Finite Group Databasebeta

← #11792024

Subgroups — #11792024

Order 1152.

885 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 885 conjugacy classes of subgroups — too many to draw. Only its 154 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×#63871C212yesmaximal
2C22×#3718C211yesmaximal, characteristic, complemented
3C2×#63647C211yesmaximal, characteristic
4C2×C3×#816C212yesmaximal, complemented
5#10589163C211yesmaximal, characteristic, complemented
6#70273—32nomaximal
7C22×C4×C3⋊D3C2211yescharacteristic
8C22×C3×Dic6C2212yes—
9C22×C32×C8C2211yescharacteristic
10C22×C3⋊(C3⋊C8)C2211yescharacteristic
11C2×C3×C4⋊Dic3C2212yes—
12C2×Dic3⋊Dic3—22no—
13C2×#3722—22no—
14#63871—24no—
15#63871—24no—
16C2×#3718—24no—
17C2×#3718—22no—
18#63647—22no—
19C3×#816—24no—
20C2×#946—32no—
21C22×#216—32no—
22C2×#897—32no—
23C2×C3×#103—32no—
24C2×(C2×D3)⋊C8—32no—
25C2×(C2×D3)⋊C8—61no—
26C2×#816D312yescomplemented
27#4529—32no—
28C23×C3⋊D3D411yescharacteristic
29C22×C32×C4C2311yescharacteristic
30C22×C3⋊Dic3D411yescharacteristic
31C2×C4×C3⋊D3—21no—
32C2×C4×C3⋊D3—22no—
33C2×C4×C3⋊D3—22no—
34C2×C4×C3⋊D3—21no—
35C22×C3×Dic3—22no—
36C22×C3×Dic3—22no—
37C2×C3×Dic6—22no—
38C2×C3×Dic6—22no—
39C2×C3×Dic6—22no—
40C2×C3×Dic6—22no—
41C2×C3×Dic6—22no—
42C2×C3×Dic6D414yes—
43C2×C32×C8—21no—
44C2×C32×C8—22no—
45C2×C3⋊(C3⋊C8)—22no—
46C2×C3⋊(C3⋊C8)D412yes—
47C3×C4⋊Dic3—22no—
48C3×C4⋊Dic3D414yes—
49Dic3⋊Dic3—28no—
50#3722—28no—
51#3718—28no—
52#2001—91noSylow
53C22×C3×Q8—32no—
54C22×C4×D3—32no—
55C22×C4×D3—61no—
56C22×Dic6C2×D312yes—
57C22×C3×C8—21no—
58C22×C3×C8C2×D312yes—
59C2×C3×C4⋊C4—32no—
60#946—64no—
61#946—64no—
62C2×#216—64no—
63C2×#216—64no—
64C2×#216—64no—
65C2×(C2×D3)⋊C4—62no—
66C2×(C2×D3)⋊C4—62no—
67C2×C4⋊Dic3—62no—
68C2×C4⋊Dic3C2×D312yes—
69C2×Dic3⋊C4—62no—
70C22×C3⋊C8—32no—
71C22×C3⋊C8—61no—
72#897—64no—
73#897—64no—
74C3×#103—64no—
75(C2×D3)⋊C8—64no—
76(C2×D3)⋊C8—64no—
77#816—24no—
78C23×C32C2×D411yescharacteristic
79C22×C3⋊D3—21no—
80C22×C3⋊D3—21no—
81C22×C3⋊D3—22no—
82C22×C3⋊D3—22no—
83C22×C3⋊D3Q1612yescomplemented
84C2×C32×C4C2×D412yes—
85C2×C32×C4C2×D411yescharacteristic
86C2×C32×C4C2×D412yes—
87C2×C32×C4#4111yescharacteristic
88C2×C3⋊Dic3—21no—
89C2×C3⋊Dic3—22no—
90C4×C3⋊D3—41no—
91C4×C3⋊D3—42no—
92C4×C3⋊D3—41no—
93C2×C3×Dic3—24no—
94C2×C3×Dic3—24no—
95C2×C3×Dic3—42no—
96C2×C3×Dic3—42no—
97C2×C3×Dic3—42no—
98C2×C3×Dic3—42no—
99C3×Dic6—24no—
100C3×Dic6—24no—
101C3×Dic6—42no—
102C3×Dic6—42no—
103C32×C8—41no—
104C3⋊(C3⋊C8)—22no—
105C22×Q16—91no—
106C2×C22⋊Q8—92no—
107C2×#103—32no—
108C2×C22⋊C8—91no—
109#340—91no—
110C23×D3—32no—
111C23×D3—61no—
112C2×C3×Q8—34no—
113C2×C3×Q8—62no—
114C2×C3×Q8—62no—
115C2×C3×Q8—62no—
116C2×C3×Q8—62no—
117C2×C3×Q8—62no—
