Finite Group Databasebeta

← #30772895

Subgroups — #30772895

Order 17152.

526 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523986C211yesmaximal, characteristic
2C2×#16863200C211yesmaximal, characteristic, complemented
3C2×#16863196C211yesmaximal, characteristic
4#22561075C211yesmaximal, characteristic
5#22558300C211yesmaximal, characteristic, complemented
6#22558287C211yesmaximal, characteristic, complemented
7C2×C4×Q16×C67C211yesmaximal, characteristic, complemented
8C22×D536C2211yescharacteristic
9C2×#14523986—22no—
10C2×#14523986—22no—
11C2×#14523986C414yescomplemented
12C22×C67⋊C16C2211yescharacteristic
13C22×C67⋊C16C2211yescharacteristic
14C2×#14523977C2211yescharacteristic
15C2×#14523965—21no—
16#16863298—21no—
17#16863200—22no—
18#16863196—22no—
19C22×Q16×C67C2211yescharacteristic
20C2×C4×Q8×C67—21no—
21C4×Q16×C67—22no—
22C4×Q16×C67—22no—
23C2×#107×C67C2211yescharacteristic
24C2×#103×C67—21no—
25C2×C4×C8×C67C2211yescharacteristic
26C22×D268—21no—
27C2×D536—21no—
28C2×D536—21no—
29C2×D536—21no—
30C2×D536—21no—
31C2×D536—21no—
32C2×D536C2×C412yes—
33C2×C4⋊Dic67—21no—
34C2×(C2×D67)⋊C4—41no—
35C2×(C2×D67)⋊C4—41no—
36#14523986—28no—
37C2×C67⋊C16—21no—
38C2×C67⋊C16—21no—
39C2×C67⋊C16—21no—
40C2×C67⋊C16—21no—
41C2×C67⋊C16—21no—
42C2×C67⋊C16C2×C412yes—
43#14523977—21no—
44#14523977—21no—
45#14523965—42no—
46C22×Q8×C67—21no—
47C2×Q16×C67—21no—
48C2×Q16×C67—21no—
49C2×Q16×C67—21no—
50C2×Q16×C67C2×C412yes—
51C2×Q16×C67D412yes—
52C2×Q16×C67D412yes—
53C22×C8×C67D411yescharacteristic
54C22×C8×C67C2311yescharacteristic
55C4×Q8×C67—24no—
56C4×Q8×C67—42no—
57C2×C4⋊C4×C67—21no—
58C2×C4⋊C4×C67—41no—
59C2×C42×C67—41no—
60C2×C42×C67D411yescharacteristic
61#107×C67—21no—
62#107×C67D412yes—
63#103×C67—24no—
64C4×C8×C67—21no—
65C4×C8×C67—21no—
66C23×D67—41no—
67C2×D268—22no—
68C2×D268—41no—
69C2×D268—41no—
70C2×D268—41no—
71C2×D268—41no—
72C2×D268—41no—
73C22×Dic67—41no—
74D536—22no—
75D536—22no—
76D536—41no—
77D536—41no—
78C4⋊Dic67—41no—
79C4⋊Dic67—41no—
80(C2×D67)⋊C4—82no—
81(C2×D67)⋊C4—82no—
82C67⋊C16—22no—
83C67⋊C16—41no—
84C2×Q8×C67—22no—
85C2×Q8×C67—22no—
86C2×Q8×C67—22no—
87C2×Q8×C67—41no—
88C2×Q8×C67—41no—
89C2×Q8×C67—41no—
90C22×C4×C67—41no—
91C22×C4×C67—41no—
92C22×C4×C67D811yescharacteristic
93C22×C4×C67D811yescharacteristic
94C22×C4×C67C2×D411yescharacteristic
95Q16×C67—22no—
96Q16×C67—22no—
97Q16×C67—22no—
98Q16×C67C22⋊C414yescomplemented
99C2×C8×C67—21no—
100C2×C8×C67—21no—
101C2×C8×C67—21no—
102C2×C8×C67C22×C411yescharacteristic
103C2×C8×C67C2×D411yescharacteristic
104C2×C8×C67C2×D411yescharacteristic
105C2×C8×C67C2×D411yescharacteristic
106C2×C8×C67#4111yescharacteristic
107C2×C8×C67#4111yescharacteristic
108C4⋊C4×C67—22no—
109C4⋊C4×C67—44no—
110C4⋊C4×C67—41no—
111C42×C67—21no—
112C42×C67—21no—
113C42×C67—44no—
114C23×C67C2×D811yescharacteristic
115C22×D67—42no—
116C22×D67—81no—
117C22×D67—81no—
118C22×D67—81no—
119C22×D67—81no—
120C22×D67—81no—
121C22×D67—81no—
122Q8×C67—24no—
123Q8×C67—42no—
124Q8×C67—42no—
125Q8×C67—42no—
126C2×C4×C67—21no—
127C2×C4×C67—21no—
128C2×C4×C67—21no—
