Finite Group Databasebeta

← #13165514

Subgroups — #13165514

Order 1728.

468 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 60 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×#88942C211yesmaximal, characteristic
2#11722945C211yesmaximal, characteristic, complemented
3#11722931C211yesmaximal, characteristic
4#11722774C211yesmaximal, characteristic, complemented
5#11722772C211yesmaximal, characteristic
6C3×#63743C211yesmaximal, characteristic, complemented
7#11722724C211yesmaximal, characteristic, complemented
8#10593431—31nomaximal
9C2×#6296C2211yescharacteristic
10C2×#6295C2211yescharacteristic
11C2×C3×#624—21no—
12C2×C3⋊#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14#88942C414yescomplemented
15#88803—21no—
16Dic3×C3⋊Dic3—21no—
17Dic3×C3⋊Dic3C2211yescharacteristic
18C3×#3724—21no—
19C3×#3724C2211yescharacteristic
20C3×#3724C2211yescharacteristic
21C2×#3775—31no—
22#63904—61no—
23C4×#625—31no—
24#63838—31no—
25#63838—61no—
26#63777—31no—
27#63743D311yescharacteristic, complemented
28#63740—31no—
29#63739—31no—
30C22×C3×C3⋊D3—21no—
31C2×C3⋊(C3⋊Dic3)D411yescharacteristic
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C2×C32×Dic3—21no—
35C2×C32×Dic3—21no—
36C2×C32×Dic3D411yescharacteristic
37#6296—21no—
38#6296C2×C412yescomplemented
39#6295—21no—
40#6295C2×C412yes—
41C3×#624—42no—
42C3⋊#622—21no—
43C3×#622C2×C412yescomplemented
44#4889—91nomaximal
45C2×C3×C4×D3—61no—
46C2×C3×C4×D3—61no—
47C2×#627—31no—
48C2×#627—61no—
49C2×#626—61no—
50C2×#625—31no—
51C2×#624—21no—
52C2×#622—31no—
53C2×#622C2×D311yescharacteristic
54#3775—34no—
55#3752—61no—
56C4×C3⋊Dic3—31no—
57C4×C3⋊Dic3—61no—
58C3×Dic3⋊C4—61no—
59C3×Dic3⋊C4—61no—
60C3×Dic3⋊C4—61no—
61C3×Dic3⋊C4—61no—
62#3724—21no—
63#3724C2×D311yescharacteristic
64#3724C2×D311yescharacteristic
65(C2×D3)⋊Dic3—61no—
66(C2×D3)⋊Dic3—61no—
67(C2×D3)⋊Dic3—121no—
68Dic32—61no—
69Dic32—61no—
70Dic32—61no—
71Dic32—61no—
72Dic32—121no—
73C22×C33C2×D411yescharacteristic
74C2×C3×C3⋊D3—22no—
75C2×C3×C3⋊D3—41no—
76C2×C3×C3⋊D3—41no—
77C3⋊(C3⋊Dic3)—21no—
78C3×C3⋊Dic3—22no—
79C3×C3⋊Dic3C22×C411yescharacteristic
80C3×C3⋊Dic3#4111yescharacteristic
81C32×Dic3—21no—
82C32×Dic3—41no—
83C32×Dic3—41no—
84C4×#67—181no—
85C2×#204—91no—
86#893—91no—
87#887—91no—
88#887—181no—
89C3×#107—91no—
90#815—91no—
91#810—91no—
92#809—91no—
93C22×C3⋊D3—21no—
94C22×C3×D3—61no—
95C22×C3×D3—61no—
96C22×C3×D3—121no—
97C2×C32×C4—31no—
98C2×C32×C4—61no—
99C2×C32×C4—61no—
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—31no—
102C2×C3⋊Dic3—61no—
103C2×C3⋊Dic3—61no—
104C2×C3⋊Dic3—121no—
105C2×C3⋊Dic3—121no—
106C2×C3⋊Dic3C22×D311yescharacteristic
