Finite Group Databasebeta

← #30766579

Subgroups — #30766579

Order 17152.

386 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 87 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#22560289C211yesmaximal, characteristic
2#22558539C211yesmaximal, characteristic
3#22558239C212yesmaximal, complemented
4#22558236C212yesmaximal, complemented
5C67×#2627C211yesmaximal, characteristic
6#16863360—21no—
7#16863360—21no—
8#16863165—21no—
9#16863165C2212yes—
10#16863122—21no—
11#16863122C2212yes—
12C2×C22⋊Q8×C67C411yescharacteristic
13C22×C4⋊C4×C67C2211yescharacteristic
14#400×C67C411yescharacteristic
15#400×C67C2211yescharacteristic
16#398×C67—21no—
17#398×C67—21no—
18#395×C67C411yescharacteristic
19#395×C67C2211yescharacteristic
20#385×C67—21no—
21#381×C67—21no—
22#380×C67C411yescharacteristic
23C2×#11761612—22no—
24C2×#11761612—41no—
25C22⋊(C67⋊C8)—22no—
26C22⋊(C67⋊C8)—41no—
27C22⋊(C67⋊C8)—41no—
28C22⋊(C67⋊C8)—41no—
29C22×Q8×C67C2×C411yescharacteristic
30C23×C4×C67C2×C411yescharacteristic
31C23×C4×C67D411yescharacteristic
32C23×C4×C67D411yescharacteristic
33C22⋊Q8×C67—41no—
34C22⋊Q8×C67—41no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67—41no—
40C2×C4⋊C4×C67—41no—
41C2×C4⋊C4×C67C2×C411yescharacteristic
42C2×C4⋊C4×C67C2×C411yescharacteristic
43C2×C4⋊C4×C67D411yescharacteristic
44C2×C4⋊C4×C67D411yescharacteristic
45C2×C4⋊C4×C67C2311yescharacteristic
46C2×C22⋊C4×C67—21no—
47C2×C22⋊C4×C67—21no—
48#95×C67—21no—
49#95×C67—21no—
50#95×C67—21no—
51#95×C67—21no—
52#95×C67—21no—
53#95×C67—21no—
54#95×C67C2×C411yescharacteristic
55#95×C67C2×C411yescharacteristic
56#11761612—42no—
57#11761612—42no—
58#11761612—82no—
59C2×C67⋊C8—42no—
60C2×C67⋊C8—42no—
61C24×C67C22⋊C411yescharacteristic
62C2×Q8×C67—41no—
63C2×Q8×C67—81no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—22no—
71C22×C4×C67—21no—
72C22×C4×C67—21no—
73C22×C4×C67—21no—
74C22×C4×C67—21no—
75C22×C4×C67—41no—
76C22×C4×C67—81no—
77C22×C4×C67—81no—
78C22×C4×C67C22⋊C411yescharacteristic
79C22×C4×C67C22⋊C411yescharacteristic
80C22×C4×C67C22⋊C411yescharacteristic
81C22×C4×C67C22×C411yescharacteristic
82C22×C4×C67C2×D411yescharacteristic
83C22×C4×C67C2×D411yescharacteristic
84C4⋊C4×C67—41no—
85C4⋊C4×C67—41no—
86C4⋊C4×C67—41no—
87C4⋊C4×C67—41no—
88C4⋊C4×C67—41no—
89C4⋊C4×C67—42no—
90C4⋊C4×C67—81no—
91C4⋊C4×C67—81no—
92C4⋊C4×C67—81no—
93C22⋊C4×C67—42no—
94C22⋊C4×C67—41no—
95C22⋊C4×C67—41no—
96C23×C67—21no—
97C23×C67—21no—
98C23×C67—41no—
99C23×C67—41no—
100C23×C67#9911yescharacteristic
101C23×C67#9911yescharacteristic
102C23×C67C2×C22⋊C411yescharacteristic
103Q8×C67—82no—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—42no—
114C2×C4×C67—42no—
115C2×C4×C67—42no—
116C2×C4×C67—41no—
117C2×C4×C67—41no—
118C2×C4×C67—41no—
119C2×C4×C67—41no—
120C2×C4×C67—41no—
121C2×C4×C67—41no—
122C2×C4×C67—41no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—42no—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67—41no—
130C2×C4×C67—41no—
131C2×C4×C67—81no—
132C2×C4×C67—81no—
133C2×C4×C67—81no—
134C2×C4×C67#13712yes—
135C67⋊C8—82no—
136C67⋊C8—82no—
137C22×C67—21no—
138C22×C67—21no—
139C22×C67—21no—
140C22×C67—21no—
141C22×C67—21no—
142C22×C67—21no—
143C22×C67—21no—
144C22×C67—21no—
145C22×C67—42no—
146C22×C67—42no—
