Finite Group Databasebeta

← C3×#69173 #11796563

Subgroups — C3×#69173 #11796563

Order 1152.

490 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 110 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×#5724C211yesmaximal, characteristic
2C3×#4751C212yesmaximal
3C32×#356C211yesmaximal, characteristic, complemented
4C3×#4548C211yesmaximal, characteristic, complemented
5C3×#4478C212yesmaximal, complemented
6C3×#2135—31nomaximal
7#69173C311yesmaximal, characteristic, direct product factor
8C2×C3×#239C2211yescharacteristic
9C2×C3×D3×Q8C411yescharacteristic
10C32×C4⋊Q8C2211yescharacteristic
11C3×#947C411yescharacteristic
12C3×#944C2211yescharacteristic
13C3×D3×M16—22no—
14C3×#896—22no—
15C3×#894—22no—
16C3×#875C411yescharacteristic
17C3×#872C411yescharacteristic
18C32×#101C2212yes—
19C3×#836C2212yes—
20C3×#823—22no—
21C3×#536—31no—
22#5724C2×C311yescharacteristic
23C3×#413—32no—
24#4751C2×C312yes—
25C3×#356—21no—
26C3×#356—31no—
27C3×#356—32no—
28C3×#356D311yescharacteristic, complemented
29C3×#356C2×C311yescharacteristic, complemented
30#4548C2×C311yescharacteristic, complemented
31#4478C2×C312yescomplemented
32C2×C32×Q8C2×C411yescharacteristic
33C2×C32×Q8C2311yescharacteristic
34C3×#239—42no—
35C3×D3×Q8—42no—
36C2×C3×D12C2×C411yescharacteristic
37C2×C3×D12D412yes—
38C2×C3×C4×D3—21no—
39C2×C3×C4×D3C2×C411yescharacteristic
40C2×C3×C4×D3D412yes—
41C2×C3×Dic6—21no—
42C2×C3×Dic6C2×C411yescharacteristic
43C32×M16—22no—
44C32×C4⋊C4—22no—
45C32×C42C2×C411yescharacteristic
46C3×(C2×D3)⋊C4—21no—
47C3×(C2×D3)⋊C4—22no—
48C3×(C2×D3)⋊C4—22no—
49C3×Dic3⋊C4—22no—
50C3×Dic3⋊C4—21no—
51C3×C4×Dic3C2×C411yescharacteristic
52C3×#208—22no—
53C3×ΓL(1,𝔽25)—42no—
54C3×D3×C8—42no—
55#2135—31noSylow
56C2×C3×#41—31no—
57C22×C3×Q8—31no—
58C2×#239C22×C311yescharacteristic
59C2×D3×Q8C3×C411yescharacteristic
60C2×C3×M16—62no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—31no—
63C3×C4⋊Q8C2×D311yescharacteristic
64C3×C4⋊Q8C22×C311yescharacteristic
65C3×#124—31no—
66C3×#124—31no—
67C3×#123—62no—
68C3×C22⋊Q8—62no—
69C3×#117—31no—
70#947C3×C411yescharacteristic
71#944C22×C311yescharacteristic
72D3×M16—22no—
73#896—22no—
74#894—22no—
75#875C3×C411yescharacteristic
76#872C3×C411yescharacteristic
77C3×#101—22no—
78C3×#101—32no—
79C3×#101C2×D312yes—
80C3×#101C22×C312yes—
81C3×#100—62no—
82#836C22×C312yes—
83#823—22no—
84C22×C3×D3C22⋊C412yes—
85C22×C3×D3C22⋊C411yescharacteristic
86C32×Q8—41no—
87C32×Q8—41no—
88C2×C32×C4—21no—
89C2×C32×C4—21no—
90C2×C32×C4C22×C411yescharacteristic
91C2×C32×C4C2×D412yes—
92C2×C3×Dic3—21no—
93C2×C3×Dic3—21no—
94C2×C3×Dic3C22⋊C411yescharacteristic
95C3×D12—41no—
96C3×D12—42no—
97C3×C4×D3—22no—
98C3×C4×D3—22no—
99C3×C4×D3—22no—
100C3×C4×D3—42no—