107C2×C3×Dic3—21no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—21no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—41no—
114C2×C3×Dic3—41no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—121no—
120C3×C4×D3—122no—
121C3×C4×D3—122no—
122#627—32no—
123#627—62no—
124#627—62no—
125#627—61no—
126#626—62no—
127#626—62no—
128#625—32no—
129#625—61no—
130#624—42no—
131#622—61no—
132#622C4×D312yescomplemented
133#440—271noSylow
134C2×C33C4×D411yescharacteristic
135C2×C33AGL(1,ℤ/8)11yescharacteristic
136C2×C33#13711yescharacteristic
137C3×C3⋊D3—42no—
138C22×C3×C4—181no—
139C2×#67—181no—
140C2×D12—91no—
141C2×C4×D3—61no—
142C2×C4×D3—61no—
143C2×Dic6—91no—
144C2×Dic6—181no—
145C2×C3×C8—91no—
146C3×C4⋊C4—91no—
147C3×C4⋊C4—91no—
148C3×C4⋊C4—181no—
149#217—181no—
150(C2×D3)⋊C4—181no—
151(C2×D3)⋊C4—181no—
152(C2×D3)⋊C4—361no—
153Dic3⋊C4—61no—
154Dic3⋊C4—61no—
155Dic3⋊C4—61no—
156Dic3⋊C4—61no—
157C4×Dic3—91no—
158C4×Dic3—181no—
159C4×Dic3—181no—
160C4×Dic3—181no—
161C4×Dic3—181no—
162C4×Dic3—181no—
163C4×Dic3—361no—
164C2×C3⋊C8—91no—
165#204—94no—
166C22×C32—21no—
167C22×C32—21no—
168C22×C32—41no—
169C22×C32—41no—
170C22×C32D3×D411yescharacteristic
171C2×C3⋊D3—22no—
172C2×C3⋊D3—41no—
173C2×C3⋊D3—41no—
174C2×C3×D3—62no—
175C2×C3×D3—62no—
176C2×C3×D3—122no—
177C2×C3×D3—121no—
178C2×C3×D3—121no—
179C2×C3×D3—121no—
180C2×C3×D3—121no—
181C2×C3×D3—241no—
182C2×C3×D3—241no—
183C32×C4—61no—
184C32×C4—121no—
185C32×C4—121no—
186C3⋊Dic3—22no—
187C3⋊Dic3—62no—
188C3⋊Dic3—61no—
189C3⋊Dic3—121no—
190C3⋊Dic3—122no—
191C3⋊Dic3—241no—
192C3⋊Dic3C2×C4×D311yescharacteristic
193C3⋊Dic3#23911yescharacteristic
194C3×Dic3—22no—
195C3×Dic3—41no—
196C3×Dic3—41no—
197C3×Dic3—41no—
198C3×Dic3—42no—
199C3×Dic3—41no—
200C3×Dic3—41no—
201C3×Dic3—61no—
202C3×Dic3—61no—
203C3×Dic3—61no—
204C3×Dic3—61no—
205C3×Dic3—62no—
206C3×Dic3—62no—
207C3×Dic3—81no—
208C3×Dic3—122no—
209C2×SD16—271no—
210C4×Q8—271no—
211C4×D4—271no—
212#107—91no—
213#103—271no—
214#102—271no—
215#97—271no—
216C33#44011yescharacteristic, complemented, Sylow
217C23×C3—181no—
218C22×D3—61no—
219C22×D3—61no—
220C22×D3—121no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—61no—
225C2×C3×C4—61no—
226C2×C3×C4—61no—
227C2×C3×C4—91no—
228C2×C3×C4—121no—
229C2×C3×C4—121no—
230C2×C3×C4—181no—
231C2×C3×C4—362no—
232#67—182no—
233#67—182no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—91no—
241C2×Dic3—121no—
242C2×Dic3—181no—
243C2×Dic3—181no—
244C2×Dic3—361no—
245C2×Dic3—361no—
246C2×Dic3GO+(4,𝔽2)11yescharacteristic
247D12—92no—
248D12—181no—
249C4×D3—122no—
250C4×D3—122no—
251Dic6—92no—
252Dic6—182no—
253Dic6—181no—
254Dic6—182no—
255C3×C8—92no—
256C3⋊C8—181no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—41no—
