Finite Group Databasebeta

← #13167015

Subgroups — #13167015

Order 1728.

438 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 57 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
1#11723685C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic, complemented
3#11723104C211yesmaximal, characteristic, complemented
4#11722779C212yesmaximal, complemented
5#11722772C212yesmaximal
6#10593367—31nomaximal
7C2×#6295C2211yescharacteristic
8C2×#6294—21no—
9C2×C3×#627—21no—
10C2×C3⋊#622C2212yes—
11C3×#3752—21no—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C32×C4⋊Dic3—21no—
14#88815C2211yescharacteristic
15C3×#3724C2212yes—
16C3×#888—61no—
17#63919D311yescharacteristic, complemented
18#63902—31no—
19#63880—61no—
20#63779—31no—
21#63743—32no—
22#63739—32no—
23C22×C3×C3⋊D3—21no—
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3—21no—
26C2×C3×C3⋊Dic3D411yescharacteristic
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C2×C32×Dic3—22no—
29#6295—21no—
30#6295Q812yescomplemented
31#6294—42no—
32C3×#627—42no—
33C3⋊#622—22no—
34#5087—91nomaximal
35C2×C3×Dic6—62no—
36C2×#627—21no—
37C2×#626—61no—
38C2×#625—31no—
39C2×#624—61no—
40C2×D3×Dic3—61no—
41C2×D3×Dic3—61no—
42C2×#622—32no—
43C32×C4⋊C4—61no—
44#3752—21no—
45C4×C3⋊Dic3C2×D311yescharacteristic
46C3×(C2×D3)⋊C4—61no—
47C3×(C2×D3)⋊C4—61no—
48C3×(C2×D3)⋊C4—121no—
49C3×C4⋊Dic3—21no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—41no—
52C3×Dic3⋊C4—62no—
53C3×C4×Dic3—61no—
54C3×C4×Dic3—61no—
55#3724—31no—
56#3724C2×D312yes—
57Dic3⋊Dic3—61no—
58C22×C33C2×D411yescharacteristic
59C2×C3×C3⋊D3—22no—
60C2×C3×C3⋊D3—41no—
61C2×C3×C3⋊D3—41no—
62C33×C4—21no—
63C3×C3⋊Dic3—41no—
64C3×C3⋊Dic3D812yes—
65C3×C3⋊Dic3C2×Q811yescharacteristic
66C3×C3⋊Dic3#4111yescharacteristic
67C32×Dic3—42no—
68C3×C4⋊Q8—91no—
69#888—61no—
70#67⋊C4—181no—
71C4×D12—91no—
72#809—92no—
73#807—92no—
74C4⋊(C3⋊C8)—91no—
75C22×C3⋊D3—21no—
76C22×C3×D3—61no—
77C22×C3×D3—61no—
78C22×C3×D3—121no—
79C2×C32×C4—21no—
80C2×C32×C4—21no—
81C2×C32×C4—41no—
82C2×C32×C4—41no—
83C2×C32×C4—62no—
84C2×C32×C4#6711yescharacteristic
85C2×C3⋊Dic3—21no—
86C2×C3⋊Dic3#6711yescharacteristic
87C2×C3⋊Dic3C22×D311yescharacteristic
88C2×C3×Dic3—22no—
89C2×C3×Dic3—22no—
90C2×C3×Dic3—42no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—121no—
98C3×Dic6—124no—
99#627—42no—
100#626—62no—
101#626—62no—
102#625—32no—
103#625—61no—
104#624—122no—
105D3×Dic3—122no—
106D3×Dic3—122no—
107#622—62no—
108#474—271noSylow
109C2×C33C22⋊Q811yescharacteristic
110C2×C33C2×D811yescharacteristic
111C2×C33#13711yescharacteristic
112C3×C3⋊D3—42no—
113C2×C3×Q8—181no—
114C2×#67—181no—
115C22×Dic3—181no—
116C2×D12—91no—
117C2×C4×D3—181no—
118C2×C4×D3—181no—
119C2×Dic6—62no—
120C3×C4⋊C4—61no—
121C3×C4⋊C4—61no—
122C3×C4⋊C4—92no—
123C3×C4⋊C4—121no—
124C3×C22⋊C4—181no—
125C3×C42—91no—
126(C2×D3)⋊C4—61no—
127(C2×D3)⋊C4—61no—
128(C2×D3)⋊C4—121no—
129C4⋊Dic3—21no—
130C4⋊Dic3—91no—
131Dic3⋊C4—62no—
132Dic3⋊C4—181no—
133C4×Dic3—61no—
134C4×Dic3—61no—
135C2×C3⋊C8—92no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32—41no—
139C22×C32—41no—
140C22×C32C2×#6711yescharacteristic
141C2×C3⋊D3—22no—
142C2×C3⋊D3—41no—
143C2×C3⋊D3—41no—
144C2×C3×D3—62no—
145C2×C3×D3—62no—
146C2×C3×D3—122no—
147C2×C3×D3—121no—
148C2×C3×D3—121no—
149C2×C3×D3—121no—
150C2×C3×D3—121no—
151C2×C3×D3—241no—
152C2×C3×D3—241no—
153C32×C4—22no—
154C32×C4—21no—
155C32×C4—41no—
156C32×C4—42no—
157C32×C4—81no—
158C32×C4—122no—