101C3×Dic6—42no—
102C3×Dic6—41no—
103C32×C8—42no—
104C3×C3⋊C8—42no—
105#536—31no—
106#413—32no—
107#356—31no—
108#356—32no—
109#356C3×D311yescharacteristic, complemented
110C3×#41—122no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—31no—
114C2×C3×Q8—61no—
115C2×C3×Q8—122no—
116C2×C3×Q8C4×D311yescharacteristic
117C2×C3×Q8C2×C3×C411yescharacteristic
118C2×C3×Q8C22×D311yescharacteristic
119C2×C3×Q8C23×C311yescharacteristic
120C2×C3×D4—31no—
121C2×C3×D4—32no—
122C22×C3×C4—31no—
123C22×C3×C4—32no—
124C22×C3×C4—61no—
125#239—42no—
126D3×Q8—42no—
127C2×D12C2×C3×C411yescharacteristic
128C2×D12C3×D412yes—
129C2×C4×D3—21no—
130C2×C4×D3C2×C3×C411yescharacteristic
131C2×C4×D3C3×D412yes—
132C2×Dic6—21no—
133C2×Dic6C2×C3×C411yescharacteristic
134C3×M16—22no—
135C3×M16—22no—
136C3×M16—42no—
137C3×M16—62no—
138C3×M16—122no—
139C2×C3×C8—122no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—42no—
143C3×C4⋊C4—61no—
144C3×C4⋊C4—62no—
145C3×C22⋊C4—61no—
146C3×C22⋊C4—62no—
147C3×C22⋊C4—62no—
148C3×C42—21no—
149C3×C42—31no—
150C3×C42C4×D311yescharacteristic
151C3×C42C2×C3×C411yescharacteristic
152(C2×D3)⋊C4—21no—
153(C2×D3)⋊C4—22no—
154(C2×D3)⋊C4—22no—
155Dic3⋊C4—22no—
156Dic3⋊C4—21no—
157C4×Dic3C2×C3×C411yescharacteristic
158#208—22no—
159ΓL(1,𝔽25)—42no—
160D3×C8—42no—
161C22×C32C2×C22⋊C411yescharacteristic
162C2×C3×D3—21no—
163C2×C3×D3—22no—
164C2×C3×D3—42no—
165C2×C3×D3—41no—
166C32×C4—22no—
167C32×C4—21no—
168C32×C4—41no—
169C32×C4—41no—
170C3×Dic3—41no—
171C3×Dic3—41no—
172C3×Dic3#9912yes—
173C2×#41—31no—
174C22×Q8—31no—
175C2×M16—62no—
176C4⋊Q8—31no—
177C4⋊Q8C2×C3×D311yescharacteristic
178#124—31no—
179#124—31no—
180#123—62no—
181C22⋊Q8—62no—
182#117—31no—
183#101—32no—
184#101C2×C3×D312yes—
185#100—62no—
186C23×C3—32no—
187C23×C3—31no—
188C22×D3C3×C22⋊C412yes—
189C22×D3C3×C22⋊C411yescharacteristic
190C3×Q8—41no—
191C3×Q8—41no—
192C3×Q8—41no—
193C3×Q8—41no—
194C3×Q8—81no—
195C3×Q8—81no—
196C3×Q8—121no—
197C3×Q8—122no—
198C3×D4—122no—
199C3×D4—121no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—22no—
206C2×C3×C4—31no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—62no—
210C2×C3×C4—62no—
211C2×C3×C4—62no—
212C2×C3×C4—61no—
213C2×C3×C4—61no—
214C2×C3×C4—122no—
215C2×C3×C4C2×C4×D311yescharacteristic
216C2×C3×C4D3×D412yes—
217C2×C3×C4C22×C3×C411yescharacteristic
218C2×C3×C4C2×C3×D412yes—
219C2×Dic3—21no—
220C2×Dic3—21no—
221C2×Dic3C3×C22⋊C411yescharacteristic
222D12—41no—
223D12—42no—
224C4×D3—22no—
225C4×D3—22no—
226C4×D3—22no—
227C4×D3—42no—
228Dic6—42no—
229Dic6—41no—
230C3×C8—42no—
231C3×C8—42no—
232C3×C8—82no—
233C3×C8—122no—
234C3⋊C8—42no—
235C2×C32—21no—
236C2×C32C2×#9911yescharacteristic
237C3×D3—41no—
238C3×D3—42no—
