Finite Group Databasebeta

← #13165583

Subgroups — #13165583

Order 1728.

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

normal subgroups only

This group has 462 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
1C2×#88937C211yesmaximal, characteristic
2C2×C3×#3775C211yesmaximal, characteristic
3#11723035C211yesmaximal, characteristic, complemented
4Dic3×#625C211yesmaximal, characteristic
5#11722794C211yesmaximal, characteristic, complemented
6#11722775C211yesmaximal, characteristic, complemented
7Dic3⋊#622C211yesmaximal, characteristic, complemented
8#10593437—31nomaximal
9C2×#6296C2211yescharacteristic
10C2×C3×#627C2211yescharacteristic
11C2×C3×#625C2211yescharacteristic
12C2×C3×D3×Dic3—21no—
13C2×C3⋊#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15#88937—22no—
16C3×#3775—22no—
17#88815C2211yescharacteristic
18#88806—21no—
19(C2×D3)⋊(C3⋊Dic3)—21no—
20Dic3×C3⋊Dic3C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C2×#3775—31no—
23C2×#3775D311yescharacteristic
24Dic3×#67—61no—
25#63919—31no—
26C4×#625—31no—
27#63839—61no—
28C4⋊#622—31no—
29#63740—31no—
30#63740—31no—
31C22×C32×D3—21no—
32C2×C3⋊(C3⋊Dic3)D411yescharacteristic
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3C2311yescharacteristic
36C2×C32×Dic3—21no—
37C2×C32×Dic3D411yescharacteristic
38#6296—21no—
39#6296—21no—
40C3×#627—21no—
41C3×#627—21no—
42C3×#625—21no—
43C3×#625D412yes—
44C3×D3×Dic3—42no—
45C3⋊#622—21no—
46C3×#622—21no—
47#5162—91nomaximal
48C2×C3×#67—61no—
49C22×C3×Dic3—61no—
50C2×C3×C4×D3—61no—
51C2×C3×Dic6—61no—
52C2×#627—31no—
53C2×#627—61no—
54C2×#627C2×D311yescharacteristic
55C2×#625C2×D311yescharacteristic
56C2×D3×Dic3—21no—
57C2×#622—31no—
58C2×#622C2×D311yescharacteristic
59#3775—22no—
60#3775—62no—
61#3757—61no—
62#3750—61no—
63C4×C3⋊Dic3—31no—
64C3×C4⋊Dic3—61no—
65C3×Dic3⋊C4—61no—
66#3724—31no—
67#3724—61no—
68#3724—61no—
69#3724—121no—
70Dic3⋊Dic3—21no—
71Dic3⋊Dic3—61no—
72(C2×D3)⋊Dic3—61no—
73(C2×D3)⋊Dic3—61no—
74(C2×D3)⋊Dic3—121no—
75Dic32—61no—
76Dic32—61no—
77C22×C33C2×D411yescharacteristic
78C2×C32×D3—22no—
79C2×C32×D3—41no—
80C2×C32×D3—41no—
81C3⋊(C3⋊Dic3)—21no—
82C3×C3⋊Dic3—41no—
83C3×C3⋊Dic3—41no—
84C3×C3⋊Dic3C2×D411yescharacteristic
85C3×C3⋊Dic3#4111yescharacteristic
86C32×Dic3—41no—
87C32×Dic3SD1612yescomplemented
88C2×C3×SD16—91no—
89#944—91no—
90D4×Dic3—91no—
91C2×#214—91no—
92C4×#67—181no—
93#888—181no—
94#835—91no—
95#816—91no—
96Dic3⋊C8—91no—
97C23×C32—61no—
98C22×C3×D3—21no—
99C22×C3×D3—21no—
100C22×C3×D3—41no—
101C2×C32×C4—31no—
102C2×C32×C4—61no—
103C2×C3⋊Dic3—21no—
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3—31no—
106C2×C3⋊Dic3—61no—
107C2×C3⋊Dic3—61no—
108C2×C3⋊Dic3—121no—
109C2×C3⋊Dic3—121no—
110C2×C3⋊Dic3C22×D311yescharacteristic
