Finite Group Databasebeta

← #30824971

Subgroups — #30824971

Order 17152.

602 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#22569320C211yesmaximal, characteristic
2#22568853C211yesmaximal, characteristic
3#22559000C212yesmaximal, complemented
4#22558995C212yesmaximal, complemented
5#22558990C212yesmaximal, complemented
6#22558985C212yesmaximal, complemented
7#22558962C211yesmaximal, characteristic, complemented
8#22558959C211yesmaximal, characteristic, complemented
9#22558955C211yesmaximal, characteristic, complemented
10C2×#16863148C211yesmaximal, characteristic, complemented
11C67×#2989C211yesmaximal, characteristic
12Q8×C67⋊C8—21no—
13D4×C67⋊C8—22no—
14D4×C67⋊C8—21no—
15#16863457C414yes—
16#16863449C2212yes—
17#16863449C2211yescharacteristic
18#16863449C2212yes—
19#16863449C2212yes—
20#16863423C412yes—
21C2×C4⋊(C67⋊C8)C2212yes—
22C2×C4⋊(C67⋊C8)C2211yescharacteristic
23#16863363C412yes—
24C4×#11761612C2211yescharacteristic
25C2×C4×C67⋊C8C2211yescharacteristic
26#16863148C2211yescharacteristic
27#16863148C2212yes—
28#16863148C2212yes—
29#16863148C2212yes—
30#16863148C2211yescharacteristic
31#16863148C2212yes—
32#16863146C2212yes—
33#16863146C2212yes—
34#16863146C2211yescharacteristic
35#16863146C2211yescharacteristic
36C4×#41×C67C2211yescharacteristic
37#446×C67—21no—
38#436×C67—21no—
39#435×C67—22no—
40#432×C67C2212yes—
41#431×C67C2211yescharacteristic
42#423×C67C2212yes—
43C2×C4⋊C8×C67C2211yescharacteristic
44C2×#11761612C2×C412yes—
45C22×C67⋊C8—21no—
46C22×C67⋊C8—22no—
47C22×C67⋊C8C2×C412yes—
48C4⋊(C67⋊C8)—21no—
49C4⋊(C67⋊C8)—22no—
50C4⋊(C67⋊C8)C2×C414yes—
51C4⋊(C67⋊C8)C2×C414yes—
52C4⋊(C67⋊C8)C2311yescharacteristic
53C4⋊(C67⋊C8)C2312yes—
54C4⋊(C67⋊C8)C2311yescharacteristic
55C22⋊(C67⋊C8)—21no—
56C22⋊(C67⋊C8)—22no—
57C22⋊(C67⋊C8)C2×C414yes—
58C22⋊(C67⋊C8)C2×C414yes—
59#14523954C2×C412yes—
60C4×C67⋊C8—21no—
61C4×C67⋊C8—22no—
62C4×C67⋊C8C2×C412yes—
63C4×C67⋊C8C2311yescharacteristic
64C4×C67⋊C8C2311yescharacteristic
65C4×C67⋊C8C2312yes—
66C2×#41×C67C2×C411yescharacteristic
67C2×M16×C67—22no—
68C22×C8×C67—21no—
69C4×Q8×C67D411yescharacteristic
70C4×Q8×C67Q811yescharacteristic
71C4×D4×C67D412yes—
72C4×D4×C67D411yescharacteristic
73C4×D4×C67Q812yes—
74C4×D4×C67Q811yescharacteristic
75#117×C67C2×C411yescharacteristic
76#117×C67C2×C412yes—
77C2×C42×C67C2311yescharacteristic
78C2×C42×C67C2312yes—
79C4⋊C8×C67—21no—
80C4⋊C8×C67—22no—
81C4⋊C8×C67C2311yescharacteristic
82C4⋊C8×C67C2312yes—
83C4⋊C8×C67C2311yescharacteristic
84C22⋊C8×C67—21no—
85C22⋊C8×C67—22no—
86#97×C67—22no—
87C4×C8×C67—21no—
88#11761612—24no—
89C2×C67⋊C8—24no—
90C2×C67⋊C8—21no—
91C2×C67⋊C8—21no—
92C2×C67⋊C8—22no—
93C2×C67⋊C8—81no—
94C2×C67⋊C8—82no—
95C2×C67⋊C8C22×C412yes—
96C2×C67⋊C8C22×C412yes—
97C2×C67⋊C8C22×C414yes—
98#41×C67—24no—
99C2×Q8×C67C4⋊C411yescharacteristic
100C2×D4×C67C4⋊C412yes—
101C2×D4×C67C4⋊C411yescharacteristic
102C22×C4×C67C22×C412yes—
103C22×C4×C67C22×C411yescharacteristic
104C22×C4×C67C2×D411yescharacteristic
105C22×C4×C67C2×D412yes—
106C22×C4×C67C2×Q812yes—
107C22×C4×C67C2×Q811yescharacteristic
108M16×C67—82no—
109C2×C8×C67—22no—
