Finite Group Databasebeta

← #11792016

Subgroups — #11792016

Order 1152.

885 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 885 conjugacy classes of subgroups — too many to draw. Only its 166 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×C3×#924C212yesmaximal
2C22×#3718C211yesmaximal, characteristic, complemented
3C2×#63657C211yesmaximal, characteristic
4C2×#63565C212yesmaximal, complemented
5#10589163C211yesmaximal, characteristic, complemented
6#71700—32nomaximal
7C23×C32×C4C2211yescharacteristic
8C22×C3×Dic6C2212yes—
9C22×C3⋊(C3⋊C8)C2211yescharacteristic
10C22×C3⋊(C3⋊C8)C2211yescharacteristic
11C2×C3×#217—22no—
12C2×C3×C4⋊Dic3C2212yes—
13C3×#924—24no—
14C3×#924—24no—
15C2×C3×Dic3⋊C4—22no—
16C2×#3718—24no—
17C2×#3718—22no—
18#63657—22no—
19#63565—24no—
20C2×C3×C22⋊Q8—32no—
21C22×#216—32no—
22C2×#924D312yes—
23C2×#835—32no—
24C2×#830—32no—
25C2×#830—61no—
26C2×#810—32no—
27#4529—32no—
28C24×C32D411yescharacteristic
29C22×C32×C4—21no—
30C22×C32×C4—22no—
31C22×C32×C4—22no—
32C22×C32×C4—21no—
33C22×C32×C4D411yescharacteristic
34C22×C32×C4C2311yescharacteristic
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×C3⋊(C3⋊C8)—22no—
44C2×C3⋊(C3⋊C8)—21no—
45C2×C3⋊(C3⋊C8)—22no—
46C2×C3⋊(C3⋊C8)D412yes—
47C3×#217—28no—
48C3×C4⋊Dic3—22no—
49C3×C4⋊Dic3D414yes—
50C3×Dic3⋊C4—28no—
51#3718—28no—
52#2001—91noSylow
53C22×C3×Q8—32no—
54C23×C3×C4—21no—
55C23×C3×C4C2×D312yes—
56C22×Dic6C2×D312yes—
57C3×C22⋊Q8—64no—
58C3×C22⋊Q8—64no—
59C2×C3×C4⋊C4—32no—
60C2×C3×C4⋊C4—62no—
61C2×C3×C22⋊C4—62no—
62C2×#217—22no—
63C2×#216—64no—
64C2×#216—64no—
65C2×#216—64no—
66C2×C4⋊Dic3C2×D312yes—
67#924—24no—
68#924—24no—
69C2×Dic3⋊C4—22no—
70C22×C3⋊C8—32no—
71C22×C3⋊C8—32no—
72C22×C3⋊C8—61no—
73C22×C3⋊C8—61no—
74#835—64no—
75#830—64no—
76#830—64no—
77#810—64no—
78C23×C32—21no—
79C23×C32—21no—
80C23×C32—22no—
81C23×C32—22no—
82C23×C32Q1612yescomplemented
83C23×C32C2×D411yescharacteristic
84C2×C32×C4—21no—
85C2×C32×C4—22no—
86C2×C32×C4—41no—
87C2×C32×C4—42no—
88C2×C32×C4—41no—
89C2×C32×C4C2×D411yescharacteristic
90C2×C32×C4C2×D412yes—
91C2×C32×C4C2×D412yes—
92C2×C32×C4#4111yescharacteristic
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—
103C3⋊(C3⋊C8)—22no—
104C3⋊(C3⋊C8)—41no—
105C22×Q16—91no—
106C2×C22⋊Q8—32no—
107C2×#103—92no—
108C2×C22⋊C8—91no—
109#340—91no—
110C24×C3—21no—
111C24×C3#6712yes—
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—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—42no—
129C22×C3×C4—42no—
130C22×C3×C4—62no—
131C22×C3×C4—62no—
132C22×C3×C4#6712yes—
133C22×C3×C4C22×D312yes—
134C22×Dic3—22no—
135C22×Dic3—22no—
136C2×Dic6—22no—
137C2×Dic6—22no—
138C2×Dic6—22no—
139C2×Dic6—22no—
140C2×Dic6—22no—
141C2×Dic6#6714yes—
142C3×C4⋊C4—34no—
143C3×C4⋊C4—68no—
144C3×C4⋊C4—62no—
145C3×C22⋊C4—68no—
146#217—28no—
147#216—616no—
148C4⋊Dic3—22no—
149C4⋊Dic3#6714yes—
150Dic3⋊C4—28no—
151C2×C3⋊C8—34no—
152C2×C3⋊C8—62no—
153C2×C3⋊C8—62no—
154C2×C3⋊C8—62no—
155C2×C3⋊C8—62no—
156C2×C3⋊C8—62no—
157C2×C3⋊C8—62no—
158C2×C3⋊C8—62no—
159C2×C3⋊C8—122no—
160C2×C3⋊C8—122no—
161C22×C32—22no—
162C22×C32—24no—
163C22×C32—41no—
164C22×C32—42no—
165C22×C32—41no—
166C22×C32#12012yes—
167C22×C32#12111yescharacteristic
168C22×C32C2×Q1611yescharacteristic
169C22×C32#13511yescharacteristic
170C22×C32#13712yes—
171C32×C4—41no—
172C32×C4#12011yescharacteristic
173C32×C4#12112yes—
174C32×C4#12711yescharacteristic
175C3×Dic3—44no—
176C3×Dic3—44no—
177C22×Q8—32no—
178C23×C4D3211yescharacteristic
179C2×Q16—182no—
180C2×Q16—184no—
181C22×C8—91no—
182C22×C8—91no—
183C22⋊Q8—64no—
184C22⋊Q8—64no—
185C2×C4⋊C4—32no—
186C2×C4⋊C4—62no—
187C2×C22⋊C4—62no—
188#103—184no—
189C22⋊C8—182no—
190C23×C3—22no—
191C23×C3—22no—
192C23×C3—22no—
193C23×C3—22no—
194C23×C3—22no—
195C23×C3—22no—
196C23×C3—22no—
197C23×C3—22no—
198C23×C3—22no—