147C22×C67C2×#9911yescharacteristic
148C22×C67#41911yescharacteristic
149C22×C67#42111yescharacteristic
150C4×C67—22no—
151C4×C67—41no—
152C4×C67—41no—
153C4×C67—41no—
154C4×C67—41no—
155C4×C67—41no—
156C4×C67—81no—
157C4×C67—81no—
158C4×C67—81no—
159C4×C67—81no—
160#11817—671nomaximal, Sylow
161C2×C67—21no—
162C2×C67—21no—
163C2×C67—41no—
164C2×C67—41no—
165C2×C67#151111yescharacteristic
166C2×C67#212111yescharacteristic
167C2×C67#213311yescharacteristic
168#2627D6711yescharacteristic
169#1821—671no—
170#1463—671no—
171#1354—672no—
172#1353—672no—
173C67#1181711yescharacteristic, complemented, Sylow
174C2×C22⋊Q8Dic6711yescharacteristic
175C22×C4⋊C4C2×D6711yescharacteristic
176#407—1341no—
177#407—1341no—
178#400Dic6711yescharacteristic
179#400C2×D6711yescharacteristic
180#398—21no—
181#398—21no—
182#395Dic6711yescharacteristic
183#395C2×D6711yescharacteristic
184#385—21no—
185#381—21no—
186#380Dic6711yescharacteristic
187#343—672no—
188#343—1341no—
189#324—672no—
190#324—1341no—
191C22×Q8C2×Dic6711yescharacteristic
192C23×C4C2×Dic6711yescharacteristic
193C23×C4C22⋊D6711yescharacteristic
194C23×C4C22⋊D6711yescharacteristic
195C2×M16—2681no—
196C2×M16—1342no—
197C22⋊Q8—41no—
198C22⋊Q8—41no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C2×Dic6711yescharacteristic
206C2×C4⋊C4C2×Dic6711yescharacteristic
207C2×C4⋊C4C22⋊D6711yescharacteristic
208C2×C4⋊C4C22⋊D6711yescharacteristic
209C2×C4⋊C4C22×D6711yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C22⋊C8—2681no—
213C22⋊C8—2681no—
214C22⋊C8—2681no—
215C22⋊C8—1342no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95C2×Dic6711yescharacteristic
223#95C2×Dic6711yescharacteristic
224C24C22⋊Dic6711yescharacteristic
225C2×Q8—41no—
226C2×Q8—81no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4C22⋊Dic6711yescharacteristic
242C22×C4C22⋊Dic6711yescharacteristic
243C22×C4C22⋊Dic6711yescharacteristic
244C22×C4C22×Dic6711yescharacteristic
245C22×C4C2×C22⋊D6711yescharacteristic
246C22×C4C2×C22⋊D6711yescharacteristic
247M16—2682no—
248M16—2682no—
249M16—5362no—
250C2×C8—2682no—
251C2×C8—2682no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—81no—
259C4⋊C4—81no—
260C4⋊C4—81no—
261C22⋊C4—42no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23#1452395811yescharacteristic
269C23#1452395811yescharacteristic
270C23C2×C22⋊Dic6711yescharacteristic
271Q8—82no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4#1452407412yes—
303C8—5362no—
304C8—5362no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—42no—
314C22—42no—
315C22C2×#1452395811yescharacteristic
316C22#1686340511yescharacteristic
317C22#1686341311yescharacteristic
318C4—22no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C2—21no—
329C2—21no—
330C2—41no—
331C2—41no—
332C2#2255874211yescharacteristic
333C2#2256185511yescharacteristic
334C2#2256189811yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2627C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2627D67
hypocenterC67#11817
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
3. #22558239C2semidirect
4. #22558236C2semidirect
173. C67#11817semidirect

Maximal subgroups

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

subgroupindex
#225602892
#225585392
#225582392
#225582362
C67×#26272
#1181767