Finite Group Databasebeta

← #30748993

Subgroups — #30748993

Order 17152.

454 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×C67⋊C8C211yesmaximal, characteristic
2#22560229C211yesmaximal, characteristic
3C4×#14523974C211yesmaximal, characteristic, complemented
4#22558964C211yesmaximal, characteristic, complemented
5#22558112C211yesmaximal, characteristic, complemented
6#22558110C211yesmaximal, characteristic, complemented
7C67×#1806C211yesmaximal, characteristic
8Q8×C67⋊C8C411yescharacteristic, complemented
9Q8×C67⋊C8C411yescharacteristic, complemented
10D4×C67⋊C8—21no—
11D4×C67⋊C8—21no—
12D4×C67⋊C8C411yescharacteristic, complemented
13D4×C67⋊C8C411yescharacteristic, complemented
14#16863449—21no—
15#16863449C2211yescharacteristic
16C2×#14523974—21no—
17#16863420C2211yescharacteristic
18C2×#14523970—21no—
19C2×C4×C67⋊C8—21no—
20C2×C4×C67⋊C8C2211yescharacteristic
21C4×C4⋊Dic67C2211yescharacteristic
22#16863158—21no—
23#16863155—21no—
24#16863146—21no—
25#16863146—21no—
26C4×#41×C67C2211yescharacteristic
27#423×C67C2211yescharacteristic
28C2×#104×C67—21no—
29C4×C4⋊C4×C67C2211yescharacteristic
30#339×C67—21no—
31C2×#11761612—21no—
32C22×C67⋊C8—21no—
33C22×C67⋊C8—21no—
34C22×C67⋊C8D411yescharacteristic
35C22×C67⋊C8D411yescharacteristic
36C2×C4⋊Dic67—21no—
37C2×C4×Dic67—21no—
38#14523974—24no—
39C4⋊(C67⋊C8)—21no—
40C4⋊(C67⋊C8)—21no—
41C4⋊(C67⋊C8)—21no—
42C4⋊(C67⋊C8)—21no—
43C4⋊(C67⋊C8)C2×C411yescharacteristic
44C4⋊(C67⋊C8)C2×C411yescharacteristic
45#14523970—42no—
46C22⋊(C67⋊C8)—21no—
47C22⋊(C67⋊C8)—21no—
48C22⋊(C67⋊C8)—21no—
49C4×C67⋊C8—21no—
50C4×C67⋊C8—21no—
51C4×C67⋊C8—21no—
52C4×C67⋊C8C2×C411yescharacteristic
53C4×C67⋊C8C2×C411yescharacteristic
54C4×C67⋊C8D411yescharacteristic
55C4×C67⋊C8D411yescharacteristic
56C67⋊#95—21no—
57C2×#41×C67D411yescharacteristic
58C2×M16×C67—21no—
59C4×Q8×C67C2×C411yescharacteristic
60C4×Q8×C67Q811yescharacteristic
61C4×D4×C67—21no—
62C4×D4×C67—21no—
63C4×D4×C67C2×C411yescharacteristic
64C4×D4×C67Q811yescharacteristic
65#117×C67—21no—
66#117×C67D411yescharacteristic
67C2×C4⋊C4×C67—21no—
68C2×C42×C67—21no—
69C2×C42×C67—21no—
70C2×C42×C67C2311yescharacteristic
71C4⋊C8×C67—21no—
72C4⋊C8×C67—21no—
73#104×C67—42no—
74#95×C67—21no—
75C22×Dic67—41no—
76#11761612—41no—
77#11761612—41no—
78C2×C67⋊C8—22no—
79C2×C67⋊C8—21no—
80C2×C67⋊C8—21no—
81C2×C67⋊C8—41no—
82C2×C67⋊C8—42no—
83C2×C67⋊C8—42no—
84C2×C67⋊C8—41no—
85C2×C67⋊C8C22⋊C411yescharacteristic
86C2×C67⋊C8C22⋊C411yescharacteristic
87C2×C67⋊C8C22⋊C411yescharacteristic
88C2×C67⋊C8C22⋊C411yescharacteristic
89C4⋊Dic67—22no—
90C4⋊Dic67—41no—
91C4×Dic67—41no—
92C4×Dic67—41no—
93#41×C67—21no—
94#41×C67—21no—
95#41×C67SD1612yes—
96#41×C67SD1612yes—
97C2×Q8×C67C4⋊C411yescharacteristic
98C2×D4×C67—21no—
99C2×D4×C67C4⋊C411yescharacteristic
100C22×C4×C67—21no—
101C22×C4×C67—21no—
102C22×C4×C67—21no—
103C22×C4×C67—41no—
104C22×C4×C67C2×D411yescharacteristic
105C22×C4×C67C2×D411yescharacteristic
106C22×C4×C67#4111yescharacteristic
107M16×C67—41no—
108M16×C67—41no—
109C2×C8×C67—41no—
110C4⋊C4×C67—21no—
111C4⋊C4×C67—21no—
112C4⋊C4×C67—22no—
113C4⋊C4×C67—41no—
114C4⋊C4×C67C4⋊C411yescharacteristic
115C4⋊C4×C67C4⋊C411yescharacteristic
116C22⋊C4×C67—21no—
117C22⋊C4×C67—21no—
118C22⋊C4×C67—21no—
119C42×C67—21no—
120C42×C67—21no—