199C23×C3—21no—
200C23×C3—42no—
201C23×C3—42no—
202C23×C3#21614yescomplemented
203C23×C3C2×#6712yes—
204C3×Q8—64no—
205C3×Q8—64no—
206C3×Q8—122no—
207C3×Q8—122no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—21no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—21no—
216C2×C3×C4—42no—
217C2×C3×C4—44no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—64no—
225C2×C3×C4—64no—
226C2×C3×C4—122no—
227C2×C3×C4—122no—
228C2×C3×C4—122no—
229C2×C3×C4—122no—
230C2×C3×C4D3×D412yes—
231C2×C3×C4D3×D412yes—
232C2×C3×C4D3×D412yes—
233C2×C3×C4#23712yes—
234C2×C3×C4C2×#6712yes—
235C2×C3×C4C2×#6712yes—
236C2×Dic3—24no—
237C2×Dic3—24no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242Dic6—24no—
243Dic6—24no—
244Dic6—42no—
245Dic6—42no—
246C3⋊C8—64no—
247C3⋊C8—122no—
248C3⋊C8—122no—
249C3⋊C8—122no—
250C2×C32—42no—
251C2×C32#39211yescharacteristic
252C2×C32#45112yes—
253C2×C32#45212yes—
254C2×C32#46711yescharacteristic
255C2×C32#47011yescharacteristic
256C24#62511yescharacteristic
257C2×Q8—34no—
258C2×Q8—62no—
259C2×Q8—62no—
260C2×Q8—62no—
261C2×Q8—62no—
262C2×Q8—62no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—62no—
268C22×C4—62no—
269C22×C4#62511yescharacteristic
270C22×C4C2×D3211yescharacteristic
271Q16—188no—
272C2×C8—92no—
273C2×C8—182no—
274C2×C8—182no—
275C2×C8—181no—
276C4⋊C4—34no—
277C4⋊C4—68no—
278C4⋊C4—62no—
279C22⋊C4—68no—
280C22×C3—24no—
281C22×C3—24no—
282C22×C3—24no—
283C22×C3—24no—
284C22×C3—21no—
285C22×C3—22no—
286C22×C3—21no—
287C22×C3—22no—
288C22×C3—21no—
289C22×C3—44no—
290C22×C3—42no—
291C22×C3—44no—
292C22×C3—42no—
293C22×C3—42no—
294C22×C3—42no—
295C22×C3—42no—
296C22×C3—42no—
297C22×C3#93712yes—
298C22×C3#93912yes—
299C22×C3#94212yes—
300C22×C3C2×#21612yes—
301C22×C3#95012yes—
302C22×C3#95112yes—
303C22×C3#95312yes—
304C3×C4—22no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—42no—
308C3×C4—42no—
309C3×C4—124no—
310C3×C4—124no—
311C3×C4#93712yes—
312C3×C4#93812yes—
313C3×C4#93912yes—
314C3×C4#94012yes—
315Dic3—44no—
316Dic3—44no—
317C32#200111yescharacteristic, complemented, Sylow
318C23—21no—
319C23—22no—
320C23—22no—
321C23—21no—
322C23#371812yescomplemented
323C23C2×#62511yescharacteristic
324Q8—64no—
325Q8—64no—
326Q8—122no—
327Q8—122no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—64no—
334C2×C4—64no—
335C2×C4—122no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4#380511yescharacteristic
340C2×C4D3×#6712yes—
341C2×C4D3×#6712yes—
342C2×C4#381111yescharacteristic
343C8—182no—
344C8—361no—
345C2×C3—22no—
346C2×C3—22no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—44no—
351C2×C3—44no—
352C2×C3#504712yes—
353C2×C3#505012yes—
354C2×C3#505112yes—
355C2×C3#505212yes—
356C2×C3#522312yes—
357C2×C3#522812yes—
358C2×C3#523912yes—
359C22—24no—
360C22—22no—
361C22—42no—
362C22—41no—
363C22—41no—
364C22#6382211yescharacteristic
365C22#6382712yes—
366C22C2×#371811yescharacteristic
367C22#6397612yes—
368C22#6398011yescharacteristic
369C4—41no—
370C4—124no—
371C4—124no—
372C4#6397012yes—
373C4#6397911yescharacteristic
374C4#6398311yescharacteristic
375C3—21no—
376C3#7170012yescomplemented
377C2—42no—
378C2#1059117012yes—
379C2#1059118512yes—
380C2#1059118811yescharacteristic
381C2#1059118911yescharacteristic
382C2#1059172311yescharacteristic
subgroupG/H
centerC23C2×#625
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC23×C32×C4C22
socleC23×C32C2×D4
prime radicalC24×C32D4
hypercenterC23×C4D32
hypocenterC32#2001
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C22×#3718C2semidirect
4. C2×#63565C2semidirect
5. #10589163C2semidirect
82. C23×C32Q16semidirect
202. C23×C3#216semidirect
317. C32#2001semidirect
322. C23#3718semidirect
376. C3#71700semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9242
C22×#37182
C2×#636572
C2×#635652
#105891632
#717003