Finite Group Databasebeta

← #34001368

Subgroups — #34001368

Order 18688.

568 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22625821C211yesmaximal, characteristic
2C2×#17989453C211yesmaximal, characteristic
3#22624749C212yesmaximal, complemented
4#22624749C212yesmaximal, complemented
5#22612415C211yesmaximal, characteristic
6C23×#10596833—21no—
7C2×C22⋊#10596833—21no—
8#17989453—21no—
9#17989453—21no—
10#17989453C2212yes—
11#17989453C2212yes—
12C2×#14556550C411yescharacteristic
13C22×C4×Dic73C2211yescharacteristic
14#17988486C411yescharacteristic
15#17988486C411yescharacteristic
16#17988486C2211yescharacteristic
17#17988486C2211yescharacteristic
18#17988462—21no—
19#17988462—21no—
20C4×C4⋊Dic73—21no—
21C4×C4⋊Dic73—21no—
22C2×#117×C73C411yescharacteristic
23C22×#10596833—24no—
24C22×#10596833—22no—
25C22×#10596833—22no—
26C22×#10596833—42no—
27C22×#10596833—42no—
28C22⋊#10596833—42no—
29C23×Dic73D411yescharacteristic
30C23×Dic73D411yescharacteristic
31#14556550—42no—
32C2×C4⋊Dic73—21no—
33C2×C4⋊Dic73—21no—
34C2×C4⋊Dic73—21no—
35C2×C4⋊Dic73—21no—
36C2×C4⋊Dic73C2×C411yescharacteristic
37C2×C4⋊Dic73C2×C411yescharacteristic
38C2×C22⋊Dic73—21no—
39C2×C4×Dic73—21no—
40C2×C4×Dic73—42no—
41C2×C4×Dic73C2×C411yescharacteristic
42C2×C4×Dic73D411yescharacteristic
43C2×C4×Dic73D411yescharacteristic
44C2×C4×Dic73C2311yescharacteristic
45#14556498—21no—
46#14556498—21no—
47#14556498—21no—
48#14556498—21no—
49C23×C4×C73C2×C411yescharacteristic
50#117×C73—22no—
51#117×C73C814yescomplemented
52C2×C4⋊C4×C73C2×C411yescharacteristic
53C2×C4⋊C4×C73C2×C411yescharacteristic
54C2×C22⋊C4×C73—21no—
55C2×C42×C73—21no—
56C2×#10596833—24no—
57C2×#10596833—48no—
58C2×#10596833—42no—
59C2×#10596833—42no—
60C2×#10596833—42no—
61C2×#10596833—42no—
62C2×#10596833—42no—
63C22×Dic73—22no—
64C22×Dic73—22no—
65C22×Dic73—21no—
66C22×Dic73—21no—
67C22×Dic73—22no—
68C22×Dic73—22no—
69C22×Dic73—21no—
70C22×Dic73—21no—
71C22×Dic73—41no—
72C22×Dic73C22⋊C411yescharacteristic
73C22×Dic73C22⋊C411yescharacteristic
74C22×Dic73C2×D411yescharacteristic
75C22×Dic73C2×D411yescharacteristic
76C4⋊Dic73—42no—
77C4⋊Dic73—42no—
78C4⋊Dic73—42no—
79C4⋊Dic73—41no—
80C4⋊Dic73—41no—
81C4⋊Dic73—41no—
82C4⋊Dic73—41no—
83C22⋊Dic73—42no—
84C4×Dic73—41no—
85C4×Dic73—41no—
86C4×Dic73—41no—
87C4×Dic73—41no—
88C4×Dic73—42no—
89C24×C73C22⋊C411yescharacteristic
90C22×C4×C73—22no—
91C22×C4×C73—22no—
92C22×C4×C73—21no—
93C22×C4×C73—21no—
94C22×C4×C73C22⋊C411yescharacteristic
95C22×C4×C73C2×C812yes—
96C22×C4×C73M1612yes—
97C22×C4×C73C22×C411yescharacteristic
98C4⋊C4×C73—21no—
99C4⋊C4×C73—21no—
100C4⋊C4×C73C2×C812yescomplemented
101C4⋊C4×C73C2×C812yescomplemented
102C22⋊C4×C73—24no—
103C42×C73—24no—
104C23×C73—21no—
105C23×C73—21no—
106C23×C73—21no—
107C23×C73—21no—
108C23×C73C22⋊C812yes—
109C23×C73#9912yes—
110C23×C73#10112yes—
111C23×C73C2×C22⋊C411yescharacteristic
