Finite Group Databasebeta

← #13165723

Subgroups — #13165723

Order 1728.

464 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 48 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#11723029C211yesmaximal, characteristic
2#11723018C211yesmaximal, characteristic, complemented
3#11722793C211yesmaximal, characteristic, complemented
4#11722773C211yesmaximal, characteristic, complemented
5#11722772C211yesmaximal, characteristic, complemented
6C3×#63739C211yesmaximal, characteristic, complemented
7#11722724C211yesmaximal, characteristic, complemented
8#10593356—31nomaximal
9C2×#6295C2211yescharacteristic
10C2×#6290—21no—
11C2×C3×#626—21no—
12C2×C3×#625C2211yescharacteristic
13C2×C3⋊#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15#88815C2211yescharacteristic
16Dic3⋊(C3⋊Dic3)—21no—
17Dic3×C3⋊Dic3C2211yescharacteristic
18C3×#3724—21no—
19C3×#3724C2211yescharacteristic
20#63979—61no—
21#63914—31no—
22#63870—31no—
23#63845—61no—
24#63777—31no—
25#63739—31no—
26#63739—31no—
27#63739—31no—
28#63739D311yescharacteristic, complemented
29C22×C3×C3⋊D3—21no—
30C22×C32×D3—21no—
31C2×C3⋊(C3⋊Dic3)D411yescharacteristic
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C2×C32×Dic3—21no—
35C2×C32×Dic3D411yescharacteristic
36#6295—21no—
37#6295—21no—
38#6290—42no—
39C3×#626—42no—
40C3×#625—21no—
41C3×#625—21no—
42C3⋊#622—21no—
43C3×#622—21no—
44#4764—91nomaximal
45C2×#638—61no—
46C2×C3×#67—61no—
47C2×C3×#67—61no—
48C2×C3×D12—61no—
49C2×#626—21no—
50C2×#626—61no—
51C2×#625—31no—
52C2×#625—61no—
53C2×#625—61no—
54C2×#625—121no—
55C2×#625C2×D311yescharacteristic
56C2×#622—31no—
57C2×#622C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)—61no—
59C4×C3⋊Dic3—31no—
60C3×Dic3⋊C4—61no—
61C3×Dic3⋊C4—61no—
62C3×Dic3⋊C4—61no—
63#3724—21no—
64#3724—31no—
65#3724C2×D311yescharacteristic
66Dic3⋊Dic3—61no—
67Dic3⋊Dic3—61no—
68Dic3⋊Dic3—61no—
69Dic3⋊Dic3—121no—
70Dic32—61no—
71Dic32—61no—
72C22×C33C2×D411yescharacteristic
73C2×C3×C3⋊D3—41no—
74C2×C3×C3⋊D3—41no—
75C2×C3×C3⋊D3—41no—
76C2×C32×D3—41no—
77C2×C32×D3—41no—
78C2×C32×D3—41no—
79C3⋊(C3⋊Dic3)—21no—
80C3×C3⋊Dic3—41no—
81C3×C3⋊Dic3#4111yescharacteristic
82C3×C3⋊Dic3#4111yescharacteristic
83C32×Dic3—21no—
84C32×Dic3—41no—
85#940—91no—
86#940—181no—
87#890—91no—
88#889—181no—
89C3×#102—91no—
90#832—91no—
91#821—91no—
92#815—91no—
93#809—91no—
94C23×C32—61no—
95C22×C3⋊D3—21no—
96C22×C3×D3—21no—
97C22×C3×D3—21no—
98C22×C3×D3—41no—
99C22×C3×D3—61no—
100C22×C3×D3—61no—
101C22×C3×D3—121no—
102C2×C32×C4—31no—
103C2×C32×C4—61no—
104#638—122no—
105C2×C3⋊Dic3—21no—
106C2×C3⋊Dic3—31no—
107C2×C3⋊Dic3—61no—
108C2×C3⋊Dic3—61no—
109C2×C3⋊Dic3—121no—
110C2×C3⋊Dic3—121no—
111C2×C3⋊Dic3C22×D311yescharacteristic
112C3×#67—121no—
113C3×#67—121no—
114C3×#67—122no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—21no—
118C2×C3×Dic3—21no—
