirect
198. #1579D67semidirect
204. C67#17858semidirect
297. Q16C8×D67semidirect

Maximal subgroups

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

subgroupindex
#225693192
C4×#145240662
#225590342
#225590322
#225590282
C8×#117616152
#225590062
#334×D672
#225585952
#225584942
C67×#15792
#1785867