Finite Group Databasebeta

← #32974167

Subgroups — #32974167

Order 18688.

550 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22609911C211yesmaximal, characteristic
2#22609625C211yesmaximal, characteristic
3#22607757C212yesmaximal, complemented
4#22607754C212yesmaximal, complemented
5C4×C22⋊Q8×C73C211yesmaximal, characteristic
6C22×#11924012—21no—
7C2×C4⋊#10596831—21no—
8C2×C22⋊#10596831—21no—
9#17988342—21no—
10#17988342C2212yes—
11#17988342C2212yes—
12C2×C22⋊Q8×C73C411yescharacteristic
13C2×C4×Q8×C73C411yescharacteristic
14C22×C42×C73C2211yescharacteristic
15#391×C73C2211yescharacteristic
16#389×C73C2211yescharacteristic
17#387×C73—21no—
18#385×C73—21no—
19#380×C73C411yescharacteristic
20C4×C4⋊C4×C73—21no—
21C4×C4⋊C4×C73C411yescharacteristic
22C4×C22⋊C4×C73—21no—
23C2×#11924012—24no—
24C2×#11924012—42no—
25C2×#11924012—42no—
26C22×#10596831—22no—
27C22×#10596831—22no—
28C4⋊#10596831—42no—
29C22⋊#10596831—42no—
30C22×Q8×C73C2×C411yescharacteristic
31C23×C4×C73C2×C411yescharacteristic
32C23×C4×C73D411yescharacteristic
33C23×C4×C73D411yescharacteristic
34C22⋊Q8×C73—22no—
35C22⋊Q8×C73—22no—
36C4×Q8×C73—42no—
37C2×C4⋊C4×C73—21no—
38C2×C4⋊C4×C73—21no—
39C2×C4⋊C4×C73—21no—
40C2×C4⋊C4×C73C2×C411yescharacteristic
41C2×C4⋊C4×C73C2×C411yescharacteristic
42C2×C4⋊C4×C73C2×C411yescharacteristic
43C2×C22⋊C4×C73—21no—
44C2×C22⋊C4×C73—21no—
45C2×C42×C73—21no—
46C2×C42×C73—21no—
47C2×C42×C73—42no—
48C2×C42×C73C2×C411yescharacteristic
49C2×C42×C73D411yescharacteristic
50C2×C42×C73D411yescharacteristic
51C2×C42×C73C2311yescharacteristic
52#95×C73—21no—
53#95×C73—21no—
54#95×C73—21no—
55#95×C73—21no—
56#95×C73—21no—
57#11924012—48no—
58C2×#10596831—24no—
59C2×#10596831—42no—
60C2×#10596831—42no—
61C2×#10596831—42no—
62C2×#10596831—42no—
63C2×#10596831—42no—
64C24×C73C22⋊C411yescharacteristic
65C2×Q8×C73—21no—
66C2×Q8×C73—41no—
67C2×Q8×C73—41no—
68C2×Q8×C73M1612yescomplemented
69C22×C4×C73—22no—
70C22×C4×C73—22no—
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—41no—
80C22×C4×C73—41no—
81C22×C4×C73—41no—
82C22×C4×C73—41no—
83C22×C4×C73C22⋊C411yescharacteristic
84C22×C4×C73C22⋊C411yescharacteristic
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73SD1612yes—
87C22×C4×C73Q1612yes—
88C22×C4×C73C22×C411yescharacteristic
89C22×C4×C73C2×D411yescharacteristic
90C22×C4×C73C2×D411yescharacteristic
91C4⋊C4×C73—24no—
92C4⋊C4×C73—21no—
93C4⋊C4×C73—41no—
94C4⋊C4×C73—41no—
95C4⋊C4×C73—42no—
96C4⋊C4×C73—41no—
97C4⋊C4×C73—41no—
98C4⋊C4×C73M1612yescomplemented
99C22⋊C4×C73—24no—
100C22⋊C4×C73—42no—
101C42×C73—22no—
102C42×C73—22no—
103C42×C73—41no—
104C42×C73—41no—
105C42×C73—42no—
106C42×C73—42no—
107C23×C73—21no—
108C23×C73—21no—
109C23×C73—21no—
110C23×C73—21no—
111C23×C73—21no—
112C23×C73—21no—
113C23×C73#10312yes—
114C23×C73C2×C22⋊C411yescharacteristic
115Q8×C73—42no—
116Q8×C73—42no—
117C2×C4×C73—21no—
118C2×C4×C73—21no—
119C2×C4×C73—21no—
120C2×C4×C73—21no—
121C2×C4×C73—21no—
122C2×C4×C73—21no—
123C2×C4×C73—24no—
124C2×C4×C73—24no—
125C2×C4×C73—22no—
126C2×C4×C73—22no—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73—21no—
130C2×C4×C73—21no—
131C2×C4×C73—21no—
132C2×C4×C73—21no—
133C2×C4×C73—42no—
134C2×C4×C73—42no—
135C2×C4×C73—42no—
136C2×C4×C73—42no—
137C2×C4×C73—41no—
138C2×C4×C73—41no—
139C2×C4×C73—41no—
140C2×C4×C73—41no—
141C2×C4×C73—41no—
142C2×C4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73#9912yes—
148C2×C4×C73#10312yes—
149C2×C4×C73#10412yes—
150C2×C4×C73C2×M1611yescharacteristic
151C2×C4×C73C2×M1611yescharacteristic
152C2×C4×C73C2×SD1611yescharacteristic
153C2×C4×C73C2×Q1611yescharacteristic
154C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C73#13711yescharacteristic
156#10596831—44no—
157#10596831—82no—
158C22×C73—22no—
159C22×C73—22no—
160C22×C73—22no—
161C22×C73—41no—
162C22×C73—41no—
163C22×C73—41no—
164C22×C73—41no—
165C22×C73#40711yescharacteristic
166C22×C73C2×#9911yescharacteristic
167C22×C73#41311yescharacteristic
168C22×C73C2×#10311yescharacteristic
169C22×C73#41711yescharacteristic
170C22×C73C2×#10411yescharacteristic
171C22×C73#42111yescharacteristic
172C4×C73—21no—
173C4×C73—21no—
174C4×C73—21no—
175C4×C73—21no—
176C4×C73—21no—
177C4×C73—21no—
178C4×C73—41no—
179C4×C73—42no—
180C4×C73—42no—
