Finite Group Databasebeta

← #13165649

Subgroups — #13165649

Order 1728.

470 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 54 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#11723037C211yesmaximal, characteristic, complemented
2#11722925C211yesmaximal, characteristic
3#11722797C211yesmaximal, characteristic, complemented
4#11722779C211yesmaximal, characteristic, complemented
5#11722773C211yesmaximal, characteristic
6C3×#63739C211yesmaximal, characteristic
7#11722732C211yesmaximal, characteristic, complemented
8#10593367—31nomaximal
9C2×#6293—21no—
10C2×D3×C3⋊Dic3—21no—
11C2×C3×#625C2211yescharacteristic
12C2×C3⋊#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14#88815C2211yescharacteristic
15#88815C2211yescharacteristic
16Dic3×C3⋊Dic3C2211yescharacteristic
17C3×#3724C2211yescharacteristic
18C3×Dic3⋊Dic3—21no—
19C3×(C2×D3)⋊Dic3—21no—
20#67⋊Dic3—61no—
21#63919—31no—
22#63902—31no—
23#63840—61no—
24#63779—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
36#6293—42no—
37D3×C3⋊Dic3—42no—
38C3×#625—21no—
39C3×#625Q812yescomplemented
40C3⋊#622—21no—
41C3×#622—21no—
42#4759—91nomaximal
43C22×C3⋊Dic3—61no—
44C2×#634—61no—
45C2×C3×#67—61no—
46C2×#627—61no—
47C2×#627—61no—
48C2×#627—121no—
49C2×#625C2×D311yescharacteristic
50C2×D3×Dic3—61no—
51C2×D3×Dic3—61no—
52C2×D3×Dic3—121no—
53C2×#622—31no—
54C2×#622C2×D311yescharacteristic
55C4×C3⋊Dic3—31no—
56C3×#217—61no—
57C3×(C2×D3)⋊C4—61no—
58C3×C4⋊Dic3—61no—
59C3×Dic3⋊C4—61no—
60C3×Dic3⋊C4—61no—
61#3724—31no—
62#3724—31no—
63#3724C2×D311yescharacteristic
64Dic3⋊Dic3—21no—
65Dic3⋊Dic3—61no—
66Dic3⋊Dic3—61no—
67(C2×D3)⋊Dic3—21no—
68Dic32—61no—
69Dic32—61no—
70C22×C33C2×D411yescharacteristic
71C2×C32×D3—22no—
72C2×C32×D3—41no—
73C2×C32×D3—41no—
74C3⋊(C3⋊Dic3)D812yes—
75C3×C3⋊Dic3—41no—
76C3×C3⋊Dic3—41no—
77C3×C3⋊Dic3C2×Q811yescharacteristic
78C3×C3⋊Dic3#4111yescharacteristic
79C32×Dic3—21no—
80C32×Dic3—41no—
81D4×Dic3—91no—
82#888—91no—
83#888—181no—
84#67⋊C4—181no—
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⋊Dic3—122no—
104C2×C3⋊Dic3C22×D311yescharacteristic
105#634—122no—
106C3×#67—62no—
107C3×#67—62no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—21no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—41no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—121no—
120C2×C3×Dic3—121no—
121#627—122no—
122#627—122no—
123#627—124no—
124#625—21no—
125#625Dic612yescomplemented
126D3×Dic3—122no—
127D3×Dic3—122no—
128D3×Dic3—124no—
129#622—21no—
130#622—61no—
131#474—271noSylow
132C2×C33C22⋊Q811yescharacteristic
133C2×C33C2×D811yescharacteristic
134C2×C33#13711yescharacteristic
135C32×D3—42no—
136C2×C3×D4—91no—
137C2×#67—61no—
138C22×Dic3—181no—
139C22×Dic3—181no—
140C22×Dic3—361no—
141C2×C4×D3—181no—
142C2×Dic6—181no—
143C2×Dic6—181no—
144C2×Dic6—181no—
145C2×Dic6—361no—
146C2×C3×C8—91no—
147C3×C4⋊C4—91no—
148C3×C4⋊C4—181no—
149C3×C22⋊C4—181no—
150#217—61no—
151(C2×D3)⋊C4—61no—
152C4⋊Dic3—61no—
153C4⋊Dic3—91no—
154C4⋊Dic3—91no—
155Dic3⋊C4—61no—
156Dic3⋊C4—61no—
157Dic3⋊C4—181no—
158Dic3⋊C4—181no—
159C4×Dic3—91no—
160C4×Dic3—181no—
161C4×Dic3—181no—
162C2×C3⋊C8—91no—
163C22×C32—21no—
164C22×C32—21no—
165C22×C32—41no—
166C22×C32—41no—
167C22×C32—62no—
168C22×C32—121no—
169C22×C32—121no—
170C22×C32D3×D411yescharacteristic
171C2×C3×D3—22no—
172C2×C3×D3—22no—
173C2×C3×D3—42no—
174C2×C3×D3—41no—
175C2×C3×D3—41no—
176C2×C3×D3—42no—
177C2×C3×D3—42no—
178C2×C3×D3—41no—
179C2×C3×D3—41no—
180C32×C4—61no—
181C32×C4—121no—
182C3⋊Dic3—32no—
183C3⋊Dic3—41no—
184C3⋊Dic3—41no—
