Finite Group Databasebeta

← #13147333

Subgroups — #13147333

Order 1728.

549 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 549 conjugacy classes of subgroups — too many to draw. Only its 63 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×C22⋊#1088C211yesmaximal, characteristic
2C2×C22⋊#1088C211yesmaximal, characteristic
3C2×C4⋊#1088C211yesmaximal, characteristic, complemented
4C2×C4×#1088C211yesmaximal, characteristic, complemented
5C22⋊C4×#272C211yesmaximal, characteristic
6#11720787C211yesmaximal, characteristic, complemented
7#11720787C211yesmaximal, characteristic, complemented
8C23×#272C2211yescharacteristic
9C22×#1088C2211yescharacteristic
10C22×#1088C2211yescharacteristic
11C22×#1088C2211yescharacteristic
12C22×#1088C2211yescharacteristic
13C2×C4×#272C2211yescharacteristic
14C2×C4×#272C2211yescharacteristic
15C22⋊#1088—22no—
16C22⋊#1088—22no—
17C4⋊#1088—22no—
18C4×#1088—22no—
19C22⋊C4×H(3,𝔽3)C411yescharacteristic, complemented
20#88552C411yescharacteristic, complemented
21#88542C411yescharacteristic, complemented
22#88542C411yescharacteristic, complemented
23C3×D3×C22⋊C4—62no—
24C23×H(3,𝔽3)C2×C411yescharacteristic
25C22×#272—21no—
26C22×#272—21no—
27C22×#272—21no—
28C22×#272—21no—
29C22×#272—21no—
30C22×#272—21no—
31C22×#272C2×C411yescharacteristic
32C22×#272C2311yescharacteristic
33C2×#1088—21no—
34C2×#1088—21no—
35C2×#1088—21no—
36C2×#1088—21no—
37C2×#1088—21no—
38C2×#1088—21no—
39C2×#1088—21no—
40C2×#1088—21no—
41C2×#1088—21no—
42C2×#1088—21no—
43C2×#1088D412yes—
44C2×#1088D412yes—
45C2×C4×H(3,𝔽3)C2×C411yescharacteristic
46C2×C4×H(3,𝔽3)C2×C411yescharacteristic
47C2×#1084C2×C411yescharacteristic
48C2×#1084C2×C411yescharacteristic
49C4×#272—21no—
50C4×#272—21no—
51C4×#272—21no—
52C4×#272—21no—
53C3×#388—91nomaximal
54C23×C3×D3—62no—
55C2×C3×C4×D3—62no—
56C2×C3×C4×D3—62no—
57C32×C22⋊C4—22no—
58C3×#217—62no—
59C3×(C2×D3)⋊C4—62no—
60C3×(C2×D3)⋊C4—62no—
61C22×H(3,𝔽3)—21no—
62C22×H(3,𝔽3)—21no—
63C22×H(3,𝔽3)—21no—
64C22×H(3,𝔽3)C22×C411yescharacteristic
65C2×#272—21no—
66C2×#272—21no—
67C2×#272—21no—
68C2×#272—41no—
69C2×#272—41no—
70C2×#272—41no—
71C2×#272—41no—
72C2×#272C2×D411yescharacteristic
73C2×#272C2×D411yescharacteristic
74C2×#272#4111yescharacteristic
75C2×#272#4111yescharacteristic
76C2×#272#4111yescharacteristic
77C2×#272#4111yescharacteristic
78#1088—22no—
79#1088—22no—
80#1088—41no—
81#1088—41no—
82C4×H(3,𝔽3)—21no—
83C4×H(3,𝔽3)—21no—
84#1084—21no—
85#1084—21no—
86C2×C3×C4⋊C4—91no—
87C2×C3×C22⋊C4—91no—
88C2×C3×C22⋊C4—91no—
89C2×C3×C22⋊C4—91no—
90C2×C3×C42—91no—
91D3×C22⋊C4—62no—
92C3×#95—91no—
93C3×#95—91no—
94C23×C32—22no—
95C22×C3×D3—62no—
96C22×C3×D3—62no—
97C22×C3×D3—64no—
98C22×C3×D3—64no—
99C22×C3×D3—122no—
100C22×C3×D3—122no—
101C22×C3×D3—122no—