121C42×C67—21no—
122C42×C67—41no—
123C42×C67—41no—
124C42×C67C22×C411yescharacteristic
125C42×C67C2×D411yescharacteristic
126C42×C67C2×Q811yescharacteristic
127C42×C67#4111yescharacteristic
128C23×C67—21no—
129C23×C67#12111yescharacteristic
130Q8×C67—21no—
131Q8×C67#10612yescomplemented
132D4×C67—22no—
133D4×C67—22no—
134D4×C67—21no—
135D4×C67#10612yescomplemented
136C2×C4×C67—21no—
137C2×C4×C67—21no—
138C2×C4×C67—21no—
139C2×C4×C67—22no—
140C2×C4×C67—22no—
141C2×C4×C67—21no—
142C2×C4×C67—21no—
143C2×C4×C67—41no—
144C2×C4×C67—42no—
145C2×C4×C67—41no—
146C2×C4×C67—41no—
147C2×C4×C67—42no—
148C2×C4×C67—81no—
149C2×C4×C67—81no—
150C2×C4×C67C2×C22⋊C411yescharacteristic
151C2×C4×C67C2×C4⋊C411yescharacteristic
152C2×C4×C67#11711yescharacteristic
153C2×C4×C67#12111yescharacteristic
154C2×C4×C67C22⋊Q811yescharacteristic
155C2×C4×C67C22⋊Q811yescharacteristic
156C2×C4×C67C2×SD1611yescharacteristic
157C2×C4×C67C2×SD1611yescharacteristic
158C2×C4×C67#13511yescharacteristic
159C2×C4×C67#13511yescharacteristic
160C2×Dic67—42no—
161C2×Dic67—81no—
162C2×Dic67—81no—
163C8×C67—81no—
164C67⋊C8—21no—
165C67⋊C8—21no—
166C67⋊C8—21no—
167C67⋊C8—41no—
168C67⋊C8—41no—
169C67⋊C8—81no—
170C67⋊C8#10412yescomplemented
171C22×C67—21no—
172C22×C67—21no—
173C22×C67—22no—
174C22×C67—41no—
175C22×C67—41no—
176C22×C67#38011yescharacteristic
177C22×C67#46511yescharacteristic
178C22×C67#46511yescharacteristic
179C4×C67—21no—
180C4×C67—22no—
181C4×C67—21no—
182C4×C67—21no—
183C4×C67—41no—
184C4×C67—41no—
185C4×C67—82no—
186C4×C67C2×#10411yescharacteristic
187C4×C67#42111yescharacteristic
188C4×C67C2×#10611yescharacteristic
189C4×C67#42711yescharacteristic
190C4×C67#46511yescharacteristic
191C4×C67#46511yescharacteristic
192Dic67—82no—
193#8666—671nomaximal, Sylow
194C2×C67—21no—
195C2×C67—21no—
196C2×C67—42no—
197C2×C67#180611yescharacteristic
198C2×C67#180711yescharacteristic
199C2×C67#181411yescharacteristic
200C8×#41—671no—
201#1806—671no—
202#1806D6711yescharacteristic
203C4×#106—671no—
204#1566—671no—
205#1303—671no—
206#1303—671no—
207C67#866611yescharacteristic, complemented, Sylow
208C4×#41C2×D6711yescharacteristic
209C8×Q8—671no—
210C8×Q8—671no—
211C8×D4—671no—
212C8×D4—671no—
213C8×D4—1341no—
214C8×D4—1341no—
215#431—671no—
216#431—1341no—
217C2×#106—1341no—
218#423—671no—
219#423C2×D6711yescharacteristic
220C2×#104—21no—
221C2×#104—1341no—
222C2×C4×C8—671no—
223C2×C4×C8—1341no—
224C4×C4⋊C4—671no—
225C4×C4⋊C4C2×D6711yescharacteristic
226#340—1341no—
227#339—21no—
228#339—1341no—
229#334—1341no—
230#334—1341no—
231C2×#41C22⋊D6711yescharacteristic
232C2×M16—21no—
233C2×M16—1341no—
234C22×C8—671no—
235C22×C8—671no—
236C22×C8—1341no—
237C22×C8—1341no—
238C4×Q8Dic13411yescharacteristic
239C4×Q8C4×D6711yescharacteristic
240C4×D4—21no—
241C4×D4—21no—
242C4×D4Dic13411yescharacteristic
243C4×D4C4×D6711yescharacteristic
244#117—21no—
245#117C22⋊D6711yescharacteristic
246C2×C4⋊C4—21no—
247C2×C4⋊C4—1341no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—1341no—
251C2×C42C22×D6711yescharacteristic
252#106—1344no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—671no—
256C4⋊C8—671no—
257C4⋊C8—1341no—
258C4⋊C8—1341no—
259C4⋊C8—1341no—
260C4⋊C8—1341no—
261#104—42no—
262#104—2682no—
263C22⋊C8—1341no—
264C22⋊C8—1341no—