185C3⋊Dic3—62no—
186C3⋊Dic3—62no—
187C3⋊Dic3—122no—
188C3⋊Dic3—122no—
189C3⋊Dic3C2×Dic611yescharacteristic
190C3⋊Dic3#23711yescharacteristic
191C3×Dic3—22no—
192C3×Dic3—41no—
193C3×Dic3—41no—
194C3×Dic3—42no—
195C3×Dic3—41no—
196C3×Dic3—61no—
197C3×Dic3—61no—
198C3×Dic3—61no—
199C3×Dic3—61no—
200C3×Dic3—121no—
201C3×Dic3—121no—
202C3×Dic3—122no—
203C3×Dic3—122no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C4⋊Q8—271no—
207C4×D4—271no—
208#107—271no—
209#107—271no—
210C4⋊C8—271no—
211#102—91no—
212#102—271no—
213C33#47411yescharacteristic, complemented, Sylow
214C23×C3—61no—
215C23×C3—61no—
216C23×C3—121no—
217C22×D3—21no—
218C3×D4—92no—
219C3×D4—181no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—91no—
225C2×C3×C4—121no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
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—181no—
241C2×Dic3—181no—
242C2×Dic3—361no—
243C2×Dic3—361no—
244C2×Dic3—362no—
245C2×Dic3—362no—
246C2×Dic3—364no—
247C2×Dic3GO+(4,𝔽2)11yescharacteristic
248C4×D3—362no—
249Dic6—362no—
250Dic6—362no—
251Dic6—362no—
252Dic6—364no—
253C3×C8—181no—
254C3⋊C8—181no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—122no—
268C2×C32#87811yescharacteristic
269C2×C32D3×D811yescharacteristic
270C2×C32#91511yescharacteristic
271C3×D3—42no—
272C3×D3—44no—
273C3×D3—42no—
274C2×Q8—541no—
275C2×D4—91no—
276C22×C4—541no—
277C2×C8—91no—
278C2×C8—271no—
279C4⋊C4—91no—
280C4⋊C4—181no—
281C4⋊C4—271no—
282C4⋊C4—271no—
283C22⋊C4—181no—
284C42—271no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—62no—
290C22×C3—62no—
291C22×C3—122no—
292C22×C3—121no—
293C22×C3—121no—
294C22×C3—122no—
295C22×C3—122no—
296C22×C3—121no—
297C22×C3—121no—
298C22×C3C2×GO+(4,𝔽2)11yescharacteristic
299C2×D3—22no—
300C2×D3—41no—
301C2×D3—41no—
302C3×C4—62no—
303C3×C4—91no—
304C3×C4—91no—
305C3×C4—121no—
306C3×C4—121no—
307C3×C4—122no—
308C3×C4—121no—
309C3×C4—361no—
310C3×C4—361no—
311Dic3—21no—
312Dic3—41no—
313Dic3—61no—
314Dic3—61no—
315Dic3—61no—
316Dic3—61no—
317Dic3—92no—
318Dic3—121no—
319Dic3—121no—
320Dic3—122no—
321Dic3—122no—
322Dic3—121no—
323Dic3—121no—
324Dic3—182no—
325Dic3—182no—
326Dic3—362no—
327Dic3—362no—
328C32—21no—
329C32—21no—
330C32—41no—
331C32—41no—
332C32#475911yescharacteristic, complemented
333C23—61no—
334Q8—1082no—
335D4—92no—
336D4—181no—
337C2×C4—31no—
338C2×C4—61no—
339C2×C4—91no—
340C2×C4—181no—
341C2×C4—181no—
342C2×C4—271no—
343C2×C4—1082no—
344C8—181no—
345C8—541no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—122no—
359C2×C3—124no—
360C2×C3—122no—
361C2×C3D32⋊Q811yescharacteristic
362C2×C3#6422511yescharacteristic
363C2×C3#6423911yescharacteristic
364D3—42no—
365C22—62no—
366C22—121no—
367C22—121no—
368C22D3×GO+(4,𝔽2)11yescharacteristic
369C4—61no—
370C4—91no—
371C4—91no—
372C4—121no—
373C4—272no—
374C4—361no—
375C4—361no—
376C3—21no—
377C3—21no—
378C3—41no—
379C3—41no—
380C3#1059336711yescharacteristic, complemented
381C2—122no—
382C2#1172411911yescharacteristic
383C2#1172412711yescharacteristic
384C2#1172414211yescharacteristic
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. #11723037C2semidirect
3. #11722797C2semidirect
4. #11722779C2semidirect
7. #11722732C2semidirect
39. C3×#625Q8semidirect
125. #625Dic6semidirect
213. C33#474semidirect
332. C32#4759semidirect
380. C3#10593367semidirect

Maximal subgroups

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

subgroupindex
#117230372
#117229252
#117227972
#117227792
#117227732
C3×#637392
#117227322
#105933673
#47599