118C22×C3×C4—21no—
119C22×C3×C4—62no—
120C22×C3×C4—62no—
121C22×C3×C4C22×D312yes—
122C22×Dic3—22no—
123C22×Dic3—22no—
124C22×Dic3—32no—
125C22×Dic3—61no—
126C2×C4×D3—62no—
127C2×C4×D3—62no—
128C2×C4×D3—62no—
129C2×C4×D3—62no—
130C2×C4×D3—62no—
131C2×C4×D3—62no—
132C2×C4×D3—62no—
133C2×C4×D3—62no—
134C2×C4×D3—122no—
135C2×C4×D3—122no—
136C2×Dic6—22no—
137C2×Dic6—22no—
138C2×Dic6—22no—
139C2×Dic6—22no—
140C2×Dic6—22no—
141C2×Dic6D1214yes—
142C2×C3×C8—22no—
143C2×C3×C8—22no—
144C2×C3×C8—22no—
145C2×C3×C8—22no—
146C2×C3×C8—42no—
147C3×C4⋊C4—34no—
148C3×C4⋊C4—62no—
149#216—616no—
150(C2×D3)⋊C4—68no—
151(C2×D3)⋊C4—68no—
152C4⋊Dic3—22no—
153C4⋊Dic3—68no—
154C4⋊Dic3#6714yes—
155Dic3⋊C4—68no—
156C2×C3⋊C8—34no—
157C2×C3⋊C8—62no—
158C2×C3⋊C8—62no—
159C2×C3⋊C8—62no—
160C2×C3⋊C8—122no—
161C22×C32#12012yes—
162C22×C32#12111yescharacteristic
163C22×C32C2×Q1611yescharacteristic
164C22×C32#13511yescharacteristic
165C22×C32#13712yes—
166C2×C3⋊D3—22no—
167C2×C3⋊D3—24no—
168C2×C3⋊D3—41no—
169C2×C3⋊D3—42no—
170C2×C3⋊D3—41no—
171C32×C4#12011yescharacteristic
172C32×C4#12112yes—
173C32×C4#12711yescharacteristic
174C3⋊Dic3—41no—
175C3×Dic3—44no—
176C3×Dic3—44no—
177C22×Q8—32no—
178C23×C4—91no—
179C2×Q16—182no—
180C2×Q16—184no—
181C22×C8—91no—
182C22×C8D3211yescharacteristic
183C22⋊Q8—184no—
184C22⋊Q8—184no—
185C2×C4⋊C4—32no—
186C2×C4⋊C4—182no—
187C2×C22⋊C4—182no—
188#103—64no—
189C22⋊C8—182no—
190C23×C3—21no—
191C23×C3D3×D412yes—
192C22×D3—34no—
193C22×D3—62no—
194C22×D3—62no—
195C22×D3—62no—
196C22×D3—62no—
197C22×D3—62no—
198C22×D3—62no—
199C22×D3—62no—
200C22×D3—62no—
201C22×D3—62no—
202C22×D3—122no—
203C22×D3—122no—
204C3×Q8—64no—
205C3×Q8—64no—
206C3×Q8—122no—
207C3×Q8—122no—
208C2×C3×C4—21no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—21no—
212C2×C3×C4—64no—
213C2×C3×C4—64no—
214C2×C3×C4—122no—
215C2×C3×C4—122no—
216C2×C3×C4—122no—
217C2×C3×C4—122no—
218C2×C3×C4C2×D1212yes—
219C2×C3×C4#23512yes—
220C2×C3×C4D3×D412yes—
221C2×C3×C4D3×D412yes—
222C2×C3×C4D3×D412yes—
223C2×C3×C4C2×#6712yes—
224C2×Dic3—24no—
225C2×Dic3—24no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—62no—
231C2×Dic3—62no—
232C2×Dic3—62no—
233C2×Dic3—62no—
234C2×Dic3—122no—
235C4×D3—124no—
236C4×D3—122no—
237C4×D3—122no—
238C4×D3—122no—
239C4×D3—122no—
240C4×D3—122no—
241C4×D3—122no—
242Dic6—24no—
243Dic6—24no—
244Dic6—42no—
245Dic6—42no—
246C3×C8—42no—
247C3×C8—42no—
248C3⋊C8—64no—
249C3⋊C8—122no—
250C2×C32#39211yescharacteristic
251C2×C32#45112yes—
252C2×C32#45212yes—
253C2×C32#46711yescharacteristic
254C2×C32#47011yescharacteristic
255C3⋊D3—42no—
256C24—91no—
257C2×Q8—34no—
258C2×Q8—62no—
259C2×Q8—62no—
260C2×Q8—62no—
261C2×Q8—62no—
262C2×Q8—62no—
263C22×C4—62no—
264C22×C4—62no—