112C2×C4×C73—24no—
113C2×C4×C73—24no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—21no—
118C2×C4×C73—22no—
119C2×C4×C73—22no—
120C2×C4×C73—41no—
121C2×C4×C73—41no—
122C2×C4×C73—41no—
123C2×C4×C73—41no—
124C2×C4×C73C22⋊C812yes—
125C2×C4×C73C22×C811yescharacteristic
126C2×C4×C73C2×M1611yescharacteristic
127C2×Dic73—24no—
128C2×Dic73—24no—
129C2×Dic73—21no—
130C2×Dic73—21no—
131C2×Dic73—21no—
132C2×Dic73—21no—
133C2×Dic73—21no—
134C2×Dic73—21no—
135C2×Dic73—21no—
136C2×Dic73—21no—
137C2×Dic73—21no—
138C2×Dic73—44no—
139C2×Dic73—41no—
140C2×Dic73—41no—
141C2×Dic73—41no—
142C2×Dic73—41no—
143C2×Dic73—41no—
144C2×Dic73—41no—
145C2×Dic73#10412yes—
146C2×Dic73#13512yes—
147C2×Dic73AGL(1,ℤ/8)11yescharacteristic
148C2×Dic73#13711yescharacteristic
149#10596833—44no—
150#10596833—82no—
151C22×C73—22no—
152C22×C73—22no—
153C22×C73—22no—
154C22×C73—22no—
155C22×C73—22no—
156C22×C73—22no—
157C22×C73#32414yes—
158C22×C73C2×C22⋊C811yescharacteristic
159C22×C73C2×#9911yescharacteristic
160C22×C73C2×#10111yescharacteristic
161C22×C73#41611yescharacteristic
162C22×C73#41711yescharacteristic
163C22×C73C2×#10411yescharacteristic
164C22×C73#42111yescharacteristic
165C4×C73—22no—
166C4×C73—21no—
167C4×C73—21no—
168C4×C73—42no—
169C4×C73—42no—
170Dic73—21no—
171Dic73—22no—
172Dic73—21no—
173Dic73—22no—
174Dic73—41no—
175Dic73—81no—
176Dic73—81no—
177#7579—731nomaximal, Sylow
178C2×C73—24no—
179C2×C73C2×#32411yescharacteristic
180C2×C73#147611yescharacteristic
181C2×C73#147711yescharacteristic
182C2×C73#150411yescharacteristic
183C2×C73#151611yescharacteristic
184C2×C73#153111yescharacteristic
185C2×C73#155211yescharacteristic
186#2296—731no—
187#1817—731no—
188C2×#336—731no—
189#1271—732no—
190#1271—732no—
191C73#757911yescharacteristic, complemented, Sylow
192C23×C8—1461no—
193C2×#117—731no—
194C2×#117#6440811yescharacteristic
195C22×C42—731no—
196C2×C22⋊C8—1461no—
197#389—731no—
198#389—731no—
199#389—731no—
200#389—731no—
201#385—1461no—
202#385—1461no—
203C4×C4⋊C4—1461no—
204C4×C4⋊C4—1461no—
205#336—732no—
206#336—732no—
207#336—1461no—
208#336—1461no—
209C23×C4—731no—
210C23×C4—731no—
211C23×C4C2×#6440811yescharacteristic
212C22×C8—2922no—
213C22×C8—2922no—
214C22×C8—1462no—
215C22×C8—1462no—
216C22×C8—1464no—
217#117—22no—
218#117—2922no—
219#117#1059683314yescomplemented
220C2×C4⋊C4—731no—
221C2×C4⋊C4—731no—
222C2×C4⋊C4—1461no—
223C2×C4⋊C4—1461no—
224C2×C4⋊C4—1461no—
225C2×C4⋊C4—1461no—
226C2×C4⋊C4C2×#6440811yescharacteristic
227C2×C4⋊C4C2×#6440811yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—1461no—
230C2×C42—21no—
231C2×C42—2922no—
232C2×C42—731no—
233C2×C42—731no—
234C2×C42—731no—
235C2×C42—731no—
236C2×C42—1461no—
237C22⋊C8—2922no—
238#95—1461no—
239#95—1461no—
240#95—1461no—
241#95—1461no—
242C24C22⋊#6440811yescharacteristic
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—2921no—