129C2×C4×C67—21no—
130C2×C4×C67—21no—
131C2×C4×C67—42no—
132C2×C4×C67—42no—
133C2×C4×C67—42no—
134C2×C4×C67—42no—
135C2×C4×C67—81no—
136C2×C4×C67—81no—
137C2×C4×C67SD3212yescomplemented
138C2×C4×C67C4×D411yescharacteristic
139C2×C4×C67#12011yescharacteristic
140C2×C4×C67#12111yescharacteristic
141C2×C4×C67#13511yescharacteristic
142C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
144C2×Dic67—81no—
145C2×Dic67—81no—
146C2×Dic67—81no—
147D268—42no—
148D268—42no—
149D268—81no—
150D268—81no—
151C8×C67—41no—
152C8×C67C2×C22⋊C411yescharacteristic
153C8×C67C4×D411yescharacteristic
154C8×C67#12111yescharacteristic
155C8×C67#12311yescharacteristic
156C22×C67C4×D811yescharacteristic
157C22×C67#44711yescharacteristic
158C22×C67#45911yescharacteristic
159C22×C67C2×SD3211yescharacteristic
160C22×C67#50811yescharacteristic
161C22×C67#50911yescharacteristic
162C22×C67#51011yescharacteristic
163C2×D67—82no—
164C2×D67—82no—
165C2×D67—82no—
166C2×D67—161no—
167C2×D67—161no—
168C2×D67—161no—
169C2×D67—161no—
170C4×C67—22no—
171C4×C67—44no—
172C4×C67—41no—
173C4×C67—82no—
174C4×C67#38611yescharacteristic
175C4×C67#44211yescharacteristic
176C4×C67#44911yescharacteristic
177C4×C67#46311yescharacteristic
178Dic67—161no—
179#13013—671nomaximal, Sylow
180C2×C67#187911yescharacteristic
181C2×C67C4×SD3211yescharacteristic
182C2×C67#217511yescharacteristic
183C2×C67#218511yescharacteristic
184C2×C67#218611yescharacteristic
185C2×C67#220711yescharacteristic
186C2×C67#221111yescharacteristic
187D67—162no—
188C22×SD32—671no—
189C2×C4×Q16D6711yescharacteristic, complemented
190C2×#358—671no—
191C2×#357—671no—
192#1970—671no—
193#1380—671no—
194#1374—671no—
195C67#1301311yescharacteristic, complemented, Sylow
196C22×Q16C2×D6711yescharacteristic
197C22×D8—671no—
198C2×C4×Q8—21no—
199C2×SD32—674no—
200C2×SD32—1342no—
201C2×SD32—1342no—
202C22×C16—671no—
203C22×C16—671no—
204C4×Q16—22no—
205C4×Q16—22no—
206C2×#107—671no—
207C2×#107C2×D6711yescharacteristic
208C2×#103—21no—
209C2×#102—1341no—
210C2×C4×C8C2×D6711yescharacteristic
211#390—1341no—
212#358—1342no—
213#357—1342no—
214C22×Q8—21no—
215C22×D4—1341no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16C4×D6712yes—
220C2×Q16D26812yes—
221C2×Q16C22⋊D6712yes—
222C2×D8—672no—
223C2×D8—1341no—
224C2×D8—1341no—
225C2×D8—1341no—
226C2×D8—1341no—
227C2×D8—1341no—
228C22×C8C22⋊D6711yescharacteristic
229C22×C8C22×D6711yescharacteristic
230C4×Q8—24no—
231C4×Q8—42no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—1341no—
235C2×C22⋊C4—2681no—
236C2×C22⋊C4—2681no—
237C2×C42—41no—
238C2×C42C22⋊D6711yescharacteristic
239SD32—1348no—
240C2×C16—672no—
241C2×C16—1341no—
242C2×C16—1341no—
243C2×C16—1341no—
244C2×C16—1341no—
245C2×C16—1341no—
246#107—21no—
247#107—1341no—
248#107—1341no—
249#107C22⋊D6712yes—
250#103—24no—
251#102—2682no—
252C4×C8—21no—
253C4×C8—21no—
254C24—2681no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—2681no—
262C2×D4—2681no—
263C2×D4—2681no—
264C2×D4—2681no—
265C2×D4—2681no—
266C2×D4—1342no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—2681no—
