Finite Group Databasebeta

← #11796376

Subgroups — #11796376

Order 1152.

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

normal subgroups only

This group has 551 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×#5668C211yesmaximal, characteristic
2#10591528C212yesmaximal, complemented
3#10589407C211yesmaximal, characteristic, complemented
4C3×#4541C211yesmaximal, characteristic, complemented
5#10589178C212yesmaximal, complemented
6#71819—31nomaximal
7#69158—31nomaximal
8C2×C3×#239C411yescharacteristic, complemented
9C2×C3×#237C2211yescharacteristic
10C32×#124C2211yescharacteristic
11C3×#944C411yescharacteristic, complemented
12C3×#940C2211yescharacteristic
13C3×#884—22no—
14C3×#67⋊C4—22no—
15C3×#875C411yescharacteristic, complemented
16C3×C42×D3C411yescharacteristic, complemented
17#63955—22no—
18#63661C2212yes—
19C3×#834C2212yes—
20#63580—22no—
21C3×#532—31no—
22#5668D311yescharacteristic
23#5240—32no—
24#4740—32no—
25C3×#354—31no—
26#4545—32no—
27#4541—31no—
28#4541—31no—
29#4541—61no—
30#4541D311yescharacteristic, complemented
31#4474—32no—
32C2×C32×Q8C2×C411yescharacteristic
33C2×C32×D4C2311yescharacteristic
34C2×C3×#67—22no—
35C2×C3×#67D412yes—
36C22×C3×Dic3D412yes—
37C3×#239—41no—
38C3×#239—41no—
39C3×#237—41no—
40C3×#237—41no—
41C2×C3×D12—21no—
42C2×C3×D12C2×C411yescharacteristic
43C2×C3×C4×D3—21no—
44C2×C3×C4×D3—21no—
45C2×C3×C4×D3C2×C411yescharacteristic
46C2×C3×Dic6C2×C411yescharacteristic
47C32×C22⋊C4—22no—
48C32×C42C2×C411yescharacteristic
49#3757—22no—
50C3×#217—22no—
51C3×(C2×D3)⋊C4—22no—
52C3×Dic3⋊C4—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×#239Dic311yescharacteristic, complemented
61C2×#237C2×D311yescharacteristic
62C3×C4⋊Q8—31no—
63C3×#127—31no—
64C3×#124—21no—
65C3×#124—31no—
66C3×#124C2×D311yescharacteristic
67C3×#124C2×D311yescharacteristic
68C3×#121—62no—
69C3×C4×D4—62no—
70C2×C3×C42—31no—
71#944Dic311yescharacteristic, complemented
72#940C2×D311yescharacteristic
73#926—62no—
74#884—22no—
75#67⋊C4—22no—
76#877—62no—
77#875Dic311yescharacteristic, complemented
78C42×D3Dic311yescharacteristic, complemented
79C3×#99—32no—
80#834—32no—
81#834—32no—
82#834—62no—
83#834—62no—
84#834C2×D312yes—
85#806—62no—
86C23×C32C2×D412yes—
87C22×C3×D3—21no—
88C22×C3×D3C22⋊C411yescharacteristic
89C32×Q8—41no—
90C32×D4—41no—
91C2×C32×C4—21no—
92C2×C32×C4—21no—
93C2×C32×C4C22×C411yescharacteristic
94C2×C3⋊Dic3—42no—
95C3×#67—42no—
96C3×#67—44no—
97C2×C3×Dic3—21no—
98C2×C3×Dic3—22no—
99C2×C3×Dic3—22no—
100C2×C3×Dic3—42no—
101C2×C3×Dic3C22⋊C412yes—
102C2×C3×Dic3C22⋊C411yescharacteristic
103C3×D12—41no—
104C3×D12—42no—
105C3×C4×D3—21no—
106C3×C4×D3—21no—
107C3×C4×D3—42no—
108C3×C4×D3—41no—
109C3×C4×D3—41no—
110C3×Dic6—41no—
111#532—31no—
112#410—92no—
113#354—31no—
114#354—91no—
115#353—92no—
116C3×#41—121no—
117C3×#41—121no—
