Finite Group Databasebeta

← #30750606

Subgroups — #30750606

Order 17152.

660 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863596C211yesmaximal, characteristic
2#22561136C211yesmaximal, characteristic
3#22561014C211yesmaximal, characteristic
4C2×Q8⋊(C67⋊C8)C211yesmaximal, characteristic, complemented
5#22558092C211yesmaximal, characteristic, complemented
6#22558050C211yesmaximal, characteristic, complemented
7Q8×C4⋊C4×C67C211yesmaximal, characteristic, complemented
8C22×#11761615—21no—
9C2×#14524040C2211yescharacteristic
10#16863596—22no—
11#16863596C414yescomplemented
12C2×C4⋊(C67⋊C8)C2211yescharacteristic
13C2×C4⋊(C67⋊C8)C2211yescharacteristic
14C2×#14523963—21no—
15C2×#14523963—21no—
16C2×#14523963—21no—
17C2×#14523963—22no—
18C2×C4×C67⋊C8C2211yescharacteristic
19Q8⋊(C67⋊C8)—22no—
20C2×C4×Q8×C67—21no—
21C2×C4×Q8×C67C2211yescharacteristic
22#389×C67—22no—
23#389×C67—21no—
24#384×C67—21no—
25#384×C67C2211yescharacteristic
26C4×C4⋊C4×C67—21no—
27C4×C4⋊C4×C67C2211yescharacteristic
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⋊C8—21no—
37#14524040—21no—
38#14524040—21no—
39#14524040—21no—
40#14524040C2×C412yes—
41C4⋊(C67⋊C8)—21no—
42C4⋊(C67⋊C8)—21no—
43C4⋊(C67⋊C8)—21no—
44C4⋊(C67⋊C8)C2×C412yes—
45#14523963—24no—
46#14523963—42no—
47#14523963—44no—
48#14523963—42no—
49C4×C67⋊C8—21no—
50C4×C67⋊C8—21no—
51C22×Q8×C67D411yescharacteristic
52C4×Q8×C67—21no—
53C4×Q8×C67—21no—
54C4×Q8×C67—42no—
55C4×Q8×C67—42no—
56C4×Q8×C67C2×C412yes—
57C4×Q8×C67D412yes—
58C2×C4⋊C4×C67—21no—
59C2×C4⋊C4×C67—22no—
60C2×C4⋊C4×C67—21no—
61C2×C4⋊C4×C67—21no—
62C2×C4⋊C4×C67—22no—
63C2×C4⋊C4×C67D411yescharacteristic
64C2×C4⋊C4×C67D411yescharacteristic
65C2×C4⋊C4×C67D411yescharacteristic
66C2×C4⋊C4×C67D412yes—
67C2×C42×C67—21no—
68C2×C42×C67—22no—
69C2×C42×C67—21no—
70C2×C42×C67C2311yescharacteristic
71#95×C67—22no—
72#95×C67—21no—
73C23×D67—41no—
74C2×D268—22no—
75C2×D268—22no—
76C2×D268—22no—
77C2×D268—42no—
78C2×D268—42no—
79C2×D268—41no—
80C2×D268—41no—
81C2×D268—41no—
82C2×D268—41no—
83C2×D268—41no—
84C2×D268—41no—
85C2×D268—41no—
86C2×D268—41no—
87C2×D268—41no—
88C2×D268—41no—
89C2×D268—41no—
90#11761615—48no—
91C2×C67⋊C8—22no—
92C2×C67⋊C8—22no—
93C2×C67⋊C8—41no—
94C2×C67⋊C8—41no—
95C2×C67⋊C8—41no—
96C2×C67⋊C8—41no—
97C2×C67⋊C8—41no—
98C2×C67⋊C8—41no—
99C2×C67⋊C8—41no—
100C2×Q8×C67—21no—
101C2×Q8×C67—21no—
102C2×Q8×C67—21no—
103C2×Q8×C67C22⋊C412yes—
104C2×Q8×C67D812yes—
105C2×Q8×C67SD1612yes—
106C22×C4×C67—21no—
107C22×C4×C67—21no—
108C22×C4×C67—21no—
109C22×C4×C67—21no—
110C22×C4×C67—22no—
111C22×C4×C67C2×D411yescharacteristic
112C22×C4×C67C2×D411yescharacteristic
113C22×C4×C67C2×D411yescharacteristic
114C4⋊C4×C67—22no—
115C4⋊C4×C67—22no—