270C22×C4#1176161311yescharacteristic
271C22×C4#1176161311yescharacteristic
272C22×C4C2×C22⋊D6711yescharacteristic
273Q16—22no—
274Q16—22no—
275Q16—22no—
276Q16(C2×D67)⋊C414yescomplemented
277D8—2681no—
278D8—2681no—
279D8—1342no—
280D8—1342no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8C2×C4×D6711yescharacteristic
285C2×C8C2×C22⋊D6711yescharacteristic
286C2×C8C2×C22⋊D6711yescharacteristic
287C2×C8C2×D26811yescharacteristic
288C2×C8#1176162811yescharacteristic
289C2×C8#1176162811yescharacteristic
290C4⋊C4—22no—
291C4⋊C4—44no—
292C4⋊C4—41no—
293C4⋊C4—2681no—
294C4⋊C4—2681no—
295C22⋊C4—5362no—
296C22⋊C4—5362no—
297C42—21no—
298C42—21no—
299C42—44no—
300C16—2681no—
301C16—1342no—
302C23—2682no—
303C23—5361no—
304C23—5361no—
305C23—5361no—
306C23—5361no—
307C23—5361no—
308C23—5361no—
309C23C2×#1176161311yescharacteristic
310Q8—24no—
311Q8—42no—
312Q8—42no—
313Q8—42no—
314D4—2682no—
315D4—2682no—
316D4—5361no—
317D4—5361no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—5361no—
330C2×C4—5361no—
331C2×C4—5361no—
332C2×C4#1452398612yescomplemented
333C2×C4C4×C22⋊D6711yescharacteristic
334C2×C4#1452401211yescharacteristic
335C2×C4#1452401711yescharacteristic
336C2×C4#1452406311yescharacteristic
337C2×C4#1452406811yescharacteristic
338C2×C4#1452406911yescharacteristic
339C8—41no—
340C8C2×(C2×D67)⋊C411yescharacteristic
341C8C4×C22⋊D6711yescharacteristic
342C8#1452401711yescharacteristic
343C8#1452402611yescharacteristic
344C22—5362no—
345C22—5362no—
346C22—5362no—
347C22—10721no—
348C22—10721no—
349C22—10721no—
350C22—10721no—
351C22C4×#1176161311yescharacteristic
352C22#1686353011yescharacteristic
353C22#1686359111yescharacteristic
354C22C2×#1452398611yescharacteristic
355C22#1686382611yescharacteristic
356C22#1686383111yescharacteristic
357C22#1686383711yescharacteristic
358C4—22no—
359C4—44no—
360C4—41no—
361C4—82no—
362C4—10721no—
363C4#1686327511yescharacteristic
364C4#1686350611yescharacteristic
365C4#1686354111yescharacteristic
366C4#1686361111yescharacteristic
367C2—10722no—
368C2#2256055711yescharacteristic
369C2C4×#1452398611yescharacteristic
370C2#2256207111yescharacteristic
371C2#2256212611yescharacteristic
372C2#2256213211yescharacteristic
373C2#2256224911yescharacteristic
374C2#2256227111yescharacteristic
subgroupG/H
centerC23C2×#11761613
commutatorC2×C8×C67C22×C4
FrattiniC22×C8C22×D67
FittingC2×C4×Q16×C67C2
socleC23×C67C2×D8
prime radicalC22×D536C22
cosocleC22×C8×C67C23
hypercenterC2×C4×Q16D67
hypocenterC67#13013
LCSG ▷ C2×C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
2. C2×#16863200C2semidirect
5. #22558300C2semidirect
6. #22558287C2semidirect
7. C2×C4×Q16×C67C2semidirect
11. C2×#14523986C4semidirect
98. Q16×C67C22⋊C4semidirect
137. C2×C4×C67SD32semidirect
189. C2×C4×Q16D67semidirect
195. C67#13013semidirect
276. Q16(C2×D67)⋊C4semidirect
332. C2×C4#14523986semidirect

Maximal subgroups

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

subgroupindex
C22×#145239862
C2×#168632002
C2×#168631962
#225610752
#225583002
#225582872
C2×C4×Q16×C672
#1301367