119C2×C3×Dic3—41no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—121no—
125C3×D12—122no—
126#626—42no—
127#626—121no—
128#626—121no—
129#625—21no—
130#625—21no—
131#625—61no—
132#625—61no—
133#625—122no—
134#625—122no—
135#625—124no—
136#622—21no—
137#622—61no—
138#490—271noSylow
139C2×C33#12411yescharacteristic
140C2×C33AGL(1,ℤ/8)11yescharacteristic
141C2×C33AGL(1,ℤ/8)11yescharacteristic
142C3×C3⋊D3—81no—
143C32×D3—81no—
144C2×C3×D4—91no—
145C2×C3×D4—181no—
146C2×#67—61no—
147C2×#67—61no—
148C2×#67—181no—
149C2×#67—181no—
150C2×#67—181no—
151C2×#67—181no—
152C2×#67—361no—
153C2×D12—61no—
154C2×D12—91no—
155C2×C3×C8—91no—
156C3×C4⋊C4—91no—
157C3×C4⋊C4—181no—
158C4⋊Dic3—91no—
159C4⋊Dic3—181no—
160C4⋊Dic3—181no—
161C4⋊Dic3—361no—
162Dic3⋊C4—61no—
163Dic3⋊C4—61no—
164Dic3⋊C4—61no—
165Dic3⋊C4—181no—
166Dic3⋊C4—181no—
167C4×Dic3—91no—
168C4×Dic3—181no—
169C4×Dic3—181no—
170C2×C3⋊C8—91no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—41no—
174C22×C32—41no—
175C22×C32—121no—
176C22×C32—121no—
177C22×C32—121no—
178C22×C32D3×D411yescharacteristic
179C2×C3⋊D3—41no—
180C2×C3⋊D3—41no—
181C2×C3⋊D3—41no—
182C2×C3×D3—41no—
183C2×C3×D3—41no—
184C2×C3×D3—41no—
185C2×C3×D3—42no—
186C2×C3×D3—41no—
187C2×C3×D3—41no—
188C2×C3×D3—41no—
189C2×C3×D3—81no—
190C2×C3×D3—81no—
191C2×C3×D3—121no—
192C2×C3×D3—121no—
193C2×C3×D3—121no—
194C2×C3×D3—122no—
195C2×C3×D3—121no—
196C2×C3×D3—121no—
197C2×C3×D3—121no—
198C2×C3×D3—241no—
199C2×C3×D3—241no—
200C32×C4—61no—
201C32×C4—121no—
202C3⋊Dic3—41no—
203C3⋊Dic3—62no—
204C3⋊Dic3—61no—
205C3⋊Dic3—121no—
206C3⋊Dic3—122no—
207C3⋊Dic3—241no—
208C3⋊Dic3#23511yescharacteristic
209C3⋊Dic3#23711yescharacteristic
210C3×Dic3—22no—
211C3×Dic3—41no—
212C3×Dic3—41no—
213C3×Dic3—42no—
214C3×Dic3—41no—
215C3×Dic3—61no—
216C3×Dic3—61no—
217C3×Dic3—61no—
218C3×Dic3—61no—
219C3×Dic3—121no—
220C3×Dic3—121no—
221C3×Dic3—241no—
222#127—271no—
223#125—271no—
224#102—91no—
225#102—271no—
226#102—271no—
227#102—271no—
228#97—271no—
229C33#49011yescharacteristic, complemented, Sylow
230C23×C3—61no—
231C23×C3—61no—
232C23×C3—121no—
233C23×C3—181no—
234C22×D3—21no—
235C22×D3—61no—
236C22×D3—61no—
237C22×D3—121no—
238C3×D4—181no—
239C3×D4—181no—
240C3×D4—362no—
241C2×C3×C4—61no—
242C2×C3×C4—61no—
243C2×C3×C4—61no—
244C2×C3×C4—61no—
245C2×C3×C4—91no—
246C2×C3×C4—121no—
247C2×C3×C4—181no—
248#67—121no—
249#67—121no—
250#67—122no—
251#67—361no—
252#67—361no—
253#67—362no—
254#67—362no—
255#67—362no—
256#67—364no—
257C2×Dic3—21no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—91no—
263C2×Dic3—121no—
264C2×Dic3—181no—
265C2×Dic3—181no—
266C2×Dic3—361no—
267C2×Dic3—361no—
268C2×Dic3GO+(4,𝔽2)11yescharacteristic
269D12—122no—
270D12—181no—
271D12—181no—
272C3×C8—181no—
273C3⋊C8—181no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—41no—
281C2×C32—41no—
282C2×C32—41no—
283C2×C32—41no—
284C2×C32—41no—
285C2×C32—41no—
