Finite Group Databasebeta

← #13167532

Subgroups — #13167532

Order 1728.

808 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C22×#157⋊D3C211yesmaximal, characteristic
2C2×C3×C4⋊#157C211yesmaximal, characteristic, complemented
3C2×#88939C212yesmaximal, complemented
4#11723700C212yesmaximal, complemented
5#11722791C211yesmaximal, characteristic, complemented
6#10593456—31nomaximal
7C22×#1113C2212yes—
8C22×C3×#157C2211yescharacteristic
9C22×C3×#157C2211yescharacteristic
10C2×#157⋊D3—24no—
11C2×#157⋊D3—22no—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×#6294C2212yes—
14C3×C4⋊#157—22no—
15#88939—24no—
16C3×#3752—22no—
17#88814—22no—
18C22×GO+(4,𝔽2)—31no—
19C2×C4⋊#157D311yescharacteristic, complemented
20C2×D32⋊C4—32no—
21#63921—32no—
22D3×(C2×D3)⋊C4—62no—
23#63737—31no—
24C22×C3×C3⋊D3—22no—
25C22×C3×C3⋊D3C2311yescharacteristic
26C2×#1113—22no—
27C2×#1113—22no—
28C2×#1113—22no—
29C2×#1113—22no—
30C2×#1113—22no—
31C2×#1113D414yes—
32C2×C3×#157—22no—
33C2×C3×#157—22no—
34C2×C3×#157—21no—
35C2×C3×#157D412yes—
36#157⋊D3—28no—
37C2×C33×C4D411yescharacteristic, complemented
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3D411yescharacteristic, complemented
40C3×C4×C3⋊D3—21no—
41C3×C4×C3⋊D3—21no—
42#6294—22no—
43#6294—22no—
44#4991—91nomaximal
45C22×D32—32no—
46C22×D32—62no—
47C22×#157C2×D311yescharacteristic
48C22×#157C2×D311yescharacteristic
49C2×GO+(4,𝔽2)—64no—
50C2×GO+(4,𝔽2)—62no—
51C2×C4×C3⋊D3C2×D311yescharacteristic
52C2×C3×C4×D3—62no—
53C2×#624—32no—
54C2×D3×Dic3—62no—
55C4⋊#157—22no—
56D32⋊C4—64no—
57#3752—22no—
58C3×(C2×D3)⋊C4—62no—
59C3×(C2×D3)⋊C4—62no—
60C3×(C2×D3)⋊C4—122no—
61#3722—62no—
62(C2×D3)⋊Dic3—62no—
63(C2×D3)⋊Dic3—62no—
64C22×C33C2×D411yescharacteristic
65C2×C3×C3⋊D3—24no—
66C2×C3×C3⋊D3—42no—
67C2×C3×C3⋊D3—42no—
68C2×C3×C3⋊D3C2×D412yes—
69C2×C3×C3⋊D3C2×D411yescharacteristic
70C2×C3×C3⋊D3#4111yescharacteristic
71C2×C3×C3⋊D3#4112yes—
72#1113—24no—
73#1113—24no—
74#1113—42no—
75#1113—42no—
76C3×#157—22no—
77C3×#157—41no—
78C33×C4—21no—
79C3×C3⋊Dic3—21no—
80C3×C3⋊Dic3—42no—
81C22×D12—91no—
82C2×C3×C4⋊C4—91no—
83C2×(C2×D3)⋊C4—92no—
84C2×(C2×D3)⋊C4—92no—
85D3×C22⋊C4—182no—
86C3⋊#95—91no—
87C22×C3⋊D3—22no—
88C22×C3⋊D3C22×D311yescharacteristic
89C22×C3×D3—62no—
90C22×C3×D3—62no—
91C22×C3×D3—62no—
92C22×C3×D3—122no—
93C2×D32—34no—
94C2×D32—64no—
95C2×D32—64no—
96C2×D32—62no—
97C2×D32—62no—
98C2×D32—62no—
99C2×D32—62no—
100C2×D32—62no—
101C2×D32—122no—
102C2×D32—122no—
103C2×D32—122no—
104C2×D32—122no—
105C2×#157—22no—
106C2×#157—22no—
107C2×#157—21no—
108C2×#157#6712yes—
109GO+(4,𝔽2)—68no—
110C2×C32×C4—22no—
111C2×C32×C4—42no—
112C2×C32×C4D1211yescharacteristic, complemented
113C2×C3⋊Dic3—22no—
114C2×C3⋊Dic3D1211yescharacteristic, complemented
115C4×C3⋊D3—21no—