265C22⋊C8—1341no—
266C4×C8—671no—
267C4×C8—671no—
268C4×C8—671no—
269C4×C8—671no—
270C4×C8—1341no—
271C4×C8—1341no—
272C4×C8—1341no—
273#95—21no—
274#95—1341no—
275#41—21no—
276#41—21no—
277#41#1176161512yes—
278#41#1176161612yes—
279C2×Q8Dic67⋊C411yescharacteristic
280C2×D4—21no—
281C2×D4Dic67⋊C411yescharacteristic
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—2681no—
287C22×C4D4×D6711yescharacteristic
288C22×C4C2×C22⋊D6711yescharacteristic
289C22×C4#1176162711yescharacteristic
290M16—41no—
291M16—41no—
292M16—2681no—
293M16—2681no—
294C2×C8—41no—
295C2×C8—2682no—
296C2×C8—2682no—
297C2×C8—2681no—
298C2×C8—2681no—
299C2×C8—671no—
300C2×C8—671no—
301C2×C8—671no—
302C2×C8—671no—
303C2×C8—1341no—
304C2×C8—1341no—
305C2×C8—1342no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—41no—
310C4⋊C4—2681no—
311C4⋊C4—1342no—
312C4⋊C4Dic67⋊C411yescharacteristic
313C4⋊C4Dic67⋊C411yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—41no—
321C42—41no—
322C42—2681no—
323C42—2681no—
324C42C2×C4×D6711yescharacteristic
325C42D4×D6711yescharacteristic
326C42C2×Dic13411yescharacteristic
327C42#1176162711yescharacteristic
328C23—21no—
329C23#1452401511yescharacteristic
330Q8—21no—
331Q8#1452397412yescomplemented
332D4—22no—
333D4—22no—
334D4—21no—
335D4#1452397412yescomplemented
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—2682no—
351C2×C4—5361no—
352C2×C4—5361no—
353C2×C4C22⋊C4×D6711yescharacteristic
354C2×C4C2×Dic67⋊C411yescharacteristic
355C2×C4#1452399911yescharacteristic
356C2×C4#1452401511yescharacteristic
357C2×C4#1452402011yescharacteristic
358C2×C4#1452402011yescharacteristic
359C2×C4C2×#1176161511yescharacteristic
360C2×C4C2×#1176161611yescharacteristic
361C2×C4#1452406311yescharacteristic
362C2×C4#1452406311yescharacteristic
363C8—81no—
364C8—2681no—
365C8—2681no—
366C8—5361no—
367C8—672no—
368C8—1341no—
369C8—1341no—
370C8—1341no—
371C22—21no—
372C22—21no—
373C22—22no—
374C22—41no—
375C22—41no—
376C22#1686325411yescharacteristic
377C22#1686362311yescharacteristic
378C22#1686362711yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—82no—
386C4—5362no—
387C4#104×D6711yescharacteristic
388C4#1686341111yescharacteristic
389C4C2×#1452397411yescharacteristic
390C4#1686343211yescharacteristic
391C4#1686362311yescharacteristic
392C4#1686362711yescharacteristic
393C2—21no—
394C2—21no—
395C2—42no—
396C2#2256022711yescharacteristic
397C2#2256023211yescharacteristic
398C2#2256026111yescharacteristic
subgroupG/H
centerC2×C4#14524015
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1806C2
socleC22×C67#380
prime radicalC2×#41×C67D4
cosocleC2×C42×C67C23
hypercenter#1806D67
hypocenterC67#8666
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
3. C4×#14523974C2semidirect
4. #22558964C2semidirect
5. #22558112C2semidirect
6. #22558110C2semidirect
8. Q8×C67⋊C8C4semidirect
9. Q8×C67⋊C8C4semidirect
12. D4×C67⋊C8C4semidirect
13. D4×C67⋊C8C4semidirect
131. Q8×C67#106semidirect
135. D4×C67#106semidirect
170. C67⋊C8#104semidirect
207. C67#8666semidirect
331. Q8#14523974semidirect
335. D4#14523974semidirect

Maximal subgroups

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

subgroupindex
#41×C67⋊C82
#225602292
C4×#145239742
#225589642
#225581122
#225581102
C67×#18062
#866667