Finite Group Databasebeta

← #30752017

Subgroups — #30752017

Order 17152.

590 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863596C211yesmaximal, characteristic
2C2×Q8×C67⋊C8C211yesmaximal, characteristic, complemented
3#22561014C211yesmaximal, characteristic
4C2×#16863139C211yesmaximal, characteristic
5#22558092C211yesmaximal, characteristic, complemented
6#22558043C211yesmaximal, characteristic, complemented
7C67×#1803C211yesmaximal, characteristic, complemented
8C22×#11761615—21no—
9C2×#14524040C2211yescharacteristic
10#16863596—22no—
11#16863596C414yescomplemented
12Q8×C67⋊C8—22no—
13Q8×C67⋊C8—22no—
14C2×C4⋊(C67⋊C8)—21no—
15C2×C4⋊(C67⋊C8)C2211yescharacteristic
16C2×C4⋊(C67⋊C8)C2211yescharacteristic
17C2×#14523963—21no—
18C2×#14523963—21no—
19C2×#14523963—21no—
20C2×C4×C67⋊C8—21no—
21C2×C4×C67⋊C8C2211yescharacteristic
22#16863139—22no—
23C2×C4×Q8×C67C2211yescharacteristic
24C2×C4⋊C8×C67C2211yescharacteristic
25C2×#103×C67—21no—
26#384×C67C2211yescharacteristic
27#340×C67—21no—
28C22×D268—21no—
29C22×D268—21no—
30C22×D268—21no—
31C2×#11761615—24no—
32C2×#11761615—42no—
33C2×#11761615—42no—
34C22×C67⋊C8—21no—
35C22×C67⋊C8—21no—
36C22×C67⋊C8D411yescharacteristic
37C22×C67⋊C8D411yescharacteristic
38#14524040—21no—
39#14524040—21no—
40#14524040—21no—
41#14524040C2×C412yes—
42C4⋊(C67⋊C8)—24no—
43C4⋊(C67⋊C8)—21no—
44C4⋊(C67⋊C8)—21no—
45C4⋊(C67⋊C8)C2×C412yes—
46C4⋊(C67⋊C8)D412yes—
47#14523963—24no—
48#14523963—42no—
49#14523963—42no—
50C4×C67⋊C8—24no—
51C4×C67⋊C8—21no—
52C4×C67⋊C8—21no—
53C22×Q8×C67D411yescharacteristic
54C22×C8×C67—21no—
55C4×Q8×C67—21no—
56C4×Q8×C67—21no—
57C4×Q8×C67C2×C412yes—
58C4×Q8×C67D412yes—
59C2×C4⋊C4×C67—21no—
60C2×C4⋊C4×C67—21no—
61C2×C4⋊C4×C67—21no—
62C2×C4⋊C4×C67—21no—
63C2×C4⋊C4×C67D411yescharacteristic
64C2×C42×C67—21no—
65C2×C42×C67C2311yescharacteristic
66C4⋊C8×C67—21no—
67C4⋊C8×C67—21no—
68#103×C67—42no—
69C23×D67—41no—
70C2×D268—22no—
71C2×D268—22no—
72C2×D268—22no—
73C2×D268—41no—
74C2×D268—41no—
75C2×D268—41no—
76C2×D268—41no—
77C2×D268—41no—
78C2×D268—41no—
79C2×D268—41no—
80C2×D268—41no—
81C2×D268—41no—
82C2×D268—41no—
83C2×D268—41no—
84C2×D268—81no—
85C2×D268—81no—
86#11761615—48no—
87C2×C67⋊C8—21no—
88C2×C67⋊C8—21no—
89C2×C67⋊C8—22no—
90C2×C67⋊C8—21no—
91C2×C67⋊C8—22no—
92C2×C67⋊C8—21no—
93C2×C67⋊C8—21no—
94C2×C67⋊C8—41no—
95C2×C67⋊C8—41no—
96C2×C67⋊C8—41no—
97C2×C67⋊C8—41no—
98C2×C67⋊C8SD1612yescomplemented
99C2×Q8×C67—21no—
100C2×Q8×C67—21no—
101C2×Q8×C67—21no—
102C2×Q8×C67C22⋊C412yes—
103C2×Q8×C67D812yes—
104C2×Q8×C67SD1612yes—
105C22×C4×C67—21no—
106C22×C4×C67—21no—
107C22×C4×C67—41no—
108C22×C4×C67C2×D411yescharacteristic
109C22×C4×C67C2×D411yescharacteristic
110C22×C4×C67#4111yescharacteristic
111C2×C8×C67—41no—
112C2×C8×C67—41no—
113C2×C8×C67—41no—
114C4⋊C4×C67—22no—
115C4⋊C4×C67—21no—
116C4⋊C4×C67—22no—
117C4⋊C4×C67—41no—
118C4⋊C4×C67—41no—
119C4⋊C4×C67—41no—
120C4⋊C4×C67—41no—
121C4⋊C4×C67—81no—
122C4⋊C4×C67C22⋊C412yes—
123C42×C67—22no—
124C42×C67—22no—
125C42×C67C22×C411yescharacteristic
126C42×C67C2×D411yescharacteristic
127C42×C67C2×D411yescharacteristic
128C42×C67#4111yescharacteristic
129C23×C67#12111yescharacteristic
130C22×D67—42no—
131C22×D67—81no—
132C22×D67—81no—
133C22×D67—81no—
134C22×D67—81no—
135C22×D67—81no—
136C22×D67—81no—
137Q8×C67—22no—
138Q8×C67—22no—
139Q8×C67—22no—
140Q8×C67#10214yescomplemented
141C2×C4×C67—22no—
142C2×C4×C67—22no—
143C2×C4×C67—22no—
144C2×C4×C67—21no—
145C2×C4×C67—21no—
146C2×C4×C67—22no—
