Finite Group Databasebeta

← #30750152

Subgroups — #30750152

Order 17152.

392 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22560229C211yesmaximal, characteristic
2#22558983C211yesmaximal, characteristic, complemented
3#22558964C211yesmaximal, characteristic, complemented
4#22558591C211yesmaximal, characteristic
5#22558110C211yesmaximal, characteristic, complemented
6#22558048C211yesmaximal, characteristic, complemented
7C67×#2925C211yesmaximal, characteristic
8#16863449C2211yescharacteristic
9#16863420C2211yescharacteristic
10C2×#14523970—21no—
11C2×C4×C67⋊C8C2211yescharacteristic
12C4×C4⋊Dic67C2211yescharacteristic
13#16863155—21no—
14#16863148—21no—
15#16863148—21no—
16#16863146—22no—
17Q8⋊(C67⋊C8)C411yescharacteristic, complemented
18Q8⋊(C67⋊C8)C411yescharacteristic, complemented
19D4⋊(C67⋊C8)C411yescharacteristic, complemented
20D4⋊(C67⋊C8)C411yescharacteristic, complemented
21C4×#41×C67C2211yescharacteristic
22#433×C67—22no—
23#432×C67—21no—
24#431×C67—21no—
25#423×C67C2211yescharacteristic
26C2×C4⋊C8×C67C2211yescharacteristic
27#408×C67—21no—
28#405×C67—21no—
29C2×#11761612—21no—
30C22×C67⋊C8—21no—
31C2×C4⋊Dic67—21no—
32C2×C4×Dic67—21no—
33C4⋊(C67⋊C8)—21no—
34C4⋊(C67⋊C8)—21no—
35C4⋊(C67⋊C8)C2×C411yescharacteristic
36C4⋊(C67⋊C8)C2×C411yescharacteristic
37#14523970—42no—
38C22⋊(C67⋊C8)—21no—
39C4×C67⋊C8—21no—
40C4×C67⋊C8—21no—
41C4×C67⋊C8C2×C411yescharacteristic
42C4×C67⋊C8C2×C411yescharacteristic
43C67⋊#95—21no—
44C2×#41×C67D411yescharacteristic
45C2×M16×C67—21no—
46C22×C8×C67—21no—
47C4×Q8×C67C2×C411yescharacteristic
48C4×Q8×C67Q811yescharacteristic
49C4×D4×C67—21no—
50C4×D4×C67—21no—
51C4×D4×C67C2×C411yescharacteristic
52C4×D4×C67Q811yescharacteristic
53#117×C67—21no—
54#117×C67D411yescharacteristic
55C2×C42×C67—21no—
56C2×C42×C67C2311yescharacteristic
57C4⋊C8×C67—21no—
58C4⋊C8×C67—21no—
59C4⋊C8×C67D412yes—
60C4⋊C8×C67D411yescharacteristic
61C4⋊C8×C67D411yescharacteristic
62C22⋊C8×C67—22no—
63C22⋊C8×C67—22no—
64#97×C67—22no—
65C4×C8×C67—22no—
66C22×Dic67—41no—
67#11761612—41no—
68#11761612—41no—
69C2×C67⋊C8—21no—
70C2×C67⋊C8—21no—
71C2×C67⋊C8—21no—
72C2×C67⋊C8—21no—
73C2×C67⋊C8—42no—
74C2×C67⋊C8—41no—
75C4⋊Dic67—22no—
76C4⋊Dic67—41no—
77C4×Dic67—41no—
78C4×Dic67—41no—
79#41×C67—21no—
80#41×C67—21no—
81#41×C67—21no—
82#41×C67—21no—
83C2×Q8×C67C4⋊C411yescharacteristic
84C2×D4×C67—21no—
85C2×D4×C67C4⋊C411yescharacteristic
86C22×C4×C67—21no—
87C22×C4×C67—21no—
88C22×C4×C67—21no—
89C22×C4×C67C2×D411yescharacteristic
90C22×C4×C67#4111yescharacteristic
91C22×C4×C67#4111yescharacteristic
92M16×C67—41no—
93M16×C67—41no—
94C2×C8×C67—21no—
95C2×C8×C67—21no—
96C2×C8×C67—22no—
97C2×C8×C67—41no—
98C2×C8×C67—41no—
99C4⋊C4×C67—21no—
100C4⋊C4×C67—21no—
101C4⋊C4×C67C4⋊C411yescharacteristic
102C4⋊C4×C67C4⋊C411yescharacteristic
103C22⋊C4×C67—21no—
104C22⋊C4×C67—21no—
105C22⋊C4×C67—21no—
106C42×C67—21no—
107C42×C67—21no—
108C42×C67—21no—
109C42×C67C22×C411yescharacteristic
110C42×C67C2×D411yescharacteristic
