Finite Group Databasebeta

← #30750129

Subgroups — #30750129

Order 17152.

438 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22568777C211yesmaximal, characteristic
2#22558985C211yesmaximal, characteristic, complemented
3#22558967C211yesmaximal, characteristic, complemented
4#22558591C211yesmaximal, characteristic
5#22558112C211yesmaximal, characteristic, complemented
6#22558050C211yesmaximal, characteristic, complemented
7C67×#1806C211yesmaximal, characteristic
8C2×#14524050—21no—
9#16863449C2211yescharacteristic
10#16863423—22no—
11#16863423—21no—
12#16863420—21no—
13#16863420C2211yescharacteristic
14C2×C4⋊(C67⋊C8)—21no—
15C2×C4⋊(C67⋊C8)C2211yescharacteristic
16C2×C4×C67⋊C8C2211yescharacteristic
17#16863148—21no—
18#16863148—21no—
19#16863148—22no—
20Q8⋊(C67⋊C8)C411yescharacteristic, complemented
21Q8⋊(C67⋊C8)C411yescharacteristic, complemented
22D4⋊(C67⋊C8)C411yescharacteristic, complemented
23D4⋊(C67⋊C8)C411yescharacteristic, complemented
24C4×#41×C67C2211yescharacteristic
25#423×C67C2211yescharacteristic
26C2×#104×C67—21no—
27C4×C4⋊C4×C67C2211yescharacteristic
28#339×C67—21no—
29#14524050—42no—
30#14524050—42no—
31C2×#11761612—22no—
32C2×#11761612—21no—
33C22×C67⋊C8—22no—
34C22×C67⋊C8—21no—
35C22×C67⋊C8—21no—
36C4⋊(C67⋊C8)—22no—
37C4⋊(C67⋊C8)—22no—
38C4⋊(C67⋊C8)—21no—
39C4⋊(C67⋊C8)—21no—
40C4⋊(C67⋊C8)C2×C411yescharacteristic
41C4⋊(C67⋊C8)C2×C411yescharacteristic
42C4⋊(C67⋊C8)D411yescharacteristic
43C4⋊(C67⋊C8)D411yescharacteristic
44C4⋊(C67⋊C8)D412yes—
45C22⋊(C67⋊C8)—21no—
46C4×C67⋊C8—21no—
47C4×C67⋊C8—21no—
48C4×C67⋊C8C2×C411yescharacteristic
49C4×C67⋊C8C2×C411yescharacteristic
50C2×#41×C67D411yescharacteristic
51C2×M16×C67—21no—
52C4×Q8×C67C2×C411yescharacteristic
53C4×Q8×C67Q811yescharacteristic
54C4×D4×C67—21no—
55C4×D4×C67—21no—
56C4×D4×C67C2×C411yescharacteristic
57C4×D4×C67Q811yescharacteristic
58#117×C67—21no—
59#117×C67D411yescharacteristic
60C2×C4⋊C4×C67—21no—
61C2×C42×C67—21no—
62C2×C42×C67—21no—
63C2×C42×C67C2311yescharacteristic
64C4⋊C8×C67—21no—
65C4⋊C8×C67—21no—
66#104×C67—42no—
67#95×C67—21no—
68#11761612—44no—
69#11761612—41no—
70#11761612—41no—
71C2×C67⋊C8—21no—
72C2×C67⋊C8—21no—
73C2×C67⋊C8—21no—
74C2×C67⋊C8—21no—
75C2×C67⋊C8—21no—
76C2×C67⋊C8—22no—
77C2×C67⋊C8—21no—
78C2×C67⋊C8—42no—
79C2×C67⋊C8—44no—
80C2×C67⋊C8—41no—
81C2×C67⋊C8—41no—
82#41×C67—21no—
83#41×C67—21no—
84#41×C67D812yes—
85#41×C67Q1612yes—
86C2×Q8×C67C4⋊C411yescharacteristic
87C2×D4×C67—21no—
88C2×D4×C67C4⋊C411yescharacteristic
89C22×C4×C67—21no—
90C22×C4×C67—21no—
91C22×C4×C67—21no—
92C22×C4×C67—41no—
93C22×C4×C67C2×D411yescharacteristic
94C22×C4×C67#4111yescharacteristic
95C22×C4×C67#4111yescharacteristic
96M16×C67—41no—
97M16×C67—41no—
98C2×C8×C67—41no—
99C4⋊C4×C67—21no—
100C4⋊C4×C67—21no—
101C4⋊C4×C67—22no—
102C4⋊C4×C67—41no—
103C4⋊C4×C67C4⋊C411yescharacteristic
104C4⋊C4×C67C4⋊C411yescharacteristic
105C22⋊C4×C67—21no—
106C22⋊C4×C67—21no—
107C22⋊C4×C67—21no—
108C42×C67—21no—
109C42×C67—21no—
110C42×C67—21no—
