Finite Group Databasebeta

← #30766142

Subgroups — #30766142

Order 17152.

452 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 115 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
2C2×#16863122C211yesmaximal, characteristic
3#22558239C212yesmaximal, complemented
4#22558236C212yesmaximal, complemented
5C67×#2627C211yesmaximal, characteristic
6C22×#11761612—21no—
7C2×C22⋊(C67⋊C8)—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—24no—
24C2×#11761612—22no—
25C2×#11761612—41no—
26C2×#11761612—42no—
27C22×C67⋊C8—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—41no—
42C2×C4⋊C4×C67—41no—
43C2×C4⋊C4×C67C2×C411yescharacteristic
44C2×C4⋊C4×C67C2×C411yescharacteristic
45C2×C4⋊C4×C67D411yescharacteristic
46C2×C4⋊C4×C67D411yescharacteristic
47C2×C4⋊C4×C67C2311yescharacteristic
48C2×C22⋊C4×C67—21no—
49C2×C22⋊C4×C67—21no—
50#95×C67—21no—
51#95×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—
61C2×C67⋊C8—44no—
62C2×C67⋊C8—42no—
63C2×C67⋊C8—42no—
64C2×C67⋊C8—82no—
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—81no—
80C22×C4×C67—81no—
81C22×C4×C67C22⋊C411yescharacteristic
82C22×C4×C67C22⋊C411yescharacteristic
83C22×C4×C67C22⋊C411yescharacteristic
84C22×C4×C67SD1612yes—
85C22×C4×C67Q1612yes—
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—81no—
96C4⋊C4×C67—81no—
97C4⋊C4×C67—81no—
98C22⋊C4×C67—41no—
99C22⋊C4×C67—41no—
100C22⋊C4×C67—42no—
101C23×C67—41no—
102C23×C67—41no—
103C23×C67#9911yescharacteristic
104C23×C67#9911yescharacteristic
105C23×C67#10312yes—
106C23×C67#10412yes—
107C23×C67C2×C22⋊C411yescharacteristic
108Q8×C67—82no—
109C2×C4×C67—22no—
110C2×C4×C67—22no—
111C2×C4×C67—22no—
112C2×C4×C67—22no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—42no—
122C2×C4×C67—41no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67—41no—
130C2×C4×C67—41no—
131C2×C4×C67—41no—
132C2×C4×C67—42no—
133C2×C4×C67—41no—
134C2×C4×C67—41no—
135C2×C4×C67—41no—
136C2×C4×C67—41no—
137C2×C4×C67—81no—
138C2×C4×C67—81no—
139C2×C4×C67—81no—
140C2×C4×C67#10312yes—
141C2×C4×C67C2×SD1611yescharacteristic
142C2×C4×C67C2×Q1611yescharacteristic
143C67⋊C8—44no—
144C67⋊C8—82no—
145C22×C67—21no—
146C22×C67—21no—
147C22×C67—21no—
148C22×C67—21no—
149C22×C67—21no—
150C22×C67—21no—
151C22×C67—42no—
152C22×C67—42no—
153C22×C67#32812yes—
154C22×C67#32812yes—
155C22×C67C2×#9911yescharacteristic
156C22×C67C2×#10311yescharacteristic
157C22×C67C2×#10411yescharacteristic
158C4×C67—21no—
159C4×C67—22no—
160C4×C67—21no—
161C4×C67—41no—
162C4×C67—41no—
163C4×C67—41no—
164C4×C67—41no—
165C4×C67—81no—
166C4×C67—81no—
167C4×C67—81no—
168C4×C67—81no—
169#11722—671nomaximal, Sylow
170C2×C67—21no—
171C2×C67—22no—
172C2×C67—22no—
173C2×C67—21no—
174C2×C67C2×#32811yescharacteristic
175C2×C67#212111yescharacteristic
176C2×C67#213311yescharacteristic
177#2627D6711yescharacteristic
178C2×#343—671no—
179C2×#324—671no—
180#1354—672no—
181#1353—672no—
182C67#1172211yescharacteristic, complemented, Sylow
183C22×M16—1341no—
184C2×C22⋊Q8Dic6711yescharacteristic
185C22×C4⋊C4C2×D6711yescharacteristic
186C2×C22⋊C8—1341no—
187#400Dic6711yescharacteristic
188#400C2×D6711yescharacteristic
189#398—21no—
190#398—21no—
191#395Dic6711yescharacteristic
192#395C2×D6711yescharacteristic
193#385—21no—
194#381—21no—
195#380Dic6711yescharacteristic
196#343—672no—
197#343—1341no—
198#324—672no—
199#324—1341no—
200C22×Q8C2×Dic6711yescharacteristic
201C23×C4C2×Dic6711yescharacteristic
202C23×C4C22⋊D6711yescharacteristic
203C23×C4C22⋊D6711yescharacteristic
204C2×M16—2681no—
205C2×M16—2682no—
206C2×M16—1344no—
207C2×M16—1342no—
208C22×C8—2681no—
209C22×C8—2681no—
210C22⋊Q8—41no—
211C22⋊Q8—41no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4C2×Dic6711yescharacteristic
219C2×C4⋊C4C2×Dic6711yescharacteristic
220C2×C4⋊C4C22⋊D6711yescharacteristic
221C2×C4⋊C4C22⋊D6711yescharacteristic
222C2×C4⋊C4C22×D6711yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C22⋊C8—2681no—
226C22⋊C8—1342no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95C2×Dic6711yescharacteristic
234#95C2×Dic6711yescharacteristic
235C24C22⋊Dic6711yescharacteristic
236C2×Q8—41no—
237C2×Q8—81no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4C22⋊Dic6711yescharacteristic
252C22×C4C22⋊Dic6711yescharacteristic
253C22×C4C22⋊Dic6711yescharacteristic
254C22×C4#1176161512yes—
255C22×C4#1176161812yes—
256C22×C4C22×Dic6711yescharacteristic
257C22×C4C2×C22⋊D6711yescharacteristic
258C22×C4C2×C22⋊D6711yescharacteristic
259M16—2684no—
260M16—2682no—
261M16—1344no—
262C2×C8—2682no—
263C2×C8—2684no—
264C2×C8—2682no—
265C2×C8—5362no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—81no—
273C4⋊C4—81no—
274C4⋊C4—81no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—42no—
278C23—41no—
279C23—41no—
280C23#1452395811yescharacteristic
281C23#1452395811yescharacteristic
282C23#1452396712yes—
283C23#1452397012yes—
284C23C2×C22⋊Dic6711yescharacteristic
285Q8—82no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4#1452396712yes—
318C2×C4C2×#1176161511yescharacteristic
319C2×C4C2×#1176161811yescharacteristic
320C8—2684no—
321C8—5362no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—42no—
329C22—42no—
330C22#1686313312yes—
331C22#1686313312yes—
332C22C2×#1452395811yescharacteristic
333C22C2×#1452396711yescharacteristic
334C22C2×#1452397011yescharacteristic
335C4—21no—
336C4—22no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—81no—
346C2—21no—
347C2—21no—
348C2—22no—
349C2—22no—
350C2C2×#1686313311yescharacteristic
351C2#2256185511yescharacteristic
352C2#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#11722
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
182. C67#11722semidirect

Maximal subgroups

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

subgroupindex
C2×#168631652
C2×#168631222
#225582392
#225582362
C67×#26272
#1172267