111C42×C67C2×D411yescharacteristic
112C42×C67C2×Q811yescharacteristic
113C23×C67—21no—
114C23×C67#12311yescharacteristic
115Q8×C67—21no—
116Q8×C67—21no—
117D4×C67—21no—
118D4×C67—21no—
119D4×C67—41no—
120D4×C67—41no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—21no—
125C2×C4×C67—21no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67—41no—
130C2×C4×C67—41no—
131C2×C4×C67—81no—
132C2×C4×C67C2×C4⋊C411yescharacteristic
133C2×C4×C67C4×D411yescharacteristic
134C2×C4×C67C4×D411yescharacteristic
135C2×C4×C67#12011yescharacteristic
136C2×C4×C67C22⋊Q811yescharacteristic
137C2×C4×C67C22⋊Q811yescharacteristic
138C2×C4×C67#13511yescharacteristic
139C2×C4×C67#13511yescharacteristic
140C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C67#13711yescharacteristic
142C2×Dic67—42no—
143C2×Dic67—81no—
144C2×Dic67—81no—
145C8×C67—41no—
146C8×C67—41no—
147C8×C67—42no—
148C67⋊C8—22no—
149C67⋊C8—41no—
150C67⋊C8—41no—
151C67⋊C8—41no—
152C67⋊C8—81no—
153C22×C67—21no—
154C22×C67—21no—
155C22×C67—41no—
156C22×C67—41no—
157C22×C67—41no—
158C22×C67#38511yescharacteristic
159C22×C67#48211yescharacteristic
160C22×C67#48511yescharacteristic
161C4×C67—21no—
162C4×C67—21no—
163C4×C67—21no—
164C4×C67—41no—
165C4×C67—41no—
166C4×C67—41no—
167C4×C67#42711yescharacteristic
168C4×C67#42811yescharacteristic
169C4×C67#44311yescharacteristic
170C4×C67#44411yescharacteristic
171C4×C67#44911yescharacteristic
172C4×C67#45211yescharacteristic
173Dic67—82no—
174#8905—671nomaximal, Sylow
175C2×C67—21no—
176C2×C67—21no—
177C2×C67—81no—
178C2×C67#185411yescharacteristic
179C2×C67#186111yescharacteristic
180C2×C67#186611yescharacteristic
181#2925D6711yescharacteristic
182#1806—671no—
183#1571—671no—
184#1566—671no—
185#1476—671no—
186#1303—671no—
187#1279—671no—
188C67#890511yescharacteristic, complemented, Sylow
189C4×#41C2×D6711yescharacteristic
190#433—22no—
191#432—21no—
192#431—21no—
193#431—671no—
194#423—671no—
195#423C2×D6711yescharacteristic
196C2×C4⋊C8C2×D6711yescharacteristic
197C2×#104—1341no—
198#408—21no—
199#405—21no—
200C2×C4×C8—671no—
201C4×C4⋊C4—671no—
202#339—1341no—
203#335—1341no—
204#335—1341no—
205#334—1342no—
206Q8⋊C8—671no—
207Q8⋊C8—671no—
208D4⋊C8—671no—
209D4⋊C8—671no—
210C2×#41C22⋊D6711yescharacteristic
211C2×M16—21no—
212C2×M16—1341no—
213C22×C8—21no—
214C22×C8—1341no—
215C4×Q8Dic13411yescharacteristic
216C4×Q8C4×D6711yescharacteristic
217C4×D4—21no—
218C4×D4—21no—
219C4×D4Dic13411yescharacteristic
220C4×D4C4×D6711yescharacteristic
221#117—21no—
222#117C22⋊D6711yescharacteristic
223C2×C4⋊C4—1341no—
224C2×C42—21no—
225C2×C42—1341no—
226C2×C42C22×D6711yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—671no—
230C4⋊C8—671no—
231C4⋊C8—1341no—
232C4⋊C8—1341no—
233C4⋊C8C22⋊D6712yes—
234C4⋊C8C22⋊D6711yescharacteristic
235C4⋊C8C22⋊D6711yescharacteristic
236#104—2682no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—1341no—
240#97—22no—
241C4×C8—22no—
242C4×C8—671no—
243C4×C8—671no—