111C42×C67—41no—
112C42×C67—41no—
113C42×C67C22×C411yescharacteristic
114C42×C67C2×D411yescharacteristic
115C42×C67C2×D411yescharacteristic
116C42×C67C2×Q811yescharacteristic
117C23×C67—21no—
118C23×C67#12311yescharacteristic
119Q8×C67—21no—
120Q8×C67#10712yes—
121D4×C67—22no—
122D4×C67—22no—
123D4×C67—21no—
124D4×C67#10712yes—
125C2×C4×C67—21no—
126C2×C4×C67—21no—
127C2×C4×C67—21no—
128C2×C4×C67—22no—
129C2×C4×C67—22no—
130C2×C4×C67—21no—
131C2×C4×C67—21no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—41no—
135C2×C4×C67—42no—
136C2×C4×C67—81no—
137C2×C4×C67—81no—
138C2×C4×C67—81no—
139C2×C4×C67C2×C4⋊C411yescharacteristic
140C2×C4×C67C4×D411yescharacteristic
141C2×C4×C67C4×D411yescharacteristic
142C2×C4×C67#12011yescharacteristic
143C2×C4×C67C22⋊Q811yescharacteristic
144C2×C4×C67C22⋊Q811yescharacteristic
145C2×C4×C67C2×D811yescharacteristic
146C2×C4×C67C2×Q1611yescharacteristic
147C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C67#13711yescharacteristic
149C8×C67—81no—
150C67⋊C8—22no—
151C67⋊C8—41no—
152C67⋊C8—41no—
153C67⋊C8—41no—
154C67⋊C8—41no—
155C67⋊C8—42no—
156C67⋊C8—41no—
157C22×C67—21no—
158C22×C67—21no—
159C22×C67—22no—
160C22×C67—41no—
161C22×C67—41no—
162C22×C67#38511yescharacteristic
163C22×C67#48011yescharacteristic
164C22×C67#48411yescharacteristic
165C4×C67—22no—
166C4×C67—21no—
167C4×C67—21no—
168C4×C67—21no—
169C4×C67—41no—
170C4×C67—41no—
171C4×C67—82no—
172C4×C67C2×#10711yescharacteristic
173C4×C67#42811yescharacteristic
174C4×C67#44311yescharacteristic
175C4×C67#44411yescharacteristic
176C4×C67#44711yescharacteristic
177C4×C67#45111yescharacteristic
178#8902—671nomaximal, Sylow
179C2×C67—21no—
180C2×C67—21no—
181C2×C67—42no—
182C2×C67#185211yescharacteristic
183C2×C67#186111yescharacteristic
184C2×C67#186611yescharacteristic
185#2906—671no—
186#1806D6711yescharacteristic
187#1571—671no—
188#1567—671no—
189#1476—671no—
190#1303—671no—
191#1279—671no—
192C67#890211yescharacteristic, complemented, Sylow
193C2×#131—1341no—
194C4×#41C2×D6711yescharacteristic
195#431—671no—
196#424—1342no—
197#424—1341no—
198#423—671no—
199#423—1341no—
200#423C2×D6711yescharacteristic
201C2×C4⋊C8—671no—
202C2×C4⋊C8—1341no—
203C2×#104—21no—
204C2×C4×C8—671no—
205C4×C4⋊C4C2×D6711yescharacteristic
206#339—21no—
207#335—1342no—
208#335—1341no—
209#335—1341no—
210Q8⋊C8—671no—
211Q8⋊C8—671no—
212D4⋊C8—671no—
213D4⋊C8—671no—
214C2×#41C22⋊D6711yescharacteristic
215#131—2682no—
216#131—2682no—
217C2×M16—21no—
218C2×M16—1342no—
219C2×M16—1341no—
220C22×C8—1342no—
221C22×C8—1341no—
222C22×C8—1341no—
223C4×Q8Dic13411yescharacteristic
224C4×Q8C4×D6711yescharacteristic
225C4×D4—21no—
226C4×D4—21no—
227C4×D4Dic13411yescharacteristic
228C4×D4C4×D6711yescharacteristic
229#117—21no—
230#117C22⋊D6711yescharacteristic
231C2×C4⋊C4—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42C22×D6711yescharacteristic
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—671no—
238C4⋊C8—671no—
239C4⋊C8—671no—
240C4⋊C8—671no—
241C4⋊C8—672no—