118C3×#41—121no—
119C3×#41—121no—
120C2×C3×Q8—21no—
121C2×C3×Q8—31no—
122C2×C3×Q8C4×D311yescharacteristic
123C2×C3×Q8C2×Dic311yescharacteristic
124C2×C3×D4—21no—
125C2×C3×D4—32no—
126C2×C3×D4—31no—
127C2×C3×D4—61no—
128C2×C3×D4—62no—
129C2×C3×D4C22×D311yescharacteristic
130C2×C3×D4C22×D311yescharacteristic
131C22×C3×C4—32no—
132C22×C3×C4—31no—
133C22×C3×C4—61no—
134C22×C3×C4—61no—
135C2×#67—22no—
136C2×#67#6712yes—
137C22×Dic3#6712yes—
138#239—41no—
139#239—41no—
140#237—41no—
141#237—41no—
142C2×D12—21no—
143C2×D12C2×Dic311yescharacteristic
144C2×C4×D3—21no—
145C2×C4×D3—21no—
146C2×C4×D3C2×Dic311yescharacteristic
147C2×Dic6C2×Dic311yescharacteristic
148C3×C4⋊C4—62no—
149C3×C22⋊C4—22no—
150C3×C22⋊C4—22no—
151C3×C22⋊C4—42no—
152C3×C22⋊C4—62no—
153C3×C22⋊C4—62no—
154C3×C42—21no—
155C3×C42—31no—
156C3×C42—61no—
157C3×C42C4×D311yescharacteristic
158C3×C42C2×Dic311yescharacteristic
159#217—22no—
160#217—62no—
161#217—62no—
162#217—122no—
163(C2×D3)⋊C4—22no—
164C4⋊Dic3—122no—
165Dic3⋊C4—22no—
166Dic3⋊C4—122no—
167C4×Dic3—21no—
168C4×Dic3—122no—
169C4×Dic3—122no—
170C4×Dic3C2×Dic311yescharacteristic
171C22×C32—22no—
172C22×C32—42no—
173C22×C32C2×C22⋊C411yescharacteristic
174C2×C3×D3—21no—
175C2×C3×D3—41no—
176C2×C3×D3—42no—
177C2×C3×D3—41no—
178C32×C4—21no—
179C32×C4—41no—
180C32×C4—41no—
181C3⋊Dic3—82no—
182C3×Dic3—22no—
183C3×Dic3—22no—
184C3×Dic3—82no—
185C3×Dic3#9911yescharacteristic
186C3×Dic3#9911yescharacteristic
187C2×#41—31no—
188C2×#41—31no—
189C4⋊Q8—31no—
190#127—31no—
191#124—31no—
192#124D3211yescharacteristic
193#121—62no—
194C4×D4—62no—
195#117—182no—
196C2×C42—31no—
197#99—32no—
198#99—92no—
199#99—182no—
200C23×C3—22no—
201C23×C3—31no—
202C23×C3—61no—
203C23×C3D3×D412yes—
204C23×C3C2×#6712yes—
205C22×D3—21no—
206C22×D3#21711yescharacteristic
207C3×Q8—41no—
208C3×Q8—41no—
209C3×Q8—81no—
210C3×Q8—121no—
211C3×D4—41no—
212C3×D4—41no—
213C3×D4—81no—
214C3×D4—121no—
215C3×D4—122no—
216C3×D4—122no—
217C3×D4—124no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—31no—
224C2×C3×C4—32no—
225C2×C3×C4—41no—
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—121no—
233C2×C3×C4—121no—
234C2×C3×C4—122no—
235C2×C3×C4—122no—
236C2×C3×C4C2×C4×D311yescharacteristic
237C2×C3×C4C22×Dic311yescharacteristic
238#67—42no—
239#67—44no—
240C2×Dic3—21no—
241C2×Dic3—22no—
242C2×Dic3—22no—
243C2×Dic3—42no—
244C2×Dic3—122no—
245C2×Dic3—122no—
246C2×Dic3—242no—
247C2×Dic3#21712yes—
248C2×Dic3#21711yescharacteristic
249D12—41no—
250D12—42no—
251C4×D3—21no—
252C4×D3—21no—
253C4×D3—42no—
254C4×D3—41no—
255C4×D3—41no—
256Dic6—41no—
257C2×C32—21no—
258C2×C32—42no—
259C2×C32C2×#9911yescharacteristic
260C3×D3—41no—
261C3×D3—81no—
262#41—121no—
263#41—121no—
264#41—121no—
265#41—121no—
266C2×Q8—31no—
267C2×Q8D3×Dic311yescharacteristic
268C2×D4—32no—
269C2×D4—31no—