110C2×C8×C67—21no—
111C2×C8×C67—21no—
112C2×C8×C67—81no—
113C4⋊C4×C67C4⋊C412yes—
114C4⋊C4×C67C4⋊C411yescharacteristic
115C4⋊C4×C67C4⋊C412yes—
116C4⋊C4×C67C4⋊C411yescharacteristic
117C22⋊C4×C67C4⋊C412yes—
118C22⋊C4×C67C4⋊C412yes—
119C22⋊C4×C67C4⋊C412yes—
120C42×C67C22×C411yescharacteristic
121C42×C67C22×C412yes—
122C42×C67C2×D411yescharacteristic
123C42×C67C2×D412yes—
124C42×C67C2×Q812yes—
125C42×C67C2×Q811yescharacteristic
126C42×C67C2411yescharacteristic
127C23×C67C2×C4⋊C412yes—
128C23×C67C2×C4⋊C411yescharacteristic
129Q8×C67—22no—
130D4×C67—24no—
131D4×C67—22no—
132C2×C4×C67—24no—
133C2×C4×C67—21no—
134C2×C4×C67—22no—
135C2×C4×C67—21no—
136C2×C4×C67—22no—
137C2×C4×C67—41no—
138C2×C4×C67—42no—
139C2×C4×C67C2×C4⋊C411yescharacteristic
140C2×C4×C67C2×C4⋊C412yes—
141C2×C4×C67C2×C4⋊C411yescharacteristic
142C2×C4×C67C2×C4⋊C412yes—
143C2×C4×C67C2×C4⋊C412yes—
144C2×C4×C67C2×C4⋊C411yescharacteristic
145C2×C4×C67#13114yes—
146C2×C4×C67C23×C411yescharacteristic
147C2×C4×C67C22×D411yescharacteristic
148C2×C4×C67C22×Q811yescharacteristic
149C8×C67—42no—
150C8×C67—41no—
151C8×C67—41no—
152C67⋊C8—22no—
153C67⋊C8—22no—
154C67⋊C8—24no—
155C67⋊C8—42no—
156C67⋊C8—41no—
157C67⋊C8—41no—
158C22×C67—21no—
159C22×C67—21no—
160C22×C67—22no—
161C22×C67—22no—
162C22×C67—22no—
163C22×C67#42412yes—
164C22×C67C22×C4⋊C411yescharacteristic
165C4×C67—22no—
166C4×C67—21no—
167C4×C67—22no—
168C4×C67—21no—
169C4×C67—21no—
170C4×C67—22no—
171C4×C67#42412yes—
172C4×C67C2×#13112yes—
173C4×C67#57611yescharacteristic
174C4×C67#57811yescharacteristic
175#17812—671nomaximal, Sylow
176C2×C67—22no—
177C2×C67—42no—
178C2×C67C2×#42411yescharacteristic
179C2×C67#291311yescharacteristic
180C2×C67#291611yescharacteristic
181#2989D6711yescharacteristic
182#2988—671no—
183#2922—671no—
184#1574—672no—
185#1573—672no—
186#1572—672no—
187#1571—672no—
188#1565—671no—
189#1565—671no—
190#1564—671no—
191C2×#335—671no—
192C67#1781211yescharacteristic, complemented, Sylow
193C4×#41C2×D6711yescharacteristic
194#446—21no—
195C8×Q8—1341no—
196#436—21no—
197#435—22no—
198C8×D4—1342no—
199C8×D4—1341no—
200#433—674no—
201#432C2×D6712yes—
202#431—672no—
203#431—672no—
204#431—671no—
205#431—672no—
206#431C2×D6711yescharacteristic
207#424—672no—
208#423C2×D6712yes—
209C2×C4⋊C8—671no—
210C2×C4⋊C8—672no—
211C2×C4⋊C8C2×D6711yescharacteristic
212#408—672no—
213C4×M16—671no—
214C2×C4×C8—671no—
215#335—672no—
216#335—672no—
217#335—671no—
218#335—671no—
219#335—672no—
220#335—672no—
221#334—671no—
222#334—671no—
223#334—672no—
224#334—672no—
225C2×#41C4×D6711yescharacteristic
226C2×M16—22no—
227C2×M16—672no—
228C22×C8—21no—
229C22×C8—672no—
230C22×C8—1341no—
231C22×C8—1342no—
232C4×Q8Dic13411yescharacteristic
233C4×Q8C22⋊D6711yescharacteristic
234C4×D4Dic13412yes—
235C4×D4Dic13411yescharacteristic
236C4×D4C22⋊D6712yes—
237C4×D4C22⋊D6711yescharacteristic
238#117C4×D6711yescharacteristic
239#117C4×D6712yes—
240C2×C42C22×D6711yescharacteristic
241C2×C42C22×D6712yes—
242C4⋊C8—21no—
243C4⋊C8—22no—
244C4⋊C8—674no—
245C4⋊C8—674no—
246C4⋊C8—671no—
247C4⋊C8—671no—
248C4⋊C8—672no—
249C4⋊C8—1341no—
