Finite Group Databasebeta

← #11798208

Subgroups — #11798208

Order 1152.

461 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 461 conjugacy classes of subgroups — too many to draw. Only its 86 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×D3×Dic12C211yesmaximal, characteristic
2C2×C3×Dic24C211yesmaximal, characteristic, complemented
3#10590851C211yesmaximal, characteristic
4C2×#63551C211yesmaximal, characteristic, complemented
5#10589369C211yesmaximal, characteristic, complemented
6C3×(C2×D3)⋊C16C211yesmaximal, characteristic
7#10589118C211yesmaximal, characteristic, complemented
8#71502—31nomaximal
9#69113—31nomaximal
10C2×D3×Dic6—21no—
11C2×#3746C2211yescharacteristic
12C2×C3×Dic12C2211yescharacteristic
13C2×C3×Dic12C2211yescharacteristic
14C2×C3×D3×C8C2211yescharacteristic
15#63856—21no—
16C2×#3719—21no—
17D3×Dic12—22no—
18D3×Dic12—22no—
19C2×C32×C16C2211yescharacteristic
20#63645C2211yescharacteristic
21C2×C3×C3⋊C16C2211yescharacteristic
22C3×Dic24—22no—
23#63566—21no—
24#63551—22no—
25C2×D3×Q16—31no—
26C22×Dic12—31no—
27C2×C3×Q32—31no—
28#5188—31no—
29C2×#829—31no—
30#5116—31no—
31C2×Dic24—31no—
32C2×Dic24D311yescharacteristic, complemented
33C3×C22⋊C16—31no—
34#4564—31no—
35(C2×D3)⋊C16D311yescharacteristic
36#4503—31no—
37#4503—31no—
38#4503—31no—
39#4503—61no—
40C2×#634—21no—
41C2×C3×C4×D3D411yescharacteristic
42C2×C3×Dic6—21no—
43C2×C3×Dic6—21no—
44C2×#627—41no—
45C2×D3×Dic3—41no—
46D3×Dic6—24no—
47D3×Dic6—42no—
48C2×C32×C8C2311yescharacteristic
49C4⋊(C3⋊Dic3)—21no—
50#3746—21no—
51#3746D412yes—
52C2×C3×C3⋊C8D411yescharacteristic
53C3×Dic12—21no—
54C3×Dic12—21no—
55C3×Dic12—21no—
56C3×Dic12D412yes—
57C3×D3×C8—21no—
58C3×D3×C8—21no—
59#3724—41no—
60(C2×D3)⋊Dic3—41no—
61#3719—24no—
62C32×C16—21no—
63C3×C3⋊C16—21no—
64#2188—91noSylow
65C2×D3×Q8—61no—
66C22×Dic6—61no—
67C2×C3×Q16—31no—
68C2×C3×Q16—31no—
69C22×C3×C8—31no—
70#946—61no—
71C2×#216—61no—
72#924—61no—
73D3×Q16—62no—
74D3×Q16—62no—
75C2×Dic12—31no—
76C2×Dic12—31no—
77C2×Dic12—61no—
78C2×Dic12—62no—
79C2×Dic12—61no—
80C2×Dic12—62no—
81C2×Dic12C2×D311yescharacteristic
82C2×Dic12C2×D311yescharacteristic
83C2×D3×C8C2×D311yescharacteristic
84C3×Q32—62no—
85C2×C3×C16—21no—
86C2×C3×C16—31no—
87C2×C3×C16C2×D311yescharacteristic
88C2×C3×C16C2×D311yescharacteristic
89#835—61no—
90#829—62no—
91#818—31no—
92#818—31no—
93#818—61no—
94#816—61no—
95C2×C3⋊C16C2×D311yescharacteristic
96Dic24—22no—
97Dic24—62no—
98C22×C3×D3D811yescharacteristic
99C2×C32×C4C2×D411yescharacteristic
100C2×C3⋊Dic3—41no—
101C2×C3⋊Dic3—41no—
102#634—22no—
103#634—41no—
104C2×C3×Dic3—41no—
105C2×C3×Dic3—41no—
106C2×C3×Dic3D811yescharacteristic
107C3×C4×D3—21no—
108C3×C4×D3—21no—
109C3×Dic6—22no—
110C3×Dic6—41no—
111C3×Dic6—41no—
112C3×Dic6—41no—
