Finite Group Databasebeta

← #13165647

Subgroups — #13165647

Order 1728.

438 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 56 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#11723032C211yesmaximal, characteristic, complemented
2Dic3×#625C211yesmaximal, characteristic
3#11722797C211yesmaximal, characteristic, complemented
4#11722779C211yesmaximal, characteristic, complemented
5#11722773C211yesmaximal, characteristic
6C3×#63739C211yesmaximal, characteristic
7Dic3⋊#622C211yesmaximal, characteristic, complemented
8#10593347—31nomaximal
9C2×C3×#627—21no—
10C2×C3×#625C2211yescharacteristic
11C2×C3×D3×Dic3—21no—
12C2×C3⋊#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14#88815C2211yescharacteristic
15#88815C2211yescharacteristic
16#88806—21no—
17(C2×D3)⋊(C3⋊Dic3)—21no—
18Dic3×C3⋊Dic3C2211yescharacteristic
19C3×#3724C2211yescharacteristic
20Dic3×#67—61no—
21C4×#625—31no—
22#63873—31no—
23#63839—61no—
24C4⋊#622—31no—
25#63743—31no—
26#63743—31no—
27#63739—31no—
28#63739D311yescharacteristic
29C22×C32×D3—21no—
30C2×C3⋊(C3⋊Dic3)D411yescharacteristic
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C2×C32×Dic3—21no—
35C2×C32×Dic3D411yescharacteristic
36C3×#627—42no—
37C3×#625—21no—
38C3×#625Q812yescomplemented
39C3×D3×Dic3—42no—
40C3⋊#622—21no—
41C3×#622—21no—
42#4759—91nomaximal
43C2×C3×#67—61no—
44C22×C3×Dic3—61no—
45C2×C3×C4×D3—61no—
46C2×C3×Dic6—61no—
47C2×C3×Dic6—61no—
48C2×#627—21no—
49C2×#625C2×D311yescharacteristic
50C2×D3×Dic3—21no—
51C2×#622—31no—
52C2×#622C2×D311yescharacteristic
53#3757—61no—
54#3750—61no—
55C4×C3⋊Dic3—31no—
56C3×Dic3⋊C4—61no—
57#3724—31no—
58#3724—31no—
59#3724—61no—
60#3724—61no—
61#3724—121no—
62#3724C2×D311yescharacteristic
63Dic3⋊Dic3—61no—
64Dic3⋊Dic3—61no—
65(C2×D3)⋊Dic3—61no—
66(C2×D3)⋊Dic3—61no—
67(C2×D3)⋊Dic3—121no—
68Dic32—61no—
69Dic32—61no—
70C22×C33C2×D411yescharacteristic
71C2×C32×D3—22no—
72C2×C32×D3—41no—
73C2×C32×D3—41no—
74C3⋊(C3⋊Dic3)—21no—
75C3×C3⋊Dic3—41no—
76C3×C3⋊Dic3—41no—
77C3×C3⋊Dic3C2×Q811yescharacteristic
78C3×C3⋊Dic3#4111yescharacteristic
79C32×Dic3—41no—
80C32×Dic3D812yes—
81#944—181no—
82D4×Dic3—91no—
83C4×#67—181no—
84#888—91no—
85C3×#102—91no—
86#832—91no—
87#818—91no—
88Dic3⋊C8—91no—
89#807—91no—
90C23×C32—61no—
91C22×C3×D3—21no—
92C22×C3×D3—21no—
93C22×C3×D3—41no—
94C2×C32×C4—31no—
95C2×C32×C4—61no—
96C2×C3⋊Dic3—21no—
97C2×C3⋊Dic3—21no—
98C2×C3⋊Dic3—31no—
99C2×C3⋊Dic3—61no—
100C2×C3⋊Dic3—61no—
101C2×C3⋊Dic3—121no—
102C2×C3⋊Dic3—121no—
103C2×C3⋊Dic3C22×D311yescharacteristic
104C3×#67—62no—
105C3×#67—62no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—41no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—121no—
118C2×C3×Dic3—121no—
119C2×C3×Dic3—122no—
120C3×C4×D3—122no—
121C3×Dic6—122no—
122C3×Dic6—122no—
123#627—42no—
124#625—21no—
125#625Dic612yescomplemented
126D3×Dic3—42no—
127#622—21no—
128#622—61no—
129#474—271noSylow
130C2×C33C22⋊Q811yescharacteristic
131C2×C33C2×D811yescharacteristic
132C2×C33#13711yescharacteristic
133C32×D3—42no—
134C2×C3×Q8—181no—
135C2×C3×D4—91no—
136C22×C3×C4—181no—
137C2×#67—61no—
138C22×Dic3—61no—
139C2×C4×D3—61no—
140C2×Dic6—61no—
141C2×Dic6—61no—
142C2×C3×C8—91no—
143C3×C4⋊C4—91no—
144#217—181no—
145#217—181no—
146#217—361no—
147(C2×D3)⋊C4—181no—
148C4⋊Dic3—91no—
149C4⋊Dic3—91no—
150Dic3⋊C4—61no—
151Dic3⋊C4—181no—
152Dic3⋊C4—181no—
153Dic3⋊C4—181no—
154Dic3⋊C4—181no—
155Dic3⋊C4—181no—
156Dic3⋊C4—361no—
157C4×Dic3—91no—
158C4×Dic3—181no—
159C4×Dic3—181no—
160C2×C3⋊C8—91no—
161C22×C32—21no—
162C22×C32—21no—
163C22×C32—41no—
164C22×C32—41no—
165C22×C32—62no—
166C22×C32—121no—
167C22×C32—121no—
168C22×C32D3×D411yescharacteristic
169C2×C3×D3—22no—
170C2×C3×D3—22no—
171C2×C3×D3—42no—
172C2×C3×D3—41no—
173C2×C3×D3—41no—
174C2×C3×D3—41no—
175C2×C3×D3—41no—
176C2×C3×D3—81no—
177C2×C3×D3—81no—
178C32×C4—32no—
179C32×C4—121no—