242C4⋊C8—1342no—
243C4⋊C8—1342no—
244C4⋊C8—1341no—
245C4⋊C8—1341no—
246#104—42no—
247C22⋊C8—1341no—
248C4×C8—671no—
249C4×C8—671no—
250C4×C8—1341no—
251C4×C8—1341no—
252#95—21no—
253#41—21no—
254#41—21no—
255#41#1176161312yes—
256#41#1176161812yes—
257C2×Q8Dic67⋊C411yescharacteristic
258C2×D4—21no—
259C2×D4Dic67⋊C411yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4C2×C22⋊D6711yescharacteristic
265C22×C4#1176162711yescharacteristic
266C22×C4#1176162711yescharacteristic
267M16—41no—
268M16—41no—
269M16—2684no—
270M16—2681no—
271M16—2681no—
272C2×C8—41no—
273C2×C8—2684no—
274C2×C8—2681no—
275C2×C8—2681no—
276C2×C8—2682no—
277C2×C8—1341no—
278C2×C8—1341no—
279C2×C8—1341no—
280C2×C8—1341no—
281C2×C8—1341no—
282C2×C8—1342no—
283C2×C8—1341no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4Dic67⋊C411yescharacteristic
289C4⋊C4Dic67⋊C411yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—41no—
297C42—41no—
298C42C2×C4×D6711yescharacteristic
299C42D4×D6711yescharacteristic
300C42D4×D6711yescharacteristic
301C42C2×Dic13411yescharacteristic
302C23—21no—
303C23#1452402411yescharacteristic
304Q8—21no—
305Q8#1452397612yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4#1452397612yes—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4C2×Dic67⋊C411yescharacteristic
325C2×C4(C22⋊D67)⋊C411yescharacteristic
326C2×C4(C22⋊D67)⋊C411yescharacteristic
327C2×C4#1452401111yescharacteristic
328C2×C4#1452402011yescharacteristic
329C2×C4#1452402011yescharacteristic
330C2×C4C2×#1176161311yescharacteristic
331C2×C4C2×#1176161811yescharacteristic
332C2×C4#1452406811yescharacteristic
333C2×C4#1452407311yescharacteristic
334C8—81no—
335C8—2681no—
336C8—2681no—
337C8—2681no—
338C8—2681no—
339C8—2682no—
340C8—2681no—
341C8—1342no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22#1686326811yescharacteristic
348C22#1686370711yescharacteristic
349C22#1686372911yescharacteristic
350C4—22no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—82no—
357C4C2×#1452397611yescharacteristic
358C4#1686343911yescharacteristic
359C4#1686351011yescharacteristic
360C4#1686351511yescharacteristic
361C4#1686352911yescharacteristic
362C4#1686355111yescharacteristic
363C2—21no—
364C2—21no—
365C2—42no—
366C2#2256039611yescharacteristic
367C2#2256044711yescharacteristic
368C2#2256048211yescharacteristic
subgroupG/H
centerC2×C4#14524011
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1806C2
socleC22×C67#385
prime radicalC2×#41×C67D4
cosocleC2×C42×C67C23
hypercenter#1806D67
hypocenterC67#8902
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
2. #22558985C2semidirect
3. #22558967C2semidirect
5. #22558112C2semidirect
6. #22558050C2semidirect
20. Q8⋊(C67⋊C8)C4semidirect
21. Q8⋊(C67⋊C8)C4semidirect
22. D4⋊(C67⋊C8)C4semidirect
23. D4⋊(C67⋊C8)C4semidirect
192. C67#8902semidirect

Maximal subgroups

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

subgroupindex
#225687772
#225589852
#225589672
#225585912
#225581122
#225580502
C67×#18062
#890267