239#41—122no—
240C2×Q8—31no—
241C2×Q8—61no—
242C2×Q8—122no—
243C2×Q8C3×C4×D311yescharacteristic
244C2×Q8C22×C3×D311yescharacteristic
245C2×D4—31no—
246C2×D4—32no—
247C22×C4—31no—
248C22×C4—32no—
249C22×C4—61no—
250M16—22no—
251M16—62no—
252M16—122no—
253C2×C8—122no—
254C4⋊C4—22no—
255C4⋊C4—61no—
256C4⋊C4—62no—
257C22⋊C4—61no—
258C22⋊C4—62no—
259C22⋊C4—62no—
260C42—31no—
261C42C3×C4×D311yescharacteristic
262C22×C3—21no—
263C22×C3—61no—
264C22×C3—62no—
265C22×C3—122no—
266C22×C3—121no—
267C22×C3D3×C22⋊C411yescharacteristic
268C22×C3C2×C3×C22⋊C411yescharacteristic
269C2×D3—21no—
270C2×D3—22no—
271C2×D3—42no—
272C2×D3—41no—
273C3×C4—22no—
274C3×C4—21no—
275C3×C4—22no—
276C3×C4—21no—
277C3×C4—32no—
278C3×C4—42no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4—41no—
282C3×C4—42no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—81no—
286C3×C4—121no—
287C3×C4—121no—
288Dic3—41no—
289Dic3—41no—
290Dic3C3×#9912yes—
291C32#213511yescharacteristic, complemented, Sylow
292C23—32no—
293C23—31no—
294Q8—41no—
295Q8—41no—
296Q8—121no—
297Q8—122no—
298D4—122no—
299D4—121no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—31no—
303C2×C4—62no—
304C2×C4—62no—
305C2×C4—62no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—122no—
309C2×C4C2×C3×C4×D311yescharacteristic
310C2×C4C3×D3×D412yes—
311C8—42no—
312C8—122no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—41no—
317C2×C3—121no—
318C2×C3—122no—
319C2×C3D3×#9911yescharacteristic
320C2×C3C2×C3×#9911yescharacteristic
321D3—41no—
322D3—42no—
323C22—61no—
324C22—62no—
325C22—122no—
326C22—121no—
327C22C3×D3×C22⋊C411yescharacteristic
328C4—22no—
329C4—21no—
330C4—32no—
331C4—41no—
332C4—41no—
333C4—121no—
334C4—121no—
335C3—21no—
336C3#6917311yescharacteristic, direct product factor
337C3C3×#213511yescharacteristic, complemented
338C2—21no—
339C2—121no—
340C2—122no—
341C2C3×D3×#9911yescharacteristic
subgroupG/H
centerC2×C3D3×#99
commutatorC2×C3×C4C22×C3×C4
FrattiniC2×Q8C22×C3×D3
FittingC32×#356C2
socleC2×C32C2×#99
prime radicalC2×C3×#239C22
cosocleC2×C3×Q8C23×C3
hypercenterC3×#356D3
hypocenterC3C3×#2135
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ C3×#356

Complements

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

NH ≅ G/Ntype
3. C32×#356C2semidirect
4. C3×#4548C2semidirect
5. C3×#4478C2semidirect
7. #69173C3direct (N × H)
28. C3×#356D3semidirect
29. C3×#356C2×C3semidirect
30. #4548C2×C3semidirect
31. #4478C2×C3semidirect
109. #356C3×D3semidirect
291. C32#2135semidirect
336. C3#69173direct (N × H)
337. C3C3×#2135semidirect

Maximal subgroups

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

subgroupindex
C3×#57242
C3×#47512
C32×#3562
C3×#45482
C3×#44782
C3×#21353
#691733