159C3⋊Dic3—41no—
160C3⋊Dic3#21312yes—
161C3⋊Dic3D3×Q811yescharacteristic
162C3⋊Dic3#23911yescharacteristic
163C3×Dic3—42no—
164C3×Dic3—42no—
165C3×Dic3—62no—
166C3×Dic3—62no—
167C3×Dic3—61no—
168C3×Dic3—61no—
169C3×Dic3—61no—
170C3×Dic3—61no—
171C3×Dic3—82no—
172C3×Dic3—121no—
173C3×Dic3—121no—
174C3×Dic3—241no—
175C4⋊Q8—91no—
176C4×D4—271no—
177#107—272no—
178C4⋊C8—271no—
179#102—272no—
180C33#47411yescharacteristic, complemented, Sylow
181C23×C3—181no—
182C22×D3—61no—
183C22×D3—61no—
184C22×D3—121no—
185C3×Q8—362no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—62no—
191C2×C3×C4—62no—
192C2×C3×C4—91no—
193C2×C3×C4—91no—
194C2×C3×C4—122no—
195C2×C3×C4—181no—
196C2×C3×C4GO+(4,𝔽2)11yescharacteristic
197#67—182no—
198#67—182no—
199C2×Dic3—22no—
200C2×Dic3—61no—
201C2×Dic3—61no—
202C2×Dic3—61no—
203C2×Dic3—61no—
204C2×Dic3—61no—
205C2×Dic3—61no—
206C2×Dic3—121no—
207C2×Dic3—362no—
208D12—92no—
209D12—181no—
210C4×D3—362no—
211C4×D3—362no—
212Dic6—124no—
213C3⋊C8—182no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32—41no—
221C2×C32—41no—
222C2×C32—41no—
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32C2×#21311yescharacteristic
227C2×C32#94211yescharacteristic
228C2×C32#94611yescharacteristic
229C3⋊D3—42no—
230C3×D3—122no—
231C3×D3—122no—
232C3×D3—242no—
233C2×Q8—181no—
234C2×D4—271no—
235C22×C4—541no—
236C2×C8—272no—
237C4⋊C4—61no—
238C4⋊C4—92no—
239C4⋊C4—271no—
240C22⋊C4—181no—
241C42—91no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3—182no—
247C22×C3—361no—
248C22×C3—361no—
249C22×C3C2×GO+(4,𝔽2)11yescharacteristic
250C2×D3—62no—
251C2×D3—62no—
252C2×D3—122no—
253C2×D3—121no—
254C2×D3—121no—
255C2×D3—121no—
256C2×D3—121no—
257C2×D3—241no—
258C2×D3—241no—
259C3×C4—22no—
260C3×C4—21no—
261C3×C4—41no—
262C3×C4—42no—
263C3×C4—81no—
264C3×C4—91no—
265C3×C4—91no—
266C3×C4—92no—
267C3×C4—122no—
268C3×C4—122no—
269C3×C4—242no—
270C3×C4—361no—
271Dic3—42no—
272Dic3—62no—
273Dic3—62no—
274Dic3—61no—
275Dic3—61no—
276Dic3—61no—
277Dic3—61no—
278Dic3—121no—
279Dic3—121no—
280Dic3—241no—
281C32—21no—
282C32—21no—
283C32—41no—
284C32—41no—
285C32#508711yescharacteristic, complemented
286C23—181no—
287Q8—362no—
288D4—272no—
289D4—541no—
290C2×C4—62no—
291C2×C4—91no—
292C2×C4—91no—
293C2×C4—181no—
294C2×C4—1082no—
295C2×C4D32⋊D311yescharacteristic
296C8—542no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—41no—
304C2×C3—41no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—362no—
310C2×C3D32⋊Q811yescharacteristic
311C2×C3#6422511yescharacteristic
312C2×C3#6423911yescharacteristic
313D3—122no—
314D3—122no—
315D3—242no—
316C22—182no—
317C22—361no—
318C22—361no—
319C22C2×D32⋊D311yescharacteristic
320C4—21no—
321C4—91no—
322C4—91no—
323C4—92no—
324C4—122no—
325C4—361no—
326C3—21no—
327C3—21no—
328C3—41no—
329C3—41no—
330C3#1059336711yescharacteristic, complemented
331C2—362no—
332C2#1172429811yescharacteristic
333C2#1172430611yescharacteristic
334C2#1172432611yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×#6295C22
hypercenterC2×C4D32⋊D3
hypocenterC33#474
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63919C2semidirect
3. #11723104C2semidirect
4. #11722779C2semidirect
17. #63919D3semidirect
30. #6295Q8semidirect
180. C33#474semidirect
285. C32#5087semidirect
330. C3#10593367semidirect

Maximal subgroups

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

subgroupindex
#117236852
C3×#639192
#117231042
#117227792
#117227722
#105933673
#50879