102C22×C3×D3—122no—
103C2×C32×C4—22no—
104C2×C32×C4—22no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—62no—
107C3×C4×D3—122no—
108C3×C4×D3—122no—
109C3×C4×D3—122no—
110C3×C4×D3—122no—
111#388—91noSylow
112C2×H(3,𝔽3)—41no—
113C2×H(3,𝔽3)#11711yescharacteristic
114C2×H(3,𝔽3)C4×D411yescharacteristic
115C2×H(3,𝔽3)C4×D411yescharacteristic
116#272—41no—
117#272#12111yescharacteristic
118#272#12311yescharacteristic
119#272#12411yescharacteristic
120#272#12611yescharacteristic
121C24×C3—91no—
122C23×D3—62no—
123C22×C3×C4—91no—
124C22×C3×C4—91no—
125C22×C3×C4—91no—
126C22×C3×C4—91no—
127C22×C3×C4—91no—
128C22×C3×C4—91no—
129C2×C4×D3—62no—
130C2×C4×D3—62no—
131C3×C4⋊C4—182no—
132C3×C22⋊C4—62no—
133C3×C22⋊C4—91no—
134C3×C22⋊C4—91no—
135C3×C22⋊C4—91no—
136C3×C22⋊C4—182no—
137C3×C22⋊C4—182no—
138C3×C22⋊C4#15711yescharacteristic
139C3×C42—182no—
140#217—62no—
141(C2×D3)⋊C4—62no—
142(C2×D3)⋊C4—62no—
143C22×C32—24no—
144C22×C32—22no—
145C22×C32—42no—
146C22×C32—42no—
147C2×C3×D3—62no—
148C2×C3×D3—62no—
149C2×C3×D3—62no—
150C2×C3×D3—62no—
151C2×C3×D3—62no—
152C2×C3×D3—62no—
153C2×C3×D3—64no—
154C2×C3×D3—122no—
155C2×C3×D3—122no—
156C2×C3×D3—124no—
157C2×C3×D3—124no—
158C2×C3×D3—124no—
159C2×C3×D3—124no—
160C32×C4—42no—
161C32×C4—42no—
162C3×Dic3—122no—
163C3×Dic3—122no—
164C2×C4⋊C4—91no—
165C2×C22⋊C4—91no—
166C2×C22⋊C4—91no—
167C2×C22⋊C4—91no—
168C2×C42—91no—
169#95—91no—
170#95—91no—
171H(3,𝔽3)#38811yescharacteristic, complemented, Sylow
172C23×C3—62no—
173C23×C3—91no—
174C23×C3—91no—
175C23×C3—181no—
176C23×C3—181no—
177C23×C3—181no—
178C23×C3—181no—
179C23×C3—181no—
180C23×C3—181no—
181C23×C3C2×#15711yescharacteristic
182C22×D3—62no—
183C22×D3—62no—
184C22×D3—64no—
185C22×D3—64no—
186C22×D3—122no—
187C22×D3—122no—
188C22×D3—122no—
189C22×D3—122no—
190C2×C3×C4—62no—
191C2×C3×C4—62no—
192C2×C3×C4—92no—
193C2×C3×C4—92no—
194C2×C3×C4—91no—
195C2×C3×C4—91no—
196C2×C3×C4—181no—
197C2×C3×C4—181no—
198C2×C3×C4—181no—
199C2×C3×C4—181no—
200C2×C3×C4—181no—
201C2×C3×C4—181no—
202C2×C3×C4—181no—
203C2×C3×C4—181no—
204C2×C3×C4—181no—
205C2×C3×C4—181no—
206C2×C3×C4—181no—
207C2×C3×C4—181no—
208C2×C3×C4—181no—
209C2×C3×C4—181no—
210C2×C3×C4C2×#15711yescharacteristic
211C2×C3×C4C2×#15711yescharacteristic
212C2×Dic3—62no—
213C2×Dic3—62no—
214C4×D3—122no—
215C4×D3—122no—
216C4×D3—122no—
217C4×D3—122no—
218C2×C32—22no—
219C2×C32—22no—
220C2×C32—22no—
221C2×C32—44no—
222C3×D3—62no—
223C3×D3—62no—
224C3×D3—62no—
225C3×D3—62no—
226C3×D3—124no—
227C24—91no—
228C22×C4—91no—
229C22×C4—91no—
230C22×C4—91no—
231C22×C4—91no—
232C22×C4—91no—
233C22×C4—91no—
234C4⋊C4—182no—
235C22⋊C4—91no—
236C22⋊C4—91no—
237C22⋊C4—91no—
238C22⋊C4—182no—
239C22⋊C4—182no—
240C22⋊C4#108811yescharacteristic, complemented