116C4×C3⋊D3—21no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—122no—
121C3×C4×D3—122no—
122C3×C4×D3—122no—
123#624—62no—
124#624—62no—
125D3×Dic3—122no—
126D3×Dic3—122no—
127#394—271noSylow
128C2×C33#12112yes—
129C2×C33#12111yescharacteristic
130C3×C3⋊D3—44no—
131C3×C3⋊D3#12011yescharacteristic
132C3×C3⋊D3#12312yes—
133C3×C3⋊D3#12411yescharacteristic
134C23×D3—92no—
135C23×D3—182no—
136C22×C3×C4—91no—
137C22×C3×C4—91no—
138C22×C3×C4—91no—
139C22×Dic3—92no—
140C2×D12—182no—
141C2×D12—184no—
142C2×C4×D3—62no—
143C2×C4×D3—182no—
144C3×C4⋊C4—182no—
145C3×C22⋊C4—182no—
146#217—182no—
147(C2×D3)⋊C4—62no—
148(C2×D3)⋊C4—62no—
149(C2×D3)⋊C4—122no—
150(C2×D3)⋊C4—182no—
151(C2×D3)⋊C4—182no—
152(C2×D3)⋊C4—184no—
153C22×C32—22no—
154C22×C32—42no—
155C22×C32C2×D1211yescharacteristic
156C2×C3⋊D3—24no—
157C2×C3⋊D3—42no—
158C2×C3⋊D3—42no—
159C2×C3⋊D3#23511yescharacteristic
160C2×C3⋊D3D3×D412yes—
161C2×C3⋊D3#23912yes—
162C2×C3⋊D3C2×#6711yescharacteristic
163C2×C3×D3—62no—
164C2×C3×D3—62no—
165C2×C3×D3—62no—
166C2×C3×D3—62no—
167C2×C3×D3—62no—
168C2×C3×D3—62no—
169C2×C3×D3—64no—
170C2×C3×D3—64no—
171C2×C3×D3—124no—
172C2×C3×D3—122no—
173C2×C3×D3—122no—
174C2×C3×D3—122no—
175C2×C3×D3—122no—
176C2×C3×D3—242no—
177C2×C3×D3—242no—
178D32—64no—
179D32—64no—
180D32—124no—
181D32—124no—
182D32—124no—
183D32—124no—
184D32—122no—
185D32—122no—
186#157—22no—
187#157—41no—
188C32×C4—21no—
189C32×C4—42no—
190C32×C4—82no—
191C3⋊Dic3—21no—
192C3⋊Dic3—42no—
193C3×Dic3—122no—
194C3×Dic3—122no—
195C3×Dic3—122no—
196C3×Dic3—242no—
197C22×D4—271no—
198C2×C4⋊C4—91no—
199C2×C22⋊C4—272no—
200C2×C22⋊C4—272no—
201#95—271no—
202C33#39411yescharacteristic, complemented, Sylow
203C23×C3—91no—
204C23×C3—182no—
205C22×D3—62no—
206C22×D3—62no—
207C22×D3—62no—
208C22×D3—94no—
209C22×D3—122no—
210C22×D3—184no—
211C22×D3—182no—
212C22×D3—184no—
213C22×D3—182no—
214C22×D3—182no—
215C22×D3—182no—
216C22×D3—182no—
217C22×D3—362no—
218C22×D3—362no—
219C22×D3—362no—
220C22×D3—362no—
221C2×C3×C4—22no—
222C2×C3×C4—42no—
223C2×C3×C4—92no—
224C2×C3×C4—91no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4—182no—
228C2×C3×C4—182no—
229C2×C3×C4—182no—
230C2×C3×C4—181no—
231C2×C3×C4GO+(4,𝔽2)11yescharacteristic, complemented
232C2×Dic3—62no—
233C2×Dic3—62no—
234C2×Dic3—62no—
235C2×Dic3—122no—
236C2×Dic3—182no—
237C2×Dic3—182no—
238D12—188no—
239C4×D3—122no—
240C4×D3—122no—
241C4×D3—362no—
242C4×D3—362no—
243C2×C32—22no—
244C2×C32—22no—
245C2×C32—22no—
246C2×C32—42no—
247C2×C32—42no—
248C2×C32—42no—
249C2×C32#89512yes—
250C2×C32#93011yescharacteristic
251C3⋊D3—44no—
252C3⋊D3#89412yes—
253C3⋊D3#93711yescharacteristic
254C3⋊D3#94711yescharacteristic
255C3×D3—62no—
256C3×D3—62no—
257C3×D3—62no—
258C3×D3—62no—
259C3×D3—124no—
260C3×D3—124no—
261C3×D3—244no—
262C24—272no—
