Finite Group Databasebeta

← #11796372

Subgroups — #11796372

Order 1152.

559 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 96 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
1C3×#5716C211yesmaximal, characteristic
2#10591528C212yesmaximal, complemented
3#10589411C211yesmaximal, characteristic, complemented
4C3×#4545C211yesmaximal, characteristic, complemented
5#10589180C212yesmaximal, complemented
6#71819—31nomaximal
7#69151—31nomaximal
8C2×C3×#237C411yescharacteristic, complemented
9C2×C3×#237C2211yescharacteristic
10C32×#127C2211yescharacteristic
11C3×#938—22no—
12C3×#936C411yescharacteristic, complemented
13C3×#936C2211yescharacteristic
14C3×D4×Dic3—22no—
15C3×C42×D3C411yescharacteristic, complemented
16C3×#869C411yescharacteristic, complemented
17#63955—22no—
18#63661C2212yes—
19C3×#834C2212yes—
20#63580—22no—
21C3×#532—31no—
22#5716D311yescharacteristic
23#5240—32no—
24#4740—32no—
25C3×#353—31no—
26#4545—31no—
27#4545—31no—
28#4545—61no—
29#4545D311yescharacteristic, complemented
30#4541—32no—
31#4473—32no—
32C2×C32×D4—22no—
33C2×C32×D4C2×C411yescharacteristic
34C2×C32×D4C2311yescharacteristic
35C2×C3×#67—21no—
36C2×C3×#67D412yes—
37C22×C3×Dic3—21no—
38C22×C3×Dic3D412yes—
39C3×#237—41no—
40C3×#237—41no—
41C3×#237—41no—
42C3×#237—41no—
43C2×C3×C4×D3—21no—
44C2×C3×C4×D3C2×C411yescharacteristic
45C2×C3×Dic6C2×C411yescharacteristic
46C2×C3×Dic6C2×C411yescharacteristic
47C32×C42C2×C411yescharacteristic
48#3757—22no—
49C3×#217—22no—
50C3×#217—22no—
51C3×#217—22no—
52C3×C4⋊Dic3—22no—
53C3×C4×Dic3—21no—
54C3×C4×Dic3C2×C411yescharacteristic
55Dic3⋊Dic3—42no—
56Dic32—42no—
57#2126—91noSylow
58C2×C3×#41—31no—
59C2×C3×#41—31no—
60C2×#237Dic311yescharacteristic, complemented
61C2×#237C2×D311yescharacteristic
62C3×C4⋊Q8—31no—
63C3×#127—21no—
64C3×#127C2×D311yescharacteristic
65C3×#127C2×D311yescharacteristic
66C3×#124—31no—
67C3×#124—31no—
68C3×#121—62no—
69C3×C4×D4—62no—
70C2×C3×C42—31no—
71#938—22no—
72#936Dic311yescharacteristic, complemented
73#936C2×D311yescharacteristic
74D4×Dic3—22no—
75#926—62no—
76#877—62no—
77C42×D3Dic311yescharacteristic, complemented
78#869Dic311yescharacteristic, complemented
79C3×#99—32no—
80#834—32no—
81#834—32no—
82#834—62no—
83#834—62no—
84#834C2×D312yes—
85#806—62no—
86C23×C32—21no—
87C23×C32C2×D412yes—
88C22×C3×D3C22⋊C411yescharacteristic
89C32×D4—41no—
90C32×D4—41no—
91C32×D4—44no—
92C2×C32×C4—21no—
93C2×C32×C4C22×C411yescharacteristic
94C2×C3⋊Dic3—42no—
95C3×#67—42no—
96C3×#67—42no—
97C2×C3×Dic3—22no—
98C2×C3×Dic3—22no—
99C2×C3×Dic3—21no—
100C2×C3×Dic3—21no—
101C2×C3×Dic3—41no—
102C2×C3×Dic3—41no—
103C2×C3×Dic3—42no—
104C2×C3×Dic3C22⋊C412yes—
105C2×C3×Dic3C22⋊C411yescharacteristic
106C3×C4×D3—21no—
107C3×C4×D3—21no—
108C3×C4×D3—42no—
109C3×Dic6—41no—
110C3×Dic6—41no—
111#532—31no—
112#410—92no—
113#354—92no—
114#353—31no—
115#353—91no—
116C3×#41—121no—
117C3×#41—121no—
118C3×#41—121no—
119C3×#41—121no—