241C42—182no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—64no—
246C22×C3—62no—
247C22×C3—91no—
248C22×C3—91no—
249C22×C3—91no—
250C22×C3—91no—
251C22×C3—91no—
252C22×C3—91no—
253C22×C3—122no—
254C22×C3—122no—
255C22×C3—181no—
256C22×C3—181no—
257C22×C3—181no—
258C22×C3—361no—
259C22×C3—361no—
260C22×C3—361no—
261C22×C3—361no—
262C22×C3C22×#15711yescharacteristic
263C2×D3—62no—
264C2×D3—62no—
265C2×D3—62no—
266C2×D3—62no—
267C2×D3—62no—
268C2×D3—62no—
269C2×D3—64no—
270C2×D3—122no—
271C2×D3—122no—
272C2×D3—124no—
273C2×D3—124no—
274C2×D3—124no—
275C2×D3—124no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—122no—
279C3×C4—122no—
280C3×C4—182no—
281C3×C4—182no—
282C3×C4—181no—
283C3×C4—181no—
284C3×C4—361no—
285C3×C4—361no—
286Dic3—122no—
287Dic3—122no—
288C32—22no—
289C23—91no—
290C23—91no—
291C23—181no—
292C23—181no—
293C23—181no—
294C23—181no—
295C23—181no—
296C23—181no—
297C23C2×#108811yescharacteristic
298C2×C4—92no—
299C2×C4—92no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—181no—
303C2×C4—181no—
304C2×C4—181no—
305C2×C4—181no—
306C2×C4—181no—
307C2×C4—181no—
308C2×C4—181no—
309C2×C4—181no—
310C2×C4—181no—
311C2×C4—181no—
312C2×C4—181no—
313C2×C4—181no—
314C2×C4—181no—
315C2×C4—181no—
316C2×C4C2×#108811yescharacteristic
317C2×C4C2×#108811yescharacteristic
318C2×C3—41no—
319C2×C3—62no—
320C2×C3—62no—
321C2×C3—62no—
322C2×C3—91no—
323C2×C3—91no—
324C2×C3—91no—
325C2×C3—91no—
326C2×C3—124no—
327C2×C3—361no—
328C2×C3#6428611yescharacteristic
329C2×C3D4×#15711yescharacteristic
330C2×C3D4×#15711yescharacteristic
331D3—62no—
332D3—62no—
333D3—62no—
334D3—62no—
335D3—124no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—91no—
340C22—91no—
341C22—91no—
342C22—91no—
343C22—91no—
344C22—91no—
345C22—181no—
346C22—181no—
347C22—181no—
348C22—361no—
349C22—361no—
350C22—361no—
351C22—361no—
352C22C22×#108811yescharacteristic
353C4—21no—
354C4—21no—
355C4—182no—
356C4—182no—
357C4—181no—
358C4—181no—
359C4—361no—
360C4—361no—
361C3—62no—
362C3#1059374711yescharacteristic
363C2—41no—
364C2—91no—
365C2—91no—
366C2—91no—
367C2—91no—
368C2—361no—
369C2#1172244811yescharacteristic
370C2D4×#108811yescharacteristic
371C2D4×#108811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
FittingC22⋊C4×H(3,𝔽3)C4
prime radicalC23×#272C22
cosocleC22×#272C23
hypercenterC3×C22⋊C4#157
hypocenterH(3,𝔽3)#388
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊#1088C2semidirect
4. C2×C4×#1088C2semidirect
6. #11720787C2semidirect
7. #11720787C2semidirect
19. C22⋊C4×H(3,𝔽3)C4semidirect
20. #88552C4semidirect
21. #88542C4semidirect
22. #88542C4semidirect
171. H(3,𝔽3)#388semidirect
240. C22⋊C4#1088semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊#10882
C2×C4⋊#10882
C2×C4×#10882
C22⋊C4×#2722
#117207872
C3×#3889