111C3×#67—62no—
112C3×#67—62no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—41no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—121no—
125C2×C3×Dic3—121no—
126C2×C3×Dic3—122no—
127C3×C4×D3—122no—
128C3×Dic6—121no—
129C3×Dic6—121no—
130#627—21no—
131#627—21no—
132#627—61no—
133#627—61no—
134#627—121no—
135#627—121no—
136#625—21no—
137#625#6712yes—
138D3×Dic3—42no—
139#622—21no—
140#622—61no—
141#461—271noSylow
142C2×C33#12111yescharacteristic
143C2×C33C2×SD1611yescharacteristic
144C2×C33#13711yescharacteristic
145C32×D3—42no—
146C2×C3×Q8—91no—
147C2×C3×D4—91no—
148C22×C3×C4—181no—
149C2×#67—61no—
150C22×Dic3—61no—
151C2×C4×D3—61no—
152C2×Dic6—61no—
153C2×Dic6—91no—
154C2×Dic6—181no—
155C3×SD16—182no—
156C2×C3×C8—91no—
157C3×C4⋊C4—181no—
158#217—181no—
159#217—181no—
160#217—361no—
161#214—182no—
162(C2×D3)⋊C4—181no—
163C4⋊Dic3—61no—
164C4⋊Dic3—91no—
165Dic3⋊C4—61no—
166Dic3⋊C4—181no—
167Dic3⋊C4—181no—
168Dic3⋊C4—181no—
169Dic3⋊C4—181no—
170Dic3⋊C4—361no—
171C4×Dic3—91no—
172C4×Dic3—181no—
173C4×Dic3—181no—
174C2×C3⋊C8—91no—
175C22×C32—21no—
176C22×C32—21no—
177C22×C32—41no—
178C22×C32—41no—
179C22×C32—62no—
180C22×C32—121no—
181C22×C32—121no—
182C22×C32D3×D411yescharacteristic
183C2×C3×D3—22no—
184C2×C3×D3—22no—
185C2×C3×D3—42no—
186C2×C3×D3—41no—
187C2×C3×D3—41no—
188C2×C3×D3—41no—
189C2×C3×D3—41no—
190C2×C3×D3—81no—
191C2×C3×D3—81no—
192C32×C4—32no—
193C32×C4—121no—
194C3⋊Dic3—41no—
195C3⋊Dic3—41no—
196C3⋊Dic3—62no—
197C3⋊Dic3—61no—
198C3⋊Dic3—121no—
199C3⋊Dic3—122no—
200C3⋊Dic3—241no—
201C3⋊Dic3#23711yescharacteristic
202C3⋊Dic3C2×#6711yescharacteristic
203C3×Dic3—22no—
204C3×Dic3—22no—
205C3×Dic3—41no—
206C3×Dic3—41no—
207C3×Dic3—61no—
208C3×Dic3—61no—
209C3×Dic3—61no—
210C3×Dic3—61no—
211C3×Dic3—81no—
212C3×Dic3—121no—
213C3×Dic3—121no—
214C3×Dic3—121no—
215C3×Dic3—121no—
216C3×Dic3—241no—
217C3×Dic3—241no—
218C2×SD16—91no—
219C2×SD16—271no—
220C4⋊Q8—271no—
221C4×D4—271no—
222C4⋊C8—271no—
223#103—271no—
224#103—271no—
225C33#46111yescharacteristic, complemented, Sylow
226C23×C3—61no—
227C23×C3—61no—
228C23×C3—121no—
229C22×D3—21no—
230C3×Q8—181no—
231C3×Q8—181no—
232C3×D4—92no—
233C3×D4—181no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—61no—
238C2×C3×C4—91no—
239C2×C3×C4—121no—
240C2×C3×C4—181no—
241C2×C3×C4—181no—
242C2×C3×C4—362no—
243#67—62no—
244#67—62no—
245C2×Dic3—21no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—91no—
253C2×Dic3—121no—
254C2×Dic3—121no—
255C2×Dic3—122no—
256C2×Dic3—181no—
257C2×Dic3—181no—
258C2×Dic3—361no—
259C2×Dic3—361no—
260C2×Dic3GO+(4,𝔽2)11yescharacteristic
261C4×D3—122no—
262Dic6—121no—
263Dic6—121no—
264Dic6—181no—
265Dic6—181no—
266Dic6—361no—
267Dic6—361no—
268C3×C8—181no—
269C3⋊C8—181no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—41no—