286C2×C32—241no—
287C2×C32#88511yescharacteristic
288C2×C32#91111yescharacteristic
289C2×C32#91411yescharacteristic
290C3⋊D3—81no—
291C3×D3—81no—
292C3×D3—82no—
293C3×D3—81no—
294C3×D3—241no—
295C3×D3—242no—
296C3×D3—241no—
297C2×D4—91no—
298C2×D4—181no—
299C2×D4—271no—
300C2×D4—541no—
301C2×C8—91no—
302C2×C8—271no—
303C4⋊C4—91no—
304C4⋊C4—181no—
305C4⋊C4—271no—
306C4⋊C4—541no—
307C42—271no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—121no—
313C22×C3—121no—
314C22×C3—121no—
315C22×C3—122no—
316C22×C3—121no—
317C22×C3—121no—
318C22×C3—121no—
319C22×C3—241no—
320C22×C3—241no—
321C22×C3—361no—
322C22×C3—361no—
323C22×C3—361no—
324C22×C3C2×GO+(4,𝔽2)11yescharacteristic
325C2×D3—41no—
326C2×D3—41no—
327C2×D3—41no—
328C2×D3—121no—
329C2×D3—121no—
330C2×D3—121no—
331C2×D3—122no—
332C2×D3—121no—
333C2×D3—121no—
334C2×D3—121no—
335C2×D3—241no—
336C2×D3—241no—
337C3×C4—62no—
338C3×C4—91no—
339C3×C4—91no—
340C3×C4—121no—
341C3×C4—121no—
342C3×C4—122no—
343C3×C4—121no—
344C3×C4—361no—
345Dic3—21no—
346Dic3—41no—
347Dic3—61no—
348Dic3—61no—
349Dic3—61no—
350Dic3—61no—
351Dic3—121no—
352Dic3—121no—
353Dic3—182no—
354Dic3—181no—
355Dic3—241no—
356Dic3—361no—
357Dic3—362no—
358Dic3—721no—
359C32—21no—
360C32—21no—
361C32—41no—
362C32—41no—
363C32#476411yescharacteristic, complemented
364C23—61no—
365C23—181no—
366D4—181no—
367D4—181no—
368D4—362no—
369D4—541no—
370D4—541no—
371D4—1082no—
372C2×C4—31no—
373C2×C4—61no—
374C2×C4—91no—
375C2×C4—181no—
376C2×C4—271no—
377C8—181no—
378C8—541no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—41no—
386C2×C3—41no—
387C2×C3—41no—
388C2×C3—41no—
389C2×C3—41no—
390C2×C3—41no—
391C2×C3—241no—
392C2×C3—242no—
393C2×C3—241no—
394C2×C3—721no—
395C2×C3#6422111yescharacteristic
396C2×C3#6422411yescharacteristic
397C2×C3#6423611yescharacteristic
398D3—81no—
399D3—241no—
400D3—242no—
401D3—241no—
402C22—121no—
403C22—121no—
404C22—121no—
405C22—361no—
406C22—361no—
407C22—361no—
408C22D3×GO+(4,𝔽2)11yescharacteristic
409C4—61no—
410C4—91no—
411C4—91no—
412C4—121no—
413C4—361no—
414C4—541no—
415C3—21no—
416C3—21no—
417C3—41no—
418C3—41no—
419C3#1059335611yescharacteristic, complemented
420C2—241no—
421C2—721no—
422C2#1172412311yescharacteristic
423C2#1172413811yescharacteristic
424C2#1172414111yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
prime radical#11723029C2
hypocenterC33#490
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. #11723018C2semidirect
3. #11722793C2semidirect
4. #11722773C2semidirect
5. #11722772C2semidirect
6. C3×#63739C2semidirect
7. #11722724C2semidirect
28. #63739D3semidirect
229. C33#490semidirect
363. C32#4764semidirect
419. C3#10593356semidirect

Maximal subgroups

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

subgroupindex
#117230292
#117230182
#117227932
#117227732
#117227722
C3×#637392
#117227242
#105933563
#47649