263C2×D4—542no—
264C2×D4—544no—
265C22×C4—91no—
266C22×C4—91no—
267C22×C4—91no—
268C22×C4—272no—
269C4⋊C4—182no—
270C22⋊C4—182no—
271C22⋊C4—542no—
272C22⋊C4—544no—
273C22×C3—22no—
274C22×C3—42no—
275C22×C3—92no—
276C22×C3—92no—
277C22×C3—91no—
278C22×C3—91no—
279C22×C3—184no—
280C22×C3—362no—
281C22×C3—362no—
282C22×C3C2×GO+(4,𝔽2)11yescharacteristic
283C2×D3—62no—
284C2×D3—62no—
285C2×D3—62no—
286C2×D3—62no—
287C2×D3—62no—
288C2×D3—62no—
289C2×D3—64no—
290C2×D3—64no—
291C2×D3—124no—
292C2×D3—122no—
293C2×D3—122no—
294C2×D3—122no—
295C2×D3—122no—
296C2×D3—184no—
297C2×D3—184no—
298C2×D3—242no—
299C2×D3—242no—
300C2×D3—364no—
301C2×D3—364no—
302C2×D3—364no—
303C2×D3—364no—
304C2×D3—362no—
305C2×D3—362no—
306C3×C4—21no—
307C3×C4—42no—
308C3×C4—82no—
309C3×C4—182no—
310C3×C4—181no—
311C3×C4—362no—
312C3×C4—361no—
313Dic3—122no—
314Dic3—122no—
315Dic3—122no—
316Dic3—242no—
317C32—22no—
318C32—42no—
319C32#499111yescharacteristic, complemented
320C23—91no—
321C23—182no—
322C23—274no—
323C23—542no—
324C23—542no—
325C23—542no—
326C23—542no—
327C23—542no—
328D4—548no—
329C2×C4—92no—
330C2×C4—91no—
331C2×C4—181no—
332C2×C4—181no—
333C2×C4—182no—
334C2×C4—182no—
335C2×C4—181no—
336C2×C4—182no—
337C2×C4—542no—
338C2×C4—542no—
339C2×C4#157⋊D311yescharacteristic, complemented
340C2×C3—22no—
341C2×C3—22no—
342C2×C3—22no—
343C2×C3—42no—
344C2×C3—42no—
345C2×C3—42no—
346C2×C3—91no—
347C2×C3—92no—
348C2×C3—91no—
349C2×C3—364no—
350C2×C3#6423012yes—
351C2×C3#6424211yescharacteristic
352D3—62no—
353D3—62no—
354D3—62no—
355D3—62no—
356D3—124no—
357D3—124no—
358D3—244no—
359C22—92no—
360C22—92no—
361C22—91no—
362C22—91no—
363C22—184no—
364C22—362no—
365C22—362no—
366C22—544no—
367C22—544no—
368C22—1082no—
369C22—1082no—
370C22C2×#157⋊D311yescharacteristic
371C4—21no—
372C4—182no—
373C4—181no—
374C4—361no—
375C4—362no—
376C3—22no—
377C3—42no—
378C3#1059345611yescharacteristic, complemented
379C2—91no—
380C2—92no—
381C2—91no—
382C2—364no—
383C2#1172434312yes—
384C2#1172435511yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC22×C3×C3⋊D3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×#157⋊D3C2
hypercenterC2×C4#157⋊D3
hypocenterC33#394
LCSG ▷ C22×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊#157C2semidirect
3. C2×#88939C2semidirect
4. #11723700C2semidirect
5. #11722791C2semidirect
19. C2×C4⋊#157D3semidirect
37. C2×C33×C4D4semidirect
39. C2×C3×C3⋊Dic3D4semidirect
112. C2×C32×C4D12semidirect
114. C2×C3⋊Dic3D12semidirect
202. C33#394semidirect
231. C2×C3×C4GO+(4,𝔽2)semidirect
319. C32#4991semidirect
339. C2×C4#157⋊D3semidirect
378. C3#10593456semidirect

Maximal subgroups

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

subgroupindex
C22×#157⋊D32
C2×C3×C4⋊#1572
C2×#889392
#117237002
#117227912
#105934563
#49919