116C4⋊C4×C67—21no—
117C4⋊C4×C67—21no—
118C4⋊C4×C67—21no—
119C4⋊C4×C67—21no—
120C4⋊C4×C67—42no—
121C4⋊C4×C67—44no—
122C4⋊C4×C67—41no—
123C4⋊C4×C67—41no—
124C4⋊C4×C67—41no—
125C4⋊C4×C67—41no—
126C4⋊C4×C67—82no—
127C4⋊C4×C67C22⋊C412yes—
128C4⋊C4×C67SD1612yescomplemented
129C42×C67—22no—
130C42×C67—22no—
131C42×C67—44no—
132C42×C67—41no—
133C42×C67—41no—
134C42×C67C22×C411yescharacteristic
135C42×C67C2×D411yescharacteristic
136C42×C67#4111yescharacteristic
137C42×C67#4111yescharacteristic
138C23×C67#12011yescharacteristic
139C22×D67—42no—
140C22×D67—81no—
141C22×D67—81no—
142C22×D67—81no—
143C22×D67—81no—
144C22×D67—81no—
145C22×D67—81no—
146Q8×C67—22no—
147Q8×C67—22no—
148Q8×C67—22no—
149Q8×C67#10214yescomplemented
150C2×C4×C67—22no—
151C2×C4×C67—22no—
152C2×C4×C67—22no—
153C2×C4×C67—21no—
154C2×C4×C67—21no—
155C2×C4×C67—22no—
156C2×C4×C67—22no—
157C2×C4×C67—41no—
158C2×C4×C67—41no—
159C2×C4×C67—41no—
160C2×C4×C67—41no—
161C2×C4×C67—42no—
162C2×C4×C67—41no—
163C2×C4×C67—41no—
164C2×C4×C67—42no—
165C2×C4×C67—42no—
166C2×C4×C67—41no—
167C2×C4×C67C2×C22⋊C411yescharacteristic
168C2×C4×C67C4×D411yescharacteristic
169C2×C4×C67C4×D411yescharacteristic
170C2×C4×C67#12111yescharacteristic
171C2×C4×C67#12111yescharacteristic
172C2×C4×C67#12311yescharacteristic
173C2×C4×C67C2×D811yescharacteristic
174C2×C4×C67C2×SD1611yescharacteristic
175C2×C4×C67C2×SD1611yescharacteristic
176C2×C4×C67#13511yescharacteristic
177C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
179C2×C4×C67#13711yescharacteristic
180C2×C4×C67#13711yescharacteristic
181D268—44no—
182D268—42no—
183D268—42no—
184D268—42no—
185D268—42no—
186D268—82no—
187D268—81no—
188D268—81no—
189D268—81no—
190D268—81no—
191C67⋊C8—42no—
192C67⋊C8—42no—
193C67⋊C8—81no—
194C22×C67#38611yescharacteristic
195C22×C67#44711yescharacteristic
196C22×C67#44811yescharacteristic
197C22×C67#44811yescharacteristic
198C22×C67#44911yescharacteristic
199C22×C67#45311yescharacteristic
200C22×C67#45411yescharacteristic
201C2×D67—82no—
202C2×D67—82no—
203C2×D67—82no—
204C2×D67—161no—
205C2×D67—161no—
206C2×D67—161no—
207C2×D67—161no—
208C4×C67—24no—
209C4×C67—21no—
210C4×C67—21no—
211C4×C67—42no—
212C4×C67—42no—
213C4×C67—81no—
214C4×C67—82no—
215C4×C67C2×#10211yescharacteristic
216C4×C67#41711yescharacteristic
217C4×C67C4×SD1611yescharacteristic
218C4×C67#44011yescharacteristic
219C4×C67#46011yescharacteristic
220C4×C67#46411yescharacteristic
221C4×C67#48011yescharacteristic
222C4×C67#48111yescharacteristic
223#8979—671nomaximal, Sylow
224C2×C67#162111yescharacteristic
225C2×C67#162311yescharacteristic
226C2×C67#162611yescharacteristic
227C2×C67#162711yescharacteristic
228C2×C67#187411yescharacteristic