113#627—44no—
114D3×Dic3—44no—
115C32×C8C2×D411yescharacteristic
116C32×C8C2×D411yescharacteristic
117C3×C3⋊C8—21no—
118C22×Q16—91no—
119C2×Q32—31no—
120C2×Q32—91no—
121#467—91no—
122#358—91no—
123#358—91no—
124C22⋊C16—31no—
125C2×C3×Q8—61no—
126C2×C3×Q8—61no—
127C22×C3×C4—31no—
128C22×Dic3—121no—
129D3×Q8—64no—
130D3×Q8—122no—
131C2×C4×D3—121no—
132C2×C4×D3D1211yescharacteristic
133C2×Dic6—21no—
134C2×Dic6—21no—
135C2×Dic6—64no—
136C2×Dic6—61no—
137C2×Dic6—61no—
138C2×Dic6—121no—
139C2×Dic6—122no—
140C2×Dic6—121no—
141C2×Dic6—121no—
142C3×Q16—32no—
143C3×Q16—61no—
144C3×Q16—61no—
145C3×Q16—61no—
146C2×C3×C8—21no—
147C2×C3×C8—31no—
148C2×C3×C8—61no—
149C2×C3×C8—61no—
150C2×C3×C8C22×D311yescharacteristic
151C2×C3×C8C22×D311yescharacteristic
152#217—121no—
153#216—64no—
154(C2×D3)⋊C4—121no—
155C4⋊Dic3—61no—
156C4⋊Dic3—61no—
157C4⋊Dic3—121no—
158Dic3⋊C4—121no—
159Dic3⋊C4—121no—
160C2×C3⋊C8D1211yescharacteristic
161Dic12—21no—
162Dic12—21no—
163Dic12—21no—
164Dic12—32no—
165Dic12—32no—
166Dic12—62no—
167Dic12—64no—
168Dic12—61no—
169Dic12—61no—
170Dic12—121no—
171Dic12#6712yes—
172D3×C8—21no—
173D3×C8—21no—
174C3×C16—21no—
175C3×C16—21no—
176C3×C16—41no—
177C3×C16—61no—
178C3⋊C16—21no—
179C22×C32C2×D811yescharacteristic
180C2×C3×D3—21no—
181C2×C3×D3—21no—
182C2×C3×D3Q3212yescomplemented
183C32×C4#12011yescharacteristic
184C32×C4AGL(1,ℤ/8)11yescharacteristic
185C3⋊Dic3—42no—
186C3⋊Dic3—81no—
187C3×Dic3—21no—
188C3×Dic3—42no—
189C3×Dic3—81no—
190C22×Q8—181no—
191C2×Q16—31no—
192C2×Q16—31no—
193C2×Q16—91no—
194C2×Q16—181no—
195C2×Q16—182no—
196C2×Q16—182no—
197C22×C8—31no—
198C22⋊Q8—181no—
199Q32—62no—
200Q32—182no—
201C2×C16—31no—
202C2×C16D3211yescharacteristic
203#107—91no—
204#103—181no—
205C23×C3—31no—
206C22×D3D2411yescharacteristic
207C3×Q8—62no—
208C3×Q8—121no—
209C3×Q8—121no—
210C3×Q8—121no—
211C2×C3×C4—21no—
212C2×C3×C4—31no—
213C2×C3×C4—61no—
214C2×C3×C4—61no—
215C2×C3×C4—121no—
216C2×C3×C4—121no—
217C2×C3×C4C2×D1211yescharacteristic
218C2×C3×C4D3×D411yescharacteristic
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—124no—
222C2×Dic3—121no—
223C2×Dic3—121no—
224C2×Dic3—121no—
225C2×Dic3—121no—
226C2×Dic3—241no—
227C2×Dic3—241no—
228C2×Dic3D2411yescharacteristic
229C4×D3—21no—
230C4×D3—21no—
231C4×D3—124no—
232Dic6—22no—
233Dic6—41no—
234Dic6—41no—
235Dic6—41no—
236Dic6—62no—
237Dic6—62no—
238Dic6—122no—
239Dic6—124no—
240Dic6—124no—
241Dic6—121no—
242Dic6—121no—
243Dic6—241no—
244C3×C8—21no—
245C3×C8—21no—
246C3×C8—61no—
247C3×C8D3×D411yescharacteristic
248C3×C8D3×D411yescharacteristic
249C3×C8D3×D411yescharacteristic
250C3×C8C2×#6711yescharacteristic
251C3⋊C8—21no—
252C2×C32#44711yescharacteristic
253C2×C32C2×Q3211yescharacteristic
254C2×C32#51011yescharacteristic
255C3×D3—22no—