265C22×C4—91no—
266C22×C4—181no—
267C22×C4—182no—
268C22×C4—182no—
269C22×C4—181no—
270C22×C4C2×D3211yescharacteristic
271Q16—188no—
272C2×C8—22no—
273C2×C8—21no—
274C2×C8—92no—
275C2×C8—182no—
276C4⋊C4—34no—
277C4⋊C4—62no—
278C4⋊C4—188no—
279C22⋊C4—188no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—21no—
284C22×C3—22no—
285C22×C3#88212yes—
286C22×C3#88412yes—
287C22×C3#91512yes—
288C22×C3#91612yes—
289C22×C3D3×Q1612yes—
290C22×C3#91812yes—
291C22×C3#93712yes—
292C2×D3—64no—
293C2×D3—64no—
294C2×D3—64no—
295C2×D3—64no—
296C2×D3—124no—
297C2×D3—122no—
298C2×D3—122no—
299C2×D3—124no—
300C2×D3—122no—
301C2×D3—122no—
302C2×D3—122no—
303C2×D3—122no—
304C3×C4—22no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—124no—
308C3×C4—124no—
309C3×C4#88212yes—
310C3×C4#88412yes—
311C3×C4#93012yes—
312C3×C4#94012yes—
313Dic3—44no—
314Dic3—44no—
315Dic3—122no—
316Dic3—122no—
317C32#200111yescharacteristic, complemented, Sylow
318C23—92no—
319C23—181no—
320C23—182no—
321C23—181no—
322C23—182no—
323C23#380211yescharacteristic
324Q8—64no—
325Q8—64no—
326Q8—122no—
327Q8—122no—
328C2×C4—64no—
329C2×C4—64no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—181no—
335C2×C4—182no—
336C2×C4—362no—
337C2×C4—361no—
338C2×C4—361no—
339C2×C4#379811yescharacteristic
340C2×C4D3×D1212yes—
341C2×C4D3×#6712yes—
342C2×C4#381111yescharacteristic
343C8—41no—
344C8—182no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—22no—
349C2×C3—22no—
350C2×C3#480812yes—
351C2×C3#480912yes—
352C2×C3#481312yes—
353C2×C3#481612yes—
354C2×C3#496012yes—
355C2×C3#517312yes—
356C2×C3#518612yes—
357D3—124no—
358D3—124no—
359C22—182no—
360C22—184no—
361C22—361no—
362C22—361no—
363C22—362no—
364C22#6380211yescharacteristic
365C22#6380911yescharacteristic
366C22#6381112yes—
367C22#6391211yescharacteristic
368C22#6392212yes—
369C4—124no—
370C4—124no—
371C4—361no—
372C4#6390612yes—
373C4#6392311yescharacteristic
374C4#6397911yescharacteristic
375C3—21no—
376C3#7027312yescomplemented
377C2—362no—
378C2#1059085511yescharacteristic
379C2#1059085611yescharacteristic
380C2#1059095012yes—
381C2#1059096112yes—
382C2#1059139311yescharacteristic
subgroupG/H
centerC23#3802
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC22×C32×C8C22
socleC23×C32C2×D4
prime radicalC23×C3⋊D3D4
hypercenterC22×C8D32
hypocenterC32#2001
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
2. C22×#3718C2semidirect
4. C2×C3×#816C2semidirect
5. #10589163C2semidirect
26. C2×#816D3semidirect
83. C22×C3⋊D3Q16semidirect
317. C32#2001semidirect
376. C3#70273semidirect

Maximal subgroups

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

subgroupindex
C2×#638712
C22×#37182
C2×#636472
C2×C3×#8162
#105891632
#702733