229C2×C67#188011yescharacteristic
230C2×C67#188311yescharacteristic
231D67—162no—
232C2×#460—671no—
233Q8×C4⋊C4D6711yescharacteristic, complemented
234#1982—671no—
235#1957—671no—
236C2×Q8⋊C8—671no—
237#1296—671no—
238#1279—671no—
239C67#897911yescharacteristic, complemented, Sylow
240C22×SD16—1341no—
241C2×#127—671no—
242C2×C4×Q8—21no—
243C2×C4×Q8C2×D6711yescharacteristic
244#460—674no—
245#460—1342no—
246C2×C4⋊C8—671no—
247C2×C4⋊C8—671no—
248C2×#102—1342no—
249C2×#102—1341no—
250C2×#102—1341no—
251C2×#102—1341no—
252C2×C4×C8—671no—
253#389—21no—
254#389—22no—
255#384—21no—
256#384C2×D6711yescharacteristic
257C4×C4⋊C4—21no—
258C4×C4⋊C4C2×D6711yescharacteristic
259Q8⋊C8—1342no—
260C22×Q8C22⋊D6711yescharacteristic
261C22×D4—1341no—
262C22×D4—1341no—
263C22×D4—1341no—
264C2×SD16—2682no—
265C2×SD16—2682no—
266C2×SD16—1344no—
267C22×C8—1341no—
268C22×C8—1341no—
269C22×C8—1341no—
270#127—672no—
271#127—1341no—
272#127—1341no—
273#127—1341no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—42no—
277C4×Q8—42no—
278C4×Q8C4×D6712yes—
279C4×Q8D26812yes—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4C22⋊D6711yescharacteristic
286C2×C4⋊C4C22⋊D6711yescharacteristic
287C2×C4⋊C4C22⋊D6711yescharacteristic
288C2×C4⋊C4C22⋊D6712yes—
289C2×C42—21no—
290C2×C42—22no—
291C2×C42—21no—
292C2×C42C22×D6711yescharacteristic
293C4⋊C8—672no—
294C4⋊C8—1341no—
295C4⋊C8—1341no—
296C4⋊C8—1341no—
297#102—2682no—
298#102—2684no—
299#102—2682no—
300#102—1344no—
301C4×C8—1341no—
302C4×C8—1341no—
303#95—22no—
304#95—21no—
305C24—2681no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8(C2×D67)⋊C412yes—
310C2×Q8#1176161312yes—
311C2×Q8#1176161512yes—
312C2×D4—2682no—
313C2×D4—2682no—
314C2×D4—2681no—
315C2×D4—2681no—
316C2×D4—2681no—
317C2×D4—2681no—
318C2×D4—2681no—
319C2×D4—2681no—
320C2×D4—2681no—
321C2×D4—2681no—
322C2×D4—2681no—
323C2×D4—2681no—
324C2×D4—2681no—
325C2×D4—1342no—
326C2×D4—1342no—
327C2×D4—1342no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4C2×C22⋊D6711yescharacteristic
334C22×C4C2×C22⋊D6711yescharacteristic
335C22×C4C2×C22⋊D6711yescharacteristic
336SD16—2688no—
337C2×C8—2681no—
338C2×C8—2681no—
339C2×C8—2681no—
340C2×C8—2681no—
341C2×C8—2681no—
342C2×C8—2681no—
343C2×C8—2681no—
344C2×C8—1342no—
345C2×C8—1342no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—42no—
353C4⋊C4—44no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—82no—
359C4⋊C4(C2×D67)⋊C412yes—
360C4⋊C4#1176161512yescomplemented
361C42—22no—
362C42—22no—
363C42—44no—
364C42—41no—
365C42—41no—
366C42C2×C4×D6711yescharacteristic
367C42C2×D26811yescharacteristic
368C42#1176162811yescharacteristic
369C42#1176162811yescharacteristic
370C23—2682no—