120C2×C3×Q8—31no—
121C2×C3×Q8—31no—
122C2×C3×D4—22no—
123C2×C3×D4—22no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—32no—
127C2×C3×D4—42no—
128C2×C3×D4—61no—
129C2×C3×D4C4×D311yescharacteristic
130C2×C3×D4C2×Dic311yescharacteristic
131C2×C3×D4C22×D311yescharacteristic
132C2×C3×D4C22×D311yescharacteristic
133C22×C3×C4—32no—
134C22×C3×C4—31no—
135C22×C3×C4—61no—
136C22×C3×C4—61no—
137C2×#67—21no—
138C2×#67#6712yes—
139C22×Dic3—21no—
140C22×Dic3#6712yes—
141#237—41no—
142#237—41no—
143#237—41no—
144#237—41no—
145C2×C4×D3—21no—
146C2×C4×D3C2×Dic311yescharacteristic
147C2×Dic6C2×Dic311yescharacteristic
148C2×Dic6C2×Dic311yescharacteristic
149C3×C4⋊C4—62no—
150C3×C22⋊C4—62no—
151C3×C22⋊C4—62no—
152C3×C22⋊C4—62no—
153C3×C42—21no—
154C3×C42—31no—
155C3×C42—61no—
156C3×C42C4×D311yescharacteristic
157C3×C42C2×Dic311yescharacteristic
158#217—22no—
159#217—22no—
160#217—22no—
161#217—62no—
162#217—62no—
163#217—122no—
164C4⋊Dic3—22no—
165C4⋊Dic3—122no—
166Dic3⋊C4—122no—
167C4×Dic3—21no—
168C4×Dic3—122no—
169C4×Dic3—122no—
170C4×Dic3C2×Dic311yescharacteristic
171C22×C32—22no—
172C22×C32—42no—
173C22×C32—42no—
174C22×C32—41no—
175C22×C32C2×C22⋊C411yescharacteristic
176C2×C3×D3—21no—
177C2×C3×D3—41no—
178C32×C4—22no—
179C32×C4—21no—
180C3⋊Dic3—82no—
181C3×Dic3—22no—
182C3×Dic3—41no—
183C3×Dic3—41no—
184C3×Dic3—82no—
185C3×Dic3#9911yescharacteristic
186C3×Dic3#9911yescharacteristic
187C2×#41—31no—
188C2×#41—31no—
189C4⋊Q8—31no—
190#127D3211yescharacteristic
191#124—31no—
192#124—31no—
193#121—62no—
194C4×D4—62no—
195#117—182no—
196C2×C42—31no—
197#99—32no—
198#99—92no—
199#99—182no—
200C23×C3—21no—
201C23×C3—21no—
202C23×C3—22no—
203C23×C3—31no—
204C23×C3—41no—
205C23×C3D3×D412yes—
206C23×C3C2×#6712yes—
207C22×D3#21711yescharacteristic
208C3×Q8—121no—
209C3×Q8—121no—
210C3×D4—41no—
211C3×D4—41no—
212C3×D4—41no—
213C3×D4—41no—
214C3×D4—44no—
215C3×D4—44no—
216C3×D4—81no—
217C3×D4—81no—
218C3×D4—84no—
219C3×D4—122no—
220C3×D4—122no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—31no—
225C2×C3×C4—32no—
226C2×C3×C4—41no—
227C2×C3×C4—61no—
228C2×C3×C4—61no—
229C2×C3×C4—62no—
230C2×C3×C4—62no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—122no—
234C2×C3×C4—121no—
235C2×C3×C4—121no—
236C2×C3×C4—122no—
237C2×C3×C4C2×C4×D311yescharacteristic
238C2×C3×C4C22×Dic311yescharacteristic
239#67—42no—
240#67—42no—
241C2×Dic3—22no—
242C2×Dic3—22no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—42no—
248C2×Dic3—122no—
249C2×Dic3—122no—
250C2×Dic3—242no—
251C2×Dic3#21712yes—
252C2×Dic3#21711yescharacteristic
253C4×D3—21no—
254C4×D3—21no—
255C4×D3—42no—
256Dic6—41no—
257Dic6—41no—
258C2×C32—21no—
259C2×C32—42no—
260C2×C32—81no—
261C2×C32C2×#9911yescharacteristic
262C3×D3—41no—
263#41—121no—
264#41—121no—
265#41—121no—
266#41—121no—
267C2×Q8—31no—
268C2×Q8—31no—
269C2×D4—22no—
270C2×D4—32no—
271C2×D4—61no—
272C2×D4D3×Dic311yescharacteristic
273C2×D4C2×D3211yescharacteristic
