Finite Group Databasebeta

← #30766362

Subgroups — #30766362

Order 17152.

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

normal subgroups only

This group has 462 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
1C2×#16863165C211yesmaximal, characteristic
2#22558539C211yesmaximal, characteristic
3#22558239C212yesmaximal, complemented
4#22558215C212yesmaximal, complemented
5C67×#2583C211yesmaximal, characteristic
6C22×#11761612—21no—
7#16863360—21no—
8#16863165—21no—
9#16863165C2212yes—
10#16863122—21no—
11#16863122C2212yes—
12C2×C22⋊Q8×C67C411yescharacteristic
13C22×C4⋊C4×C67C2211yescharacteristic
14#398×C67C411yescharacteristic
15#398×C67C2211yescharacteristic
16#395×C67C411yescharacteristic
17#395×C67C2211yescharacteristic
18#389×C67—21no—
19#389×C67—21no—
20#385×C67—21no—
21#385×C67—21no—
22C4×C22⋊C4×C67C411yescharacteristic
23C2×#11761612—24no—
24C2×#11761612—22no—
25C2×#11761612—41no—
26C2×#11761612—42no—
27C2×#11761612—41no—
28C22×C67⋊C8—41no—
29C22⋊(C67⋊C8)—22no—
30C22⋊(C67⋊C8)—41no—
31C22×Q8×C67C2×C411yescharacteristic
32C23×C4×C67C2×C411yescharacteristic
33C23×C4×C67D411yescharacteristic
34C23×C4×C67D411yescharacteristic
35C22⋊Q8×C67—41no—
36C22⋊Q8×C67—41no—
37C2×C4⋊C4×C67—21no—
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×C67C2×C411yescharacteristic
45C2×C4⋊C4×C67C2×C411yescharacteristic
46C2×C4⋊C4×C67D411yescharacteristic
47C2×C4⋊C4×C67D411yescharacteristic
48C2×C4⋊C4×C67C2311yescharacteristic
49C2×C22⋊C4×C67—21no—
50C2×C22⋊C4×C67—21no—
51C2×C42×C67—21no—
52#95×C67—21no—
53#95×C67—21no—
54#95×C67—21no—
55#95×C67—21no—
56#95×C67C2×C411yescharacteristic
57#95×C67C2×C411yescharacteristic
58#11761612—24no—
59#11761612—44no—
60#11761612—42no—
61#11761612—82no—
62C2×C67⋊C8—44no—
63C2×C67⋊C8—42no—
64C2×C67⋊C8—42no—
65C24×C67C22⋊C411yescharacteristic
66C2×Q8×C67—41no—
67C2×Q8×C67—81no—
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—21no—
76C22×C4×C67—21no—
77C22×C4×C67—21no—
78C22×C4×C67—21no—
79C22×C4×C67—21no—
80C22×C4×C67—21no—
81C22×C4×C67—81no—
82C22×C4×C67—81no—
83C22×C4×C67C22⋊C411yescharacteristic
84C22×C4×C67C22⋊C411yescharacteristic
85C22×C4×C67C22⋊C411yescharacteristic
86C22×C4×C67C22×C411yescharacteristic
87C22×C4×C67C2×D411yescharacteristic
88C22×C4×C67C2×D411yescharacteristic
89C4⋊C4×C67—41no—
90C4⋊C4×C67—41no—
91C4⋊C4×C67—41no—
92C4⋊C4×C67—41no—
93C4⋊C4×C67—41no—
94C4⋊C4×C67—42no—
95C4⋊C4×C67—42no—
96C4⋊C4×C67—41no—
97C4⋊C4×C67—41no—
98C4⋊C4×C67—81no—
99C4⋊C4×C67—81no—
100C22⋊C4×C67—24no—
101C22⋊C4×C67—41no—
102C22⋊C4×C67—41no—
103C42×C67—81no—
104C23×C67—22no—
105C23×C67—21no—
106C23×C67—21no—
107C23×C67—41no—
108C23×C67#9911yescharacteristic
109C23×C67#9911yescharacteristic
110C23×C67C2×C22⋊C411yescharacteristic
111Q8×C67—82no—
112C2×C4×C67—22no—
113C2×C4×C67—22no—
114C2×C4×C67—22no—
115C2×C4×C67—22no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—22no—
121C2×C4×C67—22no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—21no—
125C2×C4×C67—21no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67—42no—
129C2×C4×C67—41no—
130C2×C4×C67—41no—
131C2×C4×C67—41no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—41no—
135C2×C4×C67—41no—
136C2×C4×C67—41no—
137C2×C4×C67—41no—
138C2×C4×C67—41no—
139C2×C4×C67—41no—
140C2×C4×C67—41no—
141C2×C4×C67—81no—
142C2×C4×C67—81no—
143C2×C4×C67—81no—
144C2×C4×C67#10412yes—
145C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C67#13711yescharacteristic
147C67⋊C8—44no—
148C67⋊C8—82no—
149C22×C67—21no—
150C22×C67—21no—
151C22×C67—21no—
152C22×C67—21no—
153C22×C67—21no—
154C22×C67—21no—
155C22×C67—21no—
156C22×C67—42no—
157C22×C67—42no—
158C22×C67#35612yes—