371C23—5361no—
372C23—5361no—
373C23—5361no—
374C23—5361no—
375C23—5361no—
376C23—5361no—
377C23#1452401211yescharacteristic
378Q8—22no—
379Q8—22no—
380Q8—22no—
381Q8#1452396314yescomplemented
382D4—2684no—
383D4—2682no—
384D4—2682no—
385D4—2682no—
386D4—2682no—
387D4—5362no—
388D4—5361no—
389D4—5361no—
390D4—5361no—
391D4—5361no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4C2×(C2×D67)⋊C411yescharacteristic
410C2×C4C4×C22⋊D6711yescharacteristic
411C2×C4C4×C22⋊D6711yescharacteristic
412C2×C4#1452401711yescharacteristic
413C2×C4#1452401711yescharacteristic
414C2×C4#1452402611yescharacteristic
415C2×C4C2×#1176161311yescharacteristic
416C2×C4C2×#1176161511yescharacteristic
417C2×C4C2×#1176161511yescharacteristic
418C2×C4#1452406311yescharacteristic
419C2×C4#1452406811yescharacteristic
420C2×C4#1452406911yescharacteristic
421C2×C4#1452407411yescharacteristic
422C2×C4#1452407411yescharacteristic
423C8—2682no—
424C8—2682no—
425C8—5361no—
426C22—5362no—
427C22—5362no—
428C22—5362no—
429C22—10721no—
430C22—10721no—
431C22—10721no—
432C22—10721no—
433C22#1686327511yescharacteristic
434C22#1686353011yescharacteristic
435C22#1686353511yescharacteristic
436C22#1686353511yescharacteristic
437C22#1686354111yescharacteristic
438C22#1686356411yescharacteristic
439C22#1686356911yescharacteristic
440C4—24no—
441C4—21no—
442C4—21no—
443C4—42no—
444C4—42no—
445C4—81no—
446C4—82no—
447C4C2×#1452396311yescharacteristic
448C4#1686339511yescharacteristic
449C4C4×#1176161511yescharacteristic
450C4#1686349411yescharacteristic
451C4#1686359611yescharacteristic
452C4#1686361811yescharacteristic
453C4#1686370911yescharacteristic
454C4#1686371411yescharacteristic
455C2—10722no—
456C2#2255927411yescharacteristic
457C2#2255928711yescharacteristic
458C2#2255930511yescharacteristic
459C2#2255931211yescharacteristic
460C2#2256052611yescharacteristic
461C2#2256056411yescharacteristic
462C2#2256058511yescharacteristic
subgroupG/H
centerC23#14524012
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingQ8×C4⋊C4×C67C2
socleC23×C67#120
prime radicalC2×#14524040C22
cosocleC2×C42×C67C23
hypercenterQ8×C4⋊C4D67
hypocenterC67#8979
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×Q8⋊(C67⋊C8)C2semidirect
5. #22558092C2semidirect
6. #22558050C2semidirect
7. Q8×C4⋊C4×C67C2semidirect
11. #16863596C4semidirect
128. C4⋊C4×C67SD16semidirect
149. Q8×C67#102semidirect
233. Q8×C4⋊C4D67semidirect
239. C67#8979semidirect
360. C4⋊C4#11761615semidirect
381. Q8#14523963semidirect

Maximal subgroups

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

subgroupindex
C2×#168635962
#225611362
#225610142
C2×Q8⋊(C67⋊C8)2
#225580922
#225580502
Q8×C4⋊C4×C672
#897967