277C2×C32—41no—
278C2×C32—41no—
279C2×C32—41no—
280C2×C32—41no—
281C2×C32—41no—
282C2×C32—122no—
283C2×C32D3×SD1611yescharacteristic
284C2×C32#91511yescharacteristic
285C2×C32#93911yescharacteristic
286C3×D3—42no—
287C3×D3—42no—
288C3×D3—82no—
289C2×Q8—91no—
290C2×Q8—271no—
291C2×D4—91no—
292C22×C4—181no—
293SD16—182no—
294SD16—542no—
295C2×C8—91no—
296C2×C8—271no—
297C4⋊C4—181no—
298C4⋊C4—271no—
299C4⋊C4—541no—
300C22⋊C4—541no—
301C42—271no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—62no—
307C22×C3—62no—
308C22×C3—122no—
309C22×C3—121no—
310C22×C3—121no—
311C22×C3—121no—
312C22×C3—121no—
313C22×C3—241no—
314C22×C3—241no—
315C22×C3C2×GO+(4,𝔽2)11yescharacteristic
316C2×D3—22no—
317C2×D3—41no—
318C2×D3—41no—
319C3×C4—62no—
320C3×C4—62no—
321C3×C4—91no—
322C3×C4—91no—
323C3×C4—121no—
324C3×C4—121no—
325C3×C4—241no—
326C3×C4—361no—
327C3×C4—361no—
328Dic3—41no—
329Dic3—61no—
330Dic3—61no—
331Dic3—61no—
332Dic3—61no—
333Dic3—121no—
334Dic3—121no—
335Dic3—121no—
336Dic3—121no—
337Dic3—182no—
338Dic3—181no—
339Dic3—241no—
340Dic3—241no—
341Dic3—361no—
342Dic3—362no—
343Dic3—721no—
344Dic3#377512yescomplemented
345C32—21no—
346C32—21no—
347C32—41no—
348C32—41no—
349C32#516211yescharacteristic, complemented
350C23—61no—
351Q8—181no—
352Q8—181no—
353Q8—541no—
354Q8—541no—
355D4—92no—
356D4—181no—
357C2×C4—31no—
358C2×C4—61no—
359C2×C4—91no—
360C2×C4—181no—
361C2×C4—181no—
362C2×C4—271no—
363C2×C4—362no—
364C8—181no—
365C8—541no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—41no—
373C2×C3—41no—
374C2×C3—41no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—122no—
379C2×C3—122no—
380C2×C3—242no—
381C2×C3#6422611yescharacteristic
382C2×C3#6423011yescharacteristic
383C2×C3C2×#377511yescharacteristic
384D3—42no—
385C22—62no—
386C22—121no—
387C22—121no—
388C22D3×GO+(4,𝔽2)11yescharacteristic
389C4—32no—
390C4—91no—
391C4—91no—
392C4—121no—
393C4—361no—
394C4—361no—
395C4—541no—
396C3—21no—
397C3—21no—
398C3—41no—
399C3—41no—
400C3#1059343711yescharacteristic, complemented
401C2—122no—
402C2#1172411711yescharacteristic
403C2D3×#377511yescharacteristic
404C2#1172414411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socleC22×C33C2×D4
prime radicalC2×C3×#625C22
hypocenterC33#461
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
3. #11723035C2semidirect
5. #11722794C2semidirect
6. #11722775C2semidirect
7. Dic3⋊#622C2semidirect
87. C32×Dic3SD16semidirect
225. C33#461semidirect
344. Dic3#3775semidirect
349. C32#5162semidirect
400. C3#10593437semidirect

Maximal subgroups

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

subgroupindex
C2×#889372
C2×C3×#37752
#117230352
Dic3×#6252
#117227942
#117227752
Dic3⋊#6222
#105934373
#51629