181C4×C73—41no—
182C4×C73—81no—
183C4×C73—81no—
184C4×C73#32814yes—
185#7617—731nomaximal, Sylow
186C2×C73—42no—
187C2×C73#146511yescharacteristic
188C2×C73#148711yescharacteristic
189C2×C73#148911yescharacteristic
190C2×C73C2×#32811yescharacteristic
191C2×C73#151311yescharacteristic
192C2×C73#152911yescharacteristic
193C2×C73#155011yescharacteristic
194C4×C22⋊Q8D7311yescharacteristic
195#1797—731no—
196#1728—731no—
197#1272—732no—
198#1271—732no—
199C73#761711yescharacteristic, complemented, Sylow
200C22×M16—1461no—
201C2×C22⋊Q8Dic7311yescharacteristic
202C2×C4×Q8Dic7311yescharacteristic
203C22×C42C2×D7311yescharacteristic
204C2×C4⋊C8—1461no—
205C2×C22⋊C8—1461no—
206#391C2×D7311yescharacteristic
207#389C2×D7311yescharacteristic
208#387—21no—
209#385—21no—
210#380Dic7311yescharacteristic
211C4×C4⋊C4—21no—
212C4×C4⋊C4Dic7311yescharacteristic
213C4×C22⋊C4—21no—
214#336—732no—
215#336—732no—
216#336—1461no—
217C22×Q8C2×Dic7311yescharacteristic
218C23×C4C2×Dic7311yescharacteristic
219C23×C4C22⋊D7311yescharacteristic
220C23×C4C22⋊D7311yescharacteristic
221C2×M16—2922no—
222C2×M16—2922no—
223C2×M16—1464no—
224C22×C8—1462no—
225C22×C8—1462no—
226C22⋊Q8—22no—
227C22⋊Q8—22no—
228C4×Q8—42no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4C2×Dic7311yescharacteristic
233C2×C4⋊C4C2×Dic7311yescharacteristic
234C2×C4⋊C4C2×Dic7311yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—42no—
240C2×C42C2×Dic7311yescharacteristic
241C2×C42C22⋊D7311yescharacteristic
242C2×C42C22⋊D7311yescharacteristic
243C2×C42C22×D7311yescharacteristic
244C4⋊C8—2922no—
245C22⋊C8—2922no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251C24C22⋊Dic7311yescharacteristic
252C2×Q8—21no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8#1192401212yescomplemented
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4C22⋊Dic7311yescharacteristic
271C22×C4C22⋊Dic7311yescharacteristic
272C22×C4C22⋊Dic7311yescharacteristic
273C22×C4#1192401512yes—
274C22×C4#1192401812yes—
275C22×C4C22×Dic7311yescharacteristic
276C22×C4C2×C22⋊D7311yescharacteristic
277C22×C4C2×C22⋊D7311yescharacteristic
278M16—2928no—
279C2×C8—2922no—
280C2×C8—2922no—
281C2×C8—2922no—
282C2×C8—2922no—
283C2×C8—2922no—
284C2×C8—1464no—
285C4⋊C4—24no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4#1192401212yescomplemented
293C22⋊C4—24no—
294C22⋊C4—42no—
295C42—22no—
296C42—22no—
297C42—41no—
298C42—41no—
299C42—42no—
300C42—42no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23#1455651512yes—
308C23C2×C22⋊Dic7311yescharacteristic
309Q8—42no—
310Q8—42no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4#1455650612yes—
342C2×C4#1455651512yes—
343C2×C4#1455651812yes—
344C2×C4C2×#1192401211yescharacteristic
345C2×C4C2×#1192401211yescharacteristic
346C2×C4C2×#1192401511yescharacteristic
347C2×C4C2×#1192401811yescharacteristic
348C2×C4#1455661611yescharacteristic
349C2×C4#1455662111yescharacteristic
350C8—2924no—
351C8—5842no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22#1798855111yescharacteristic
360C22C2×#1455650611yescharacteristic
361C22#1798857111yescharacteristic
362C22C2×#1455651511yescharacteristic
363C22#1798858811yescharacteristic
364C22C2×#1455651811yescharacteristic
365C22#1798860411yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—42no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4#1798832414yes—
379C2—42no—
380C2#2260826911yescharacteristic
381C2#2260836011yescharacteristic
382C2#2260837011yescharacteristic
383C2C2×#1798832411yescharacteristic
384C2#2260847411yescharacteristic
385C2#2260853811yescharacteristic
386C2#2260862111yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic73
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC4×C22⋊Q8×C73C2
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C22⋊C4
cosocleC2×C42×C73C23
hypercenterC4×C22⋊Q8D73
hypocenterC73#7617
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22607757C2semidirect
4. #22607754C2semidirect
68. C2×Q8×C73M16semidirect
98. C4⋊C4×C73M16semidirect
199. C73#7617semidirect
255. C2×Q8#11924012semidirect
292. C4⋊C4#11924012semidirect

Maximal subgroups

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

subgroupindex
#226099112
#226096252
#226077572
#226077542
C4×C22⋊Q8×C732
#761773