270C2×D4—61no—
271C2×D4—62no—
272C2×D4C2×D3211yescharacteristic
273C22×C4—32no—
274C22×C4—31no—
275C22×C4—61no—
276C22×C4—61no—
277C4⋊C4—62no—
278C4⋊C4—362no—
279C22⋊C4—22no—
280C22⋊C4—62no—
281C22⋊C4—62no—
282C22⋊C4—182no—
283C42—31no—
284C42—61no—
285C42—362no—
286C42D3×Dic311yescharacteristic
287C22×C3—22no—
288C22×C3—22no—
289C22×C3—21no—
290C22×C3—42no—
291C22×C3—42no—
292C22×C3—42no—
293C22×C3—61no—
294C22×C3—82no—
295C22×C3—121no—
296C22×C3—122no—
297C22×C3—121no—
298C22×C3D3×C22⋊C411yescharacteristic
299C22×C3C2×#21711yescharacteristic
300C2×D3—21no—
301C2×D3—41no—
302C2×D3—42no—
303C2×D3—41no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—41no—
309C3×C4—41no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—62no—
314C3×C4—62no—
315C3×C4—81no—
316C3×C4—81no—
317C3×C4—242no—
318Dic3—22no—
319Dic3—22no—
320Dic3—82no—
321Dic3—242no—
322Dic3—242no—
323Dic3—482no—
324Dic3#83411yescharacteristic
325Dic3#83411yescharacteristic
326C32#212611yescharacteristic, complemented, Sylow
327C23—31no—
328C23—61no—
329C23D3×#6712yes—
330Q8—41no—
331Q8—121no—
332D4—41no—
333D4—121no—
334D4—122no—
335D4—122no—
336D4—124no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—31no—
340C2×C4—32no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—61no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—362no—
351C2×C4C2×D3×Dic311yescharacteristic
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—41no—
356C2×C3—42no—
357C2×C3—42no—
358C2×C3—82no—
359C2×C3—121no—
360C2×C3—241no—
361C2×C3D3×#9911yescharacteristic
362C2×C3C2×#83411yescharacteristic
363D3—41no—
364D3—81no—
365C22—22no—
366C22—42no—
367C22—61no—
368C22—121no—
369C22—122no—
370C22—121no—
371C22D3×#21711yescharacteristic
372C4—21no—
373C4—31no—
374C4—31no—
375C4—41no—
376C4—41no—
377C4—62no—
378C4—62no—
379C4—242no—
380C4—722no—
381C3—21no—
382C3#6915811yescharacteristic, complemented
383C3#7181911yescharacteristic, complemented
384C2—21no—
385C2—42no—
386C2—121no—
387C2—241no—
388C2D3×#83411yescharacteristic
subgroupG/H
centerC2D3×#834
commutatorC2×C32×C4C22×C4
FrattiniC2×D4C2×D32
FittingC32×#124C22
socleC2×C32C2×#99
prime radicalC3×#5668C2
cosocleC2×C32×D4C23
hypercenter#124D32
hypocenterC32#2126
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #10591528C2semidirect
3. #10589407C2semidirect
4. C3×#4541C2semidirect
5. #10589178C2semidirect
8. C2×C3×#239C4semidirect
11. C3×#944C4semidirect
15. C3×#875C4semidirect
16. C3×C42×D3C4semidirect
30. #4541D3semidirect
60. C2×#239Dic3semidirect
71. #944Dic3semidirect
77. #875Dic3semidirect
78. C42×D3Dic3semidirect
326. C32#2126semidirect
382. C3#69158semidirect
383. C3#71819semidirect

Maximal subgroups

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

subgroupindex
C3×#56682
#105915282
#105894072
C3×#45412
#105891782
#718193
#691583