269C2×C32#89311yescharacteristic
270C2×C32#91111yescharacteristic
271C2×C32#91811yescharacteristic
272C3⋊D3—42no—
273C3×D3—122no—
274C3×D3—122no—
275C3×D3—242no—
276C2×Q8—271no—
277C2×D4—271no—
278C22×C4—181no—
279SD16—274no—
280C2×C8—91no—
281C2×C8—271no—
282C4⋊C4—91no—
283C4⋊C4—91no—
284C4⋊C4—181no—
285C22⋊C4—541no—
286C42—271no—
287C42—541no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—182no—
293C22×C3—361no—
294C22×C3—361no—
295C22×C3C2×GO+(4,𝔽2)11yescharacteristic
296C2×D3—62no—
297C2×D3—62no—
298C2×D3—122no—
299C2×D3—121no—
300C2×D3—121no—
301C2×D3—121no—
302C2×D3—121no—
303C2×D3—241no—
304C2×D3—241no—
305C3×C4—62no—
306C3×C4—91no—
307C3×C4—91no—
308C3×C4—121no—
309C3×C4—121no—
310C3×C4—121no—
311C3×C4—122no—
312C3×C4—121no—
313C3×C4—121no—
314C3×C4—182no—
315C3×C4—241no—
316Dic3—21no—
317Dic3—41no—
318Dic3—41no—
319Dic3—61no—
320Dic3—61no—
321Dic3—61no—
322Dic3—61no—
323Dic3—62no—
324Dic3—62no—
325Dic3—122no—
326Dic3—182no—
327Dic3—181no—
328Dic3—361no—
329Dic3—362no—
330Dic3—721no—
331C32—21no—
332C32—21no—
333C32—41no—
334C32—41no—
335C32#488911yescharacteristic, complemented
336C23—181no—
337Q8—272no—
338Q8—541no—
339D4—272no—
340D4—541no—
341C2×C4—31no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—91no—
345C2×C4—181no—
346C2×C4—271no—
347C2×C4—362no—
348C8—92no—
349C8—541no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3—362no—
363C2×C3#6422611yescharacteristic
364C2×C3C4×GO+(4,𝔽2)11yescharacteristic
365C2×C3#6423611yescharacteristic
366D3—122no—
367D3—122no—
368D3—242no—
369C22—182no—
370C22—361no—
371C22—361no—
372C22D3×GO+(4,𝔽2)11yescharacteristic
373C4—61no—
374C4—91no—
375C4—91no—
376C4—121no—
377C4—121no—
378C4—182no—
379C4—541no—
380C3—21no—
381C3—21no—
382C3—41no—
383C3—41no—
384C3#1059343111yescharacteristic, complemented
385C2—362no—
386C2#1172411111yescharacteristic
387C2#1172413811yescharacteristic
388C2#1172414511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutatorC3×C3⋊Dic3C22×C4
Fitting, socleC22×C33C2×D4
prime radicalC2×#6295C22
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33#440
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. #11722945C2semidirect
4. #11722774C2semidirect
6. C3×#63743C2semidirect
7. #11722724C2semidirect
14. #88942C4semidirect
27. #63743D3semidirect
38. #6296C2×C4semidirect
43. C3×#622C2×C4semidirect
132. #622C4×D3semidirect
216. C33#440semidirect
335. C32#4889semidirect
384. C3#10593431semidirect

Maximal subgroups

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

subgroupindex
C2×#889422
#117229452
#117229312
#117227742
#117227722
C3×#637432
#117227242
#105934313
#48899