Finite Group Databasebeta

← #30765294

Subgroups — #30765294

Order 17152.

442 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22558550C211yesmaximal, characteristic
2#22558547C211yesmaximal, characteristic
3#22558157C212yesmaximal, complemented
4#22558139C212yesmaximal, complemented
5C67×#2322C211yesmaximal, characteristic
6#16863453—21no—
7#16863360—21no—
8#16863122—21no—
9#16863120—21no—
10#16863120C2212yes—
11#16863120C2212yes—
12C2×#123×C67C411yescharacteristic
13C2×C4×D4×C67C2211yescharacteristic
14C22×C4⋊C4×C67C411yescharacteristic
15#389×C67—21no—
16#388×C67—21no—
17#387×C67—21no—
18#386×C67C411yescharacteristic
19#385×C67—21no—
20#385×C67C2211yescharacteristic
21#381×C67C2211yescharacteristic
22C4×C22⋊C4×C67C411yescharacteristic
23C2×#11761612—41no—
24C4⋊(C67⋊C8)—41no—
25C22⋊(C67⋊C8)—22no—
26C22⋊(C67⋊C8)—22no—
27C22⋊(C67⋊C8)—41no—
28#14523954—41no—
29C22×D4×C67C2×C411yescharacteristic
30C23×C4×C67C2×C411yescharacteristic
31C23×C4×C67C2×C411yescharacteristic
32C23×C4×C67C2×C411yescharacteristic
33C23×C4×C67D411yescharacteristic
34#123×C67—22no—
35#123×C67—22no—
36C4×D4×C67—41no—
37C4×D4×C67—41no—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67—21no—
41C2×C4⋊C4×C67—21no—
42C2×C4⋊C4×C67—41no—
43C2×C4⋊C4×C67—41no—
44C2×C4⋊C4×C67D411yescharacteristic
45C2×C22⋊C4×C67—21no—
46C2×C22⋊C4×C67—21no—
47C2×C22⋊C4×C67C2×C411yescharacteristic
48C2×C22⋊C4×C67C2×C411yescharacteristic
49C2×C22⋊C4×C67D411yescharacteristic
50C2×C22⋊C4×C67C2311yescharacteristic
51C2×C42×C67—21no—
52C2×C42×C67D411yescharacteristic
53#95×C67—21no—
54#95×C67—21no—
55#95×C67—21no—
56#95×C67—21no—
57#95×C67—21no—
58#11761612—82no—
59C2×C67⋊C8—42no—
60C2×C67⋊C8—42no—
61C24×C67C22⋊C411yescharacteristic
62C24×C67C22×C411yescharacteristic
63C2×D4×C67—21no—
64C2×D4×C67—21no—
65C2×D4×C67—41no—
66C2×D4×C67—41no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—21no—
71C22×C4×C67—21no—
72C22×C4×C67—21no—
73C22×C4×C67—21no—
74C22×C4×C67—21no—
75C22×C4×C67—21no—
76C22×C4×C67—41no—
77C22×C4×C67—41no—
78C22×C4×C67—41no—
79C22×C4×C67—81no—
80C22×C4×C67—81no—
81C22×C4×C67—81no—
82C22×C4×C67C22⋊C411yescharacteristic
83C22×C4×C67C22⋊C411yescharacteristic
84C22×C4×C67C22⋊C411yescharacteristic
85C22×C4×C67M1612yescomplemented
86C22×C4×C67C2×D411yescharacteristic
87C22×C4×C67C2×D411yescharacteristic
88C4⋊C4×C67—24no—
89C4⋊C4×C67—41no—
90C4⋊C4×C67—42no—
91C4⋊C4×C67—41no—
92C4⋊C4×C67—41no—
93C4⋊C4×C67—81no—
94C22⋊C4×C67—24no—
95C22⋊C4×C67—21no—
96C22⋊C4×C67—41no—
97C22⋊C4×C67—41no—
98C22⋊C4×C67—41no—
99C22⋊C4×C67—81no—
100C22⋊C4×C67M1612yescomplemented
101C42×C67—21no—
102C42×C67—21no—
103C42×C67—81no—
104C23×C67—21no—
105C23×C67—21no—
106C23×C67—21no—
107C23×C67—21no—
108C23×C67—21no—
109C23×C67—41no—
110C23×C67—41no—
111C23×C67—41no—
112C23×C67—41no—
113C23×C67C2×C22⋊C411yescharacteristic
114C23×C67C2×M1611yescharacteristic
115C23×C67C2×M1611yescharacteristic
116D4×C67—42no—
117D4×C67—42no—
118C2×C4×C67—22no—
119C2×C4×C67—22no—
120C2×C4×C67—21no—
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—42no—
128C2×C4×C67—41no—
129C2×C4×C67—41no—
130C2×C4×C67—42no—
131C2×C4×C67—41no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—41no—
135C2×C4×C67—42no—
136C2×C4×C67—41no—
137C2×C4×C67—41no—
138C2×C4×C67—41no—
139C2×C4×C67—41no—
140C2×C4×C67—41no—
141C2×C4×C67—41no—
142C2×C4×C67—41no—
143C2×C4×C67—81no—
144C2×C4×C67—81no—
145C2×C4×C67—81no—
146C2×C4×C67—82no—
147C2×C4×C67#9911yescharacteristic
148C2×C4×C67#9911yescharacteristic
149C67⋊C8—82no—
150C67⋊C8—82no—
151C22×C67—21no—
152C22×C67—21no—
153C22×C67—21no—
154C22×C67—21no—
155C22×C67—41no—
156C22×C67—41no—
157C22×C67—42no—
158C22×C67—42no—
159C22×C67—41no—
160C22×C67—41no—
161C22×C67—41no—
162C22×C67—41no—
163C22×C67—41no—
164C22×C67—81no—
165C22×C67#40711yescharacteristic
166C22×C67C2×#9911yescharacteristic
167C22×C67#41311yescharacteristic