274C22×C4—32no—
275C22×C4—31no—
276C22×C4—61no—
277C22×C4—61no—
278C4⋊C4—62no—
279C4⋊C4—362no—
280C22⋊C4—62no—
281C22⋊C4—62no—
282C22⋊C4—62no—
283C22⋊C4—182no—
284C42—31no—
285C42—61no—
286C42—362no—
287C42D3×Dic311yescharacteristic
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—21no—
291C22×C3—42no—
292C22×C3—42no—
293C22×C3—42no—
294C22×C3—42no—
295C22×C3—42no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—61no—
299C22×C3—81no—
300C22×C3—82no—
301C22×C3—82no—
302C22×C3—121no—
303C22×C3D3×C22⋊C411yescharacteristic
304C22×C3C2×#21711yescharacteristic
305C2×D3—21no—
306C2×D3—41no—
307C3×C4—22no—
308C3×C4—22no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—31no—
312C3×C4—31no—
313C3×C4—41no—
314C3×C4—42no—
315C3×C4—62no—
316C3×C4—121no—
317C3×C4—121no—
318C3×C4—242no—
319Dic3—22no—
320Dic3—41no—
321Dic3—41no—
322Dic3—82no—
323Dic3—242no—
324Dic3—242no—
325Dic3—482no—
326Dic3#83411yescharacteristic
327Dic3#83411yescharacteristic
328C32#212611yescharacteristic, complemented, Sylow
329C23—21no—
330C23—31no—
331C23D3×#6712yes—
332Q8—121no—
333Q8—121no—
334D4—41no—
335D4—41no—
336D4—44no—
337D4—122no—
338D4—122no—
339C2×C4—21no—
340C2×C4—31no—
341C2×C4—32no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—122no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—122no—
352C2×C4—362no—
353C2×C4C2×D3×Dic311yescharacteristic
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—41no—
358C2×C3—42no—
359C2×C3—42no—
360C2×C3—82no—
361C2×C3—81no—
362C2×C3—81no—
363C2×C3—121no—
364C2×C3—161no—
365C2×C3D3×#9911yescharacteristic
366C2×C3C2×#83411yescharacteristic
367D3—41no—
368C22—22no—
369C22—42no—
370C22—42no—
371C22—41no—
372C22—61no—
373C22—121no—
374C22D3×#21711yescharacteristic
375C4—22no—
376C4—21no—
377C4—31no—
378C4—31no—
379C4—62no—
380C4—121no—
381C4—121no—
382C4—242no—
383C4—722no—
384C3—21no—
385C3#6915111yescharacteristic, complemented
386C3#7181911yescharacteristic, complemented
387C2—21no—
388C2—42no—
389C2—81no—
390C2—121no—
391C2D3×#83411yescharacteristic
subgroupG/H
centerC2D3×#834
commutatorC2×C32×C4C22×C4
FrattiniC2×D4C2×D32
FittingC32×#127C22
socleC2×C32C2×#99
prime radicalC3×#5716C2
cosocleC2×C32×D4C23
hypercenter#127D32
hypocenterC32#2126
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #127

Complements

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

NH ≅ G/Ntype
2. #10591528C2semidirect
3. #10589411C2semidirect
4. C3×#4545C2semidirect
5. #10589180C2semidirect
8. C2×C3×#237C4semidirect
12. C3×#936C4semidirect
15. C3×C42×D3C4semidirect
16. C3×#869C4semidirect
29. #4545D3semidirect
60. C2×#237Dic3semidirect
72. #936Dic3semidirect
77. C42×D3Dic3semidirect
78. #869Dic3semidirect
328. C32#2126semidirect
385. C3#69151semidirect
386. C3#71819semidirect

Maximal subgroups

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

subgroupindex
C3×#57162
#105915282
#105894112
C3×#45452
#105891802
#718193
#691513