180C3⋊Dic3—41no—
181C3⋊Dic3—41no—
182C3⋊Dic3—62no—
183C3⋊Dic3—61no—
184C3⋊Dic3—121no—
185C3⋊Dic3—122no—
186C3⋊Dic3—241no—
187C3⋊Dic3C2×Dic611yescharacteristic
188C3⋊Dic3#23711yescharacteristic
189C3×Dic3—22no—
190C3×Dic3—22no—
191C3×Dic3—41no—
192C3×Dic3—41no—
193C3×Dic3—61no—
194C3×Dic3—61no—
195C3×Dic3—61no—
196C3×Dic3—61no—
197C3×Dic3—81no—
198C3×Dic3—121no—
199C3×Dic3—121no—
200C3×Dic3—121no—
201C3×Dic3—121no—
202C3×Dic3—241no—
203C3×Dic3—241no—
204C4⋊Q8—271no—
205C4×D4—271no—
206#107—271no—
207#107—271no—
208C4⋊C8—271no—
209#102—91no—
210#102—271no—
211C33#47411yescharacteristic, complemented, Sylow
212C23×C3—61no—
213C23×C3—61no—
214C23×C3—121no—
215C22×D3—21no—
216C3×Q8—362no—
217C3×D4—92no—
218C3×D4—181no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—91no—
224C2×C3×C4—121no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4—362no—
228#67—62no—
229#67—62no—
230C2×Dic3—21no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—91no—
238C2×Dic3—121no—
239C2×Dic3—121no—
240C2×Dic3—122no—
241C2×Dic3—181no—
242C2×Dic3—181no—
243C2×Dic3—361no—
244C2×Dic3—361no—
245C2×Dic3GO+(4,𝔽2)11yescharacteristic
246C4×D3—122no—
247Dic6—122no—
248Dic6—122no—
249C3×C8—181no—
250C3⋊C8—181no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—41no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—122no—
264C2×C32#87811yescharacteristic
265C2×C32D3×D811yescharacteristic
266C2×C32#91511yescharacteristic
267C3×D3—42no—
268C3×D3—42no—
269C3×D3—82no—
270C2×Q8—181no—
271C2×D4—91no—
272C22×C4—181no—
273C2×C8—91no—
274C2×C8—271no—
275C4⋊C4—91no—
276C4⋊C4—271no—
277C4⋊C4—271no—
278C4⋊C4—541no—
279C22⋊C4—541no—
280C42—271no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—62no—
286C22×C3—62no—
287C22×C3—122no—
288C22×C3—121no—
289C22×C3—121no—
290C22×C3—121no—
291C22×C3—121no—
292C22×C3—241no—
293C22×C3—241no—
294C22×C3C2×GO+(4,𝔽2)11yescharacteristic
295C2×D3—22no—
296C2×D3—41no—
297C2×D3—41no—
298C3×C4—62no—
299C3×C4—62no—
300C3×C4—91no—
301C3×C4—91no—
302C3×C4—121no—
303C3×C4—121no—
304C3×C4—241no—
305C3×C4—361no—
306C3×C4—361no—
307Dic3—41no—
308Dic3—61no—
309Dic3—61no—
310Dic3—61no—
311Dic3—61no—
312Dic3—121no—
313Dic3—121no—
314Dic3—121no—
315Dic3—121no—
316Dic3—182no—
317Dic3—181no—
318Dic3—241no—
319Dic3—241no—
320Dic3—361no—
321Dic3—362no—
322Dic3—721no—
323Dic3#377412yes—
324C32—21no—
325C32—21no—
326C32—41no—
327C32—41no—
328C32#475911yescharacteristic, complemented
329C23—61no—
330Q8—362no—
331D4—92no—
332D4—181no—
333C2×C4—31no—
334C2×C4—61no—
335C2×C4—91no—
336C2×C4—181no—
337C2×C4—181no—
338C2×C4—271no—
339C2×C4—362no—
340C8—181no—
341C8—541no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—41no—
349C2×C3—41no—
350C2×C3—41no—
351C2×C3—41no—
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—122no—
355C2×C3—122no—
356C2×C3—242no—
357C2×C3D32⋊Q811yescharacteristic
358C2×C3#6422611yescharacteristic
359C2×C3C2×#377411yescharacteristic
360D3—42no—
361C22—62no—
362C22—121no—
363C22—121no—
364C22D3×GO+(4,𝔽2)11yescharacteristic
365C4—32no—
366C4—91no—
367C4—91no—
368C4—121no—
369C4—361no—
370C4—361no—
371C4—541no—
372C3—21no—
373C3—21no—
374C3—41no—
375C3—41no—
376C3#1059334711yescharacteristic, complemented
377C2—122no—
378C2#1172411911yescharacteristic
379C2D3×#377411yescharacteristic
380C2#1172414411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
prime radicalC2×C3×#625C22
hypocenterC33#474
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #11723032C2semidirect
3. #11722797C2semidirect
4. #11722779C2semidirect
7. Dic3⋊#622C2semidirect
38. C3×#625Q8semidirect
125. #625Dic6semidirect
211. C33#474semidirect
328. C32#4759semidirect
376. C3#10593347semidirect

Maximal subgroups

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

subgroupindex
#117230322
Dic3×#6252
#117227972
#117227792
#117227732
C3×#637392
Dic3⋊#6222
#105933473
#47599