256C2×Q8—61no—
257C2×Q8—61no—
258C2×Q8—184no—
259C2×Q8—181no—
260C2×Q8—361no—
261C2×Q8—362no—
262C22×C4—31no—
263C22×C4—361no—
264Q16—32no—
265Q16—61no—
266Q16—61no—
267Q16—61no—
268Q16—92no—
269Q16—184no—
270Q16—181no—
271C2×C8—31no—
272C2×C8—61no—
273C2×C8—61no—
274C2×C8C2×D3211yescharacteristic
275C4⋊C4—181no—
276C4⋊C4—361no—
277C22⋊C4—361no—
278C16—21no—
279C16—61no—
280C22×C3—21no—
281C22×C3—32no—
282C22×C3—61no—
283C22×C3—61no—
284C22×C3C2×D2411yescharacteristic
285C22×C3D3×D811yescharacteristic
286C2×D3—21no—
287C2×D3—21no—
288C2×D3Dic2412yescomplemented
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—61no—
292C3×C4—122no—
293C3×C4—241no—
294C3×C4#88211yescharacteristic
295C3×C4#90811yescharacteristic
296C3×C4#91111yescharacteristic
297C3×C4#93711yescharacteristic
298Dic3—21no—
299Dic3—42no—
300Dic3—81no—
301Dic3—122no—
302Dic3—122no—
303Dic3—242no—
304Dic3—241no—
305Dic3—241no—
306Dic3—481no—
307C32#218811yescharacteristic, complemented, Sylow
308C23—31no—
309Q8—62no—
310Q8—121no—
311Q8—121no—
312Q8—121no—
313Q8—182no—
314Q8—364no—
315Q8—361no—
316C2×C4—31no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—364no—
322C2×C4—361no—
323C2×C4—361no—
324C2×C4D3×D1211yescharacteristic
325C8—61no—
326C8#380211yescharacteristic
327C8D3×#6711yescharacteristic
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—62no—
332C2×C3#473211yescharacteristic
333C2×C3C2×Dic2411yescharacteristic
334C2×C3#490911yescharacteristic
335C2×C3D3×Q3211yescharacteristic
336C2×C3#491811yescharacteristic
337C2×C3#515611yescharacteristic
338D3—22no—
339C22—32no—
340C22—61no—
341C22—61no—
342C22D3×D2411yescharacteristic
343C4—61no—
344C4—122no—
345C4—241no—
346C4—362no—
347C4—721no—
348C4#6379511yescharacteristic
349C4#6392211yescharacteristic
350C3—21no—
351C3#6911311yescharacteristic, complemented
352C3#7150211yescharacteristic, complemented
353C2—62no—
354C2D3×Dic2411yescharacteristic
355C2#1059023811yescharacteristic
356C2#1059094411yescharacteristic
subgroupG/H
centerC22D3×D24
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
FittingC2×C32×C16C22
socleC22×C32C2×D8
prime radicalC22×C3×D3D8
hypercenterC2×C16D32
hypocenterC32#2188
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16

Complements

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

NH ≅ G/Ntype
2. C2×C3×Dic24C2semidirect
4. C2×#63551C2semidirect
5. #10589369C2semidirect
7. #10589118C2semidirect
32. C2×Dic24D3semidirect
182. C2×C3×D3Q32semidirect
288. C2×D3Dic24semidirect
307. C32#2188semidirect
351. C3#69113semidirect
352. C3#71502semidirect

Maximal subgroups

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

subgroupindex
C2×D3×Dic122
C2×C3×Dic242
#105908512
C2×#635512
#105893692
C3×(C2×D3)⋊C162
#105891182
#715023
#691133