244C4×C8—1341no—
245C4×C8—1341no—
246#95—1341no—
247#41—21no—
248#41—21no—
249#41—21no—
250#41—21no—
251C2×Q8Dic67⋊C411yescharacteristic
252C2×D4—21no—
253C2×D4Dic67⋊C411yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—2681no—
258C22×C4C2×C22⋊D6711yescharacteristic
259C22×C4#1176162811yescharacteristic
260C22×C4#1176162811yescharacteristic
261M16—41no—
262M16—41no—
263M16—2681no—
264M16—2681no—
265C2×C8—21no—
266C2×C8—22no—
267C2×C8—21no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—2682no—
271C2×C8—2681no—
272C2×C8—1341no—
273C2×C8—1341no—
274C2×C8—1341no—
275C2×C8—1341no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—2681no—
279C4⋊C4—1342no—
280C4⋊C4Dic67⋊C411yescharacteristic
281C4⋊C4Dic67⋊C411yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—2681no—
289C42—2681no—
290C42C2×C4×D6711yescharacteristic
291C42C2×C22⋊D6711yescharacteristic
292C42C2×C22⋊D6711yescharacteristic
293C42C2×Dic13411yescharacteristic
294C23—21no—
295C23#1452402611yescharacteristic
296Q8—21no—
297Q8—21no—
298D4—21no—
299D4—21no—
300D4—41no—
301D4—41no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—2682no—
314C2×C4—5361no—
315C2×C4—5361no—
316C2×C4C2×Dic67⋊C411yescharacteristic
317C2×C4C4×C22⋊D6711yescharacteristic
318C2×C4C4×C22⋊D6711yescharacteristic
319C2×C4#1452401211yescharacteristic
320C2×C4#1452402211yescharacteristic
321C2×C4#1452402211yescharacteristic
322C2×C4#1452406311yescharacteristic
323C2×C4#1452406311yescharacteristic
324C2×C4#1452406811yescharacteristic
325C2×C4#1452407311yescharacteristic
326C8—41no—
327C8—41no—
328C8—42no—
329C8—2681no—
330C8—2681no—
331C8—2681no—
332C8—5361no—
333C8—1342no—
334C22—21no—
335C22—21no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22#1686326911yescharacteristic
340C22#1686372011yescharacteristic
341C22#1686373711yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—5362no—
349C4#1686343211yescharacteristic
350C4#1686343911yescharacteristic
351C4#1686351111yescharacteristic
352C4#1686351611yescharacteristic
353C4#1686354111yescharacteristic
354C4#1686355811yescharacteristic
355C2—21no—
356C2—21no—
357C2—81no—
358C2#2256040811yescharacteristic
359C2#2256044911yescharacteristic
360C2#2256048411yescharacteristic
subgroupG/H
centerC2×C4#14524012
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2925C2
socleC22×C67#385
prime radicalC2×#41×C67D4
cosocleC2×C42×C67C23
hypercenter#2925D67
hypocenterC67#8905
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2925

Complements

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

NH ≅ G/Ntype
2. #22558983C2semidirect
3. #22558964C2semidirect
5. #22558110C2semidirect
6. #22558048C2semidirect
17. Q8⋊(C67⋊C8)C4semidirect
18. Q8⋊(C67⋊C8)C4semidirect
19. D4⋊(C67⋊C8)C4semidirect
20. D4⋊(C67⋊C8)C4semidirect
188. C67#8905semidirect

Maximal subgroups

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

subgroupindex
#225602292
#225589832
#225589642
#225585912
#225581102
#225580482
C67×#29252
#890567