248C22×C4—731no—
249C22×C4—731no—
250C22×C4—731no—
251C22×C4—731no—
252C22×C4—1461no—
253C22×C4—1461no—
254C22×C4—1462no—
255C22×C4—1462no—
256C22×C4—1461no—
257C22×C4—1462no—
258C22×C4—1461no—
259C22×C4—1462no—
260C22×C4C22⋊#6440811yescharacteristic
261C22×C4C2×#1059683312yes—
262C22×C4#1192403712yes—
263C22×C4C22×#6440811yescharacteristic
264C2×C8—2922no—
265C2×C8—2928no—
266C2×C8—2922no—
267C2×C8—2922no—
268C2×C8—2922no—
269C2×C8—2922no—
270C2×C8—1464no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—2921no—
274C4⋊C4—2921no—
275C4⋊C4—2922no—
276C4⋊C4—2921no—
277C4⋊C4—2921no—
278C4⋊C4—2922no—
279C4⋊C4—2922no—
280C4⋊C4C2×#1059683312yescomplemented
281C4⋊C4C2×#1059683312yescomplemented
282C22⋊C4—24no—
283C22⋊C4—2922no—
284C42—24no—
285C42—2922no—
286C42—2921no—
287C42—2921no—
288C42—2921no—
289C42—2921no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23C22⋊#1059683312yes—
295C23#1455665012yes—
296C23#1455665212yes—
297C23C2×C22⋊#6440811yescharacteristic
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—2924no—
311C2×C4—2921no—
312C2×C4—2921no—
313C2×C4—2921no—
314C2×C4—2921no—
315C2×C4—2921no—
316C2×C4—2921no—
317C2×C4—732no—
318C2×C4—731no—
319C2×C4—731no—
320C2×C4—732no—
321C2×C4—1461no—
322C2×C4—1461no—
323C2×C4—1461no—
324C2×C4—1464no—
325C2×C4—1464no—
326C2×C4—1461no—
327C2×C4—1461no—
328C2×C4—1461no—
329C2×C4—1461no—
330C2×C4—1461no—
331C2×C4—1461no—
332C2×C4C22⋊#1059683312yes—
333C2×C4C22×#1059683311yescharacteristic
334C2×C4C2×#1192403711yescharacteristic
335C8—2924no—
336C8—5842no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22#1798943214yes—
344C22C2×C22⋊#1059683311yescharacteristic
345C22C2×#1455665011yescharacteristic
346C22C2×#1455665211yescharacteristic
347C22#1798957711yescharacteristic
348C22#1798957811yescharacteristic
349C22D73⋊#10411yescharacteristic
350C22#1798958411yescharacteristic
351C4—21no—
352C4—21no—
353C4—22no—
354C4—42no—
355C4—42no—
356C4—2921no—
357C4—5841no—
358C4—5841no—
359C4—1461no—
360C4—1461no—
361C4—1462no—
362C4—1462no—
363C2—24no—
364C2C2×#1798943211yescharacteristic
365C2#2262522211yescharacteristic
366C2#2262522611yescharacteristic
367C2#2262531911yescharacteristic
368C2#2262535911yescharacteristic
369C2#2262540311yescharacteristic
370C2#2262545211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC2×#117×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC2×#117#64408
hypocenterC73#7579
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #22624749C2semidirect
4. #22624749C2semidirect
51. #117×C73C8semidirect
100. C4⋊C4×C73C2×C8semidirect
101. C4⋊C4×C73C2×C8semidirect
191. C73#7579semidirect
219. #117#10596833semidirect
280. C4⋊C4C2×#10596833semidirect
281. C4⋊C4C2×#10596833semidirect

Maximal subgroups

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

subgroupindex
#226258212
C2×#179894532
#226247492
#226124152
#757973