168C4×C67—21no—
169C4×C67—21no—
170C4×C67—42no—
171C4×C67—41no—
172C4×C67—41no—
173C4×C67—41no—
174C4×C67—41no—
175C4×C67—42no—
176C4×C67—81no—
177C4×C67—81no—
178#11528—671nomaximal, Sylow
179C2×C67—21no—
180C2×C67—21no—
181C2×C67—41no—
182C2×C67—41no—
183C2×C67—81no—
184C2×C67#146511yescharacteristic
185C2×C67#212111yescharacteristic
186C2×C67#212311yescharacteristic
187#2322D6711yescharacteristic
188#1466—671no—
189#1465—671no—
190#1326—672no—
191#1317—672no—
192C67#1152811yescharacteristic, complemented, Sylow
193C2×#123Dic6711yescharacteristic
194C2×C4×D4C2×D6711yescharacteristic
195C22×C4⋊C4Dic6711yescharacteristic
196#432—1341no—
197#407—1341no—
198#389—21no—
199#388—21no—
200#387—21no—
201#386Dic6711yescharacteristic
202#385—21no—
203#385C2×D6711yescharacteristic
204#381C2×D6711yescharacteristic
205C4×C22⋊C4Dic6711yescharacteristic
206#324—1341no—
207#323—672no—
208#323—672no—
209#323—1341no—
210C22×D4C2×Dic6711yescharacteristic
211C23×C4C2×Dic6711yescharacteristic
212C23×C4C2×Dic6711yescharacteristic
213C23×C4C2×Dic6711yescharacteristic
214C23×C4C22⋊D6711yescharacteristic
215C2×M16—2681no—
216#123—22no—
217#123—22no—
218C4×D4—41no—
219C4×D4—41no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4C22⋊D6711yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4C2×Dic6711yescharacteristic
230C2×C22⋊C4C2×Dic6711yescharacteristic
231C2×C22⋊C4C22⋊D6711yescharacteristic
232C2×C22⋊C4C22×D6711yescharacteristic
233C2×C42—21no—
234C2×C42C22⋊D6711yescharacteristic
235C4⋊C8—2681no—
236C22⋊C8—2681no—
237C22⋊C8—1342no—
238C22⋊C8—1342no—
239#97—2681no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245C24C22⋊Dic6711yescharacteristic
246C24C22×Dic6711yescharacteristic
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4C22⋊Dic6711yescharacteristic
267C22×C4C22⋊Dic6711yescharacteristic
268C22×C4C22⋊Dic6711yescharacteristic
269C22×C4#1176161212yescomplemented
270C22×C4C2×C22⋊D6711yescharacteristic
271C22×C4C2×C22⋊D6711yescharacteristic
272M16—5362no—
273C2×C8—2682no—
274C2×C8—2682no—
275C4⋊C4—24no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—81no—
281C22⋊C4—24no—
282C22⋊C4—21no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—81no—
287C22⋊C4#1176161212yescomplemented
288C42—21no—
289C42—21no—
290C42—81no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23C2×C22⋊Dic6711yescharacteristic
301C23C2×#1176161211yescharacteristic
302C23C2×#1176161211yescharacteristic
303D4—42no—
304D4—42no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4#1452395811yescharacteristic
335C2×C4#1452395811yescharacteristic
336C8—5362no—
337C8—5362no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—42no—
345C22—42no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—81no—
352C22#1686336011yescharacteristic
353C22C2×#1452395811yescharacteristic
354C22#1686338011yescharacteristic
355C4—21no—
356C4—21no—
357C4—42no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—42no—
363C4—81no—
364C4—81no—
365C2—21no—
366C2—21no—
367C2—41no—
368C2—41no—
369C2—81no—
370C2#2255854711yescharacteristic
371C2#2256185511yescharacteristic
372C2#2256186311yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC24×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2322C2
socleC22×C67C2×#99
prime radicalC22×D4×C67C2×C4
cosocleC2×C22⋊C4×C67C23
hypercenter#2322D67
hypocenterC67#11528
LCSG ▷ C24×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2322

Complements

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

NH ≅ G/Ntype
3. #22558157C2semidirect
4. #22558139C2semidirect
85. C22×C4×C67M16semidirect
100. C22⋊C4×C67M16semidirect
192. C67#11528semidirect
269. C22×C4#11761612semidirect
287. C22⋊C4#11761612semidirect

Maximal subgroups

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

subgroupindex
#225585502
#225585472
#225581572
#225581392
C67×#23222
#1152867