147C2×C4×C67—41no—
148C2×C4×C67—41no—
149C2×C4×C67—81no—
150C2×C4×C67—81no—
151C2×C4×C67—81no—
152C2×C4×C67C2×C22⋊C411yescharacteristic
153C2×C4×C67C4×D411yescharacteristic
154C2×C4×C67C4×D411yescharacteristic
155C2×C4×C67#12011yescharacteristic
156C2×C4×C67#12111yescharacteristic
157C2×C4×C67#12311yescharacteristic
158C2×C4×C67C2×D811yescharacteristic
159C2×C4×C67C2×SD1611yescharacteristic
160C2×C4×C67C2×SD1611yescharacteristic
161C2×C4×C67#13511yescharacteristic
162C2×C4×C67#13511yescharacteristic
163C2×C4×C67#13511yescharacteristic
164C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C67#13711yescharacteristic
166D268—42no—
167D268—42no—
168D268—42no—
169D268—42no—
170D268—82no—
171D268—82no—
172D268—81no—
173D268—81no—
174D268—81no—
175D268—81no—
176C8×C67—81no—
177C67⋊C8—22no—
178C67⋊C8—42no—
179C67⋊C8—41no—
180C67⋊C8—42no—
181C22×C67#38611yescharacteristic
182C22×C67#46011yescharacteristic
183C22×C67#46411yescharacteristic
184C22×C67#46511yescharacteristic
185C22×C67#46611yescharacteristic
186C22×C67#47111yescharacteristic
187C22×C67#47211yescharacteristic
188C2×D67—82no—
189C2×D67—82no—
190C2×D67—82no—
191C2×D67—161no—
192C2×D67—161no—
193C2×D67—161no—
194C2×D67—161no—
195C4×C67—24no—
196C4×C67—21no—
197C4×C67—21no—
198C4×C67—42no—
199C4×C67—161no—
200C4×C67C2×#10211yescharacteristic
201C4×C67#41611yescharacteristic
202C4×C67C4×SD1611yescharacteristic
203C4×C67#44011yescharacteristic
204C4×C67#44811yescharacteristic
205C4×C67#44911yescharacteristic
206C4×C67#46511yescharacteristic
207C4×C67#46611yescharacteristic
208#9222—671nomaximal, Sylow
209C2×C67#166611yescharacteristic
210C2×C67#166811yescharacteristic
211C2×C67#167811yescharacteristic
212C2×C67#168211yescharacteristic
213C2×C67#188011yescharacteristic
214C2×C67#189311yescharacteristic
215C2×C67#190711yescharacteristic
216D67—162no—
217C2×#460—671no—
218C2×C8×Q8—671no—
219#1957—671no—
220#1803D6711yescharacteristic, complemented
221C2×#331—671no—
222#1296—671no—
223#1277—671no—
224C67#922211yescharacteristic, complemented, Sylow
225C22×SD16—1341no—
226C2×#127—671no—
227C2×C4×Q8C2×D6711yescharacteristic
228#460—674no—
229#460—1342no—
230C8×Q8—1342no—
231C8×Q8—1342no—
232C2×C4⋊C8—671no—
233C2×C4⋊C8—671no—
234C2×C4⋊C8—1341no—
235C2×C4⋊C8C2×D6711yescharacteristic
236C2×#103—21no—
237C2×#102—1341no—
238C2×#102—1341no—
239C2×#102—1341no—
240C2×C4×C8—671no—
241C2×C4×C8—1341no—
242#384C2×D6711yescharacteristic
243#340—21no—
244#331—1342no—
245C22×Q8C22⋊D6711yescharacteristic
246C22×D4—1341no—
247C22×D4—1341no—
248C22×D4—1341no—
249C2×SD16—2682no—
250C2×SD16—2682no—
251C2×SD16—1344no—
252C22×C8—21no—
253C22×C8—671no—
254C22×C8—671no—
255C22×C8—1341no—
256C22×C8—1341no—
257#127—672no—
258#127—1341no—
259#127—1341no—
260#127—1341no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8C4×D6712yes—
264C4×Q8D26812yes—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4C22⋊D6711yescharacteristic
270C2×C42—21no—
271C2×C42C22×D6711yescharacteristic
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—672no—
275C4⋊C8—672no—
276C4⋊C8—1344no—
277C4⋊C8—1341no—
278C4⋊C8—1341no—
279#103—42no—
280#102—2682no—
281#102—2682no—
282#102—1344no—
283C4×C8—1344no—
284C4×C8—1341no—
285C4×C8—1341no—
286C24—2681no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8(C2×D67)⋊C412yes—
291C2×Q8#1176161312yes—
292C2×Q8#1176161512yes—
293C2×D4—2681no—
294C2×D4—2681no—
295C2×D4—2681no—
296C2×D4—2681no—
297C2×D4—2681no—
298C2×D4—2681no—
299C2×D4—2681no—