250C4⋊C8—1342no—
251C4⋊C8C22×D6711yescharacteristic
252C4⋊C8C22×D6712yes—
253C4⋊C8C22×D6711yescharacteristic
254C22⋊C8—21no—
255C22⋊C8—22no—
256C22⋊C8—674no—
257C22⋊C8—674no—
258C22⋊C8—1341no—
259C22⋊C8—1342no—
260#97—22no—
261#97—672no—
262C4×C8—21no—
263C4×C8—672no—
264C4×C8—671no—
265C4×C8—672no—
266C4×C8—671no—
267C4×C8—1341no—
268C4×C8—1342no—
269#41—24no—
270C2×Q8Dic67⋊C411yescharacteristic
271C2×D4Dic67⋊C412yes—
272C2×D4Dic67⋊C411yescharacteristic
273C22×C4C2×C4×D6712yes—
274C22×C4C2×C4×D6711yescharacteristic
275C22×C4C2×C22⋊D6711yescharacteristic
276C22×C4C2×C22⋊D6712yes—
277C22×C4C2×Dic13412yes—
278C22×C4C2×Dic13411yescharacteristic
279M16—82no—
280M16—1344no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—22no—
284C2×C8—81no—
285C2×C8—5361no—
286C2×C8—5362no—
287C2×C8—672no—
288C2×C8—672no—
289C2×C8—674no—
290C2×C8—1341no—
291C2×C8—1341no—
292C2×C8—1342no—
293C2×C8—1344no—
294C4⋊C4Dic67⋊C412yes—
295C4⋊C4Dic67⋊C411yescharacteristic
296C4⋊C4Dic67⋊C412yes—
297C4⋊C4Dic67⋊C411yescharacteristic
298C22⋊C4Dic67⋊C412yes—
299C22⋊C4Dic67⋊C412yes—
300C22⋊C4Dic67⋊C412yes—
301C42C2×C4×D6711yescharacteristic
302C42C2×C4×D6712yes—
303C42C2×C22⋊D6711yescharacteristic
304C42C2×C22⋊D6712yes—
305C42C2×Dic13412yes—
306C42C2×Dic13411yescharacteristic
307C42C23×D6711yescharacteristic
308C23C2×Dic67⋊C412yes—
309C23C2×Dic67⋊C411yescharacteristic
310Q8—22no—
311D4—24no—
312D4—22no—
313C2×C4—24no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4C2×Dic67⋊C411yescharacteristic
321C2×C4C2×Dic67⋊C412yes—
322C2×C4C2×Dic67⋊C411yescharacteristic
323C2×C4C2×Dic67⋊C412yes—
324C2×C4C2×Dic67⋊C412yes—
325C2×C4C2×Dic67⋊C411yescharacteristic
326C2×C4#1452405114yes—
327C2×C4C22×C4×D6711yescharacteristic
328C2×C4C22×C22⋊D6711yescharacteristic
329C2×C4C22×Dic13411yescharacteristic
330C8—42no—
331C8—41no—
332C8—41no—
333C8—2682no—
334C8—2681no—
335C8—2681no—
336C8—1342no—
337C8—1342no—
338C8—1344no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22#1686342412yes—
345C22C22×Dic67⋊C411yescharacteristic
346C4—22no—
347C4—21no—
348C4—22no—
349C4—21no—
350C4—21no—
351C4—22no—
352C4#1686342412yes—
353C4C2×#1452405112yes—
354C4#1686419911yescharacteristic
355C4#1686421111yescharacteristic
356C2—22no—
357C2—42no—
358C2C2×#1686342411yescharacteristic
359C2#2256880711yescharacteristic
360C2#2256881811yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#2989C2
socleC22×C67C22×C4⋊C4
prime radicalC2×#41×C67C2×C4
cosocleC42×C67C24
hypercenter#2989D67
hypocenterC67#17812
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2989

Complements

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

NH ≅ G/Ntype
3. #22559000C2semidirect
4. #22558995C2semidirect
5. #22558990C2semidirect
6. #22558985C2semidirect
7. #22558962C2semidirect
8. #22558959C2semidirect
9. #22558955C2semidirect
10. C2×#16863148C2semidirect
192. C67#17812semidirect

Maximal subgroups

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

subgroupindex
#225693202
#225688532
#225590002
#225589952
#225589902
#225589852
#225589622
#225589592
#225589552
C2×#168631482
C67×#29892
#1781267