159C22×C67C2×#9911yescharacteristic
160C22×C67#41711yescharacteristic
161C22×C67C2×#10411yescharacteristic
162C4×C67—21no—
163C4×C67—22no—
164C4×C67—21no—
165C4×C67—42no—
166C4×C67—42no—
167C4×C67—41no—
168C4×C67—41no—
169C4×C67—41no—
170C4×C67—41no—
171C4×C67—81no—
172C4×C67—81no—
173#11768—671nomaximal, Sylow
174C2×C67—21no—
175C2×C67—21no—
176C2×C67—22no—
177C2×C67—41no—
178C2×C67#150411yescharacteristic
179C2×C67#212111yescharacteristic
180C2×C67C2×#35611yescharacteristic
181#2583D6711yescharacteristic
182C2×#343—671no—
183#1463—671no—
184#1354—672no—
185#1346—672no—
186C67#1176811yescharacteristic, complemented, Sylow
187C22×M16—1341no—
188C2×C22⋊Q8Dic6711yescharacteristic
189C22×C4⋊C4C2×D6711yescharacteristic
190#407—1341no—
191#398Dic6711yescharacteristic
192#398C2×D6711yescharacteristic
193#395Dic6711yescharacteristic
194#395C2×D6711yescharacteristic
195#389—21no—
196#389—21no—
197#385—21no—
198#385—21no—
199C4×C22⋊C4Dic6711yescharacteristic
200#343—672no—
201#343—1341no—
202#324—672no—
203#324—1341no—
204C22×Q8C2×Dic6711yescharacteristic
205C23×C4C2×Dic6711yescharacteristic
206C23×C4C22⋊D6711yescharacteristic
207C23×C4C22⋊D6711yescharacteristic
208C2×M16—2681no—
209C2×M16—2682no—
210C2×M16—2681no—
211C2×M16—1344no—
212C2×M16—1342no—
213C22×C8—2681no—
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4C2×Dic6711yescharacteristic
224C2×C4⋊C4C2×Dic6711yescharacteristic
225C2×C4⋊C4C22⋊D6711yescharacteristic
226C2×C4⋊C4C22⋊D6711yescharacteristic
227C2×C4⋊C4C22×D6711yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—21no—
231C22⋊C8—2681no—
232C22⋊C8—1342no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95C2×Dic6711yescharacteristic
238#95C2×Dic6711yescharacteristic
239C24C22⋊Dic6711yescharacteristic
240C2×Q8—41no—
241C2×Q8—81no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4C22⋊Dic6711yescharacteristic
258C22×C4C22⋊Dic6711yescharacteristic
259C22×C4C22⋊Dic6711yescharacteristic
260C22×C4C22×Dic6711yescharacteristic
261C22×C4C2×C22⋊D6711yescharacteristic
262C22×C4C2×C22⋊D6711yescharacteristic
263M16—2684no—
264M16—2682no—
265M16—5362no—
266M16—1344no—
267C2×C8—2684no—
268C2×C8—2682no—
269C2×C8—2682no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—81no—
280C4⋊C4—81no—
281C22⋊C4—24no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C42—81no—
285C23—21no—
286C23—21no—
287C23—22no—
288C23—41no—
289C23#1452395811yescharacteristic
290C23#1452395811yescharacteristic
291C23C2×C22⋊Dic6711yescharacteristic
292Q8—82no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4#1452397012yes—
326C2×C4#1452406811yescharacteristic
327C2×C4#1452407311yescharacteristic
328C8—2684no—
329C8—5362no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—42no—
338C22—42no—
339C22#1686319512yes—
340C22C2×#1452395811yescharacteristic
341C22#1686339711yescharacteristic
342C22C2×#1452397011yescharacteristic
343C4—21no—
344C4—22no—
345C4—21no—
346C4—42no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—81no—
353C4—81no—
354C2—21no—
355C2—21no—
356C2—22no—
357C2—41no—
358C2#2255871311yescharacteristic
359C2#2256185511yescharacteristic
360C2C2×#1686319511yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2583C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2583D67
hypocenterC67#11768
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2583

Complements

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

NH ≅ G/Ntype
3. #22558239C2semidirect
4. #22558215C2semidirect
186. C67#11768semidirect

Maximal subgroups

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

subgroupindex
C2×#168631652
#225585392
#225582392
#225582152
C67×#25832
#1176867