300C2×D4—2681no—
301C2×D4—2681no—
302C2×D4—2681no—
303C2×D4—2681no—
304C2×D4—5361no—
305C2×D4—5361no—
306C2×D4—1342no—
307C2×D4—1342no—
308C2×D4—1342no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4D4×D6711yescharacteristic
313C22×C4C2×C22⋊D6711yescharacteristic
314C22×C4#1176162711yescharacteristic
315SD16—2688no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—2681no—
320C2×C8—2681no—
321C2×C8—2681no—
322C2×C8—2681no—
323C2×C8—672no—
324C2×C8—1342no—
325C2×C8—1342no—
326C2×C8—1341no—
327C2×C8—1341no—
328C2×C8—1341no—
329C2×C8—1341no—
330C2×C8—1341no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—81no—
339C4⋊C4(C2×D67)⋊C412yes—
340C42—22no—
341C42—22no—
342C42C2×C4×D6711yescharacteristic
343C42D4×D6711yescharacteristic
344C42C2×D26811yescharacteristic
345C42#1176162711yescharacteristic
346C23—2682no—
347C23—5361no—
348C23—5361no—
349C23—5361no—
350C23—5361no—
351C23—5361no—
352C23—5361no—
353C23#1452401511yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—22no—
357Q8#1452396314yescomplemented
358D4—2682no—
359D4—2682no—
360D4—2682no—
361D4—2682no—
362D4—5362no—
363D4—5362no—
364D4—5361no—
365D4—5361no—
366D4—5361no—
367D4—5361no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4C2×(C2×D67)⋊C411yescharacteristic
380C2×C4(C22⋊D67)⋊C411yescharacteristic
381C2×C4(C22⋊D67)⋊C411yescharacteristic
382C2×C4#1452401311yescharacteristic
383C2×C4#1452401511yescharacteristic
384C2×C4#1452402711yescharacteristic
385C2×C4C2×#1176161311yescharacteristic
386C2×C4SD16×D6711yescharacteristic
387C2×C4C2×#1176161511yescharacteristic
388C2×C4#1452406311yescharacteristic
389C2×C4#1452406311yescharacteristic
390C2×C4#1452406611yescharacteristic
391C2×C4#1452407011yescharacteristic
392C2×C4#1452407511yescharacteristic
393C8—81no—
394C8—2682no—
395C8—2681no—
396C8—2682no—
397C8—1342no—
398C22—5362no—
399C22—5362no—
400C22—5362no—
401C22—10721no—
402C22—10721no—
403C22—10721no—
404C22—10721no—
405C22#1686327311yescharacteristic
406C22#1686359411yescharacteristic
407C22#1686362011yescharacteristic
408C22#1686362711yescharacteristic
409C22#1686363011yescharacteristic
410C22#1686366111yescharacteristic
411C22#1686366411yescharacteristic
412C4—24no—
413C4—21no—
414C4—21no—
415C4—42no—
416C4—161no—
417C4C2×#1452396311yescharacteristic
418C4#1686339011yescharacteristic
419C4#1686348111yescharacteristic
420C4#1686349911yescharacteristic
421C4#1686353611yescharacteristic
422C4#1686354411yescharacteristic
423C4#1686362711yescharacteristic
424C4#1686363011yescharacteristic
425C2—10722no—
426C2#2255955911yescharacteristic
427C2#2255956711yescharacteristic
428C2#2255962711yescharacteristic
429C2#2255965511yescharacteristic
430C2#2256056211yescharacteristic
431C2#2256063511yescharacteristic
432C2#2256071511yescharacteristic
subgroupG/H
centerC23#14524015
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1803C2
socleC23×C67#121
prime radicalC2×#14524040C22
cosocleC2×C42×C67C23
hypercenter#1803D67
hypocenterC67#9222
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C67⋊C8C2semidirect
5. #22558092C2semidirect
6. #22558043C2semidirect
7. C67×#1803C2semidirect
11. #16863596C4semidirect
98. C2×C67⋊C8SD16semidirect
140. Q8×C67#102semidirect
220. #1803D67semidirect
224. C67#9222semidirect
357. Q8#14523963semidirect

Maximal subgroups

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

subgroupindex
C2×#168635962
C2×Q8×C67⋊C82
#225610142
C2×#168631392
#225580922
#225580432
C67×#18032
#922267