Finite Group Databasebeta

← #13165375

Subgroups — #13165375

Order 1728.

800 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C22×D3×#157C211yesmaximal, characteristic
2C2×#88940C211yesmaximal, characteristic, complemented
3C2×C3×#3773C211yesmaximal, characteristic, complemented
4#11722904C212yesmaximal
5#11722769C212yesmaximal, complemented
6#10593516—31nomaximal
7C22×D3×C3⋊D3C2211yescharacteristic
8C22×C3⋊#157C2211yescharacteristic
9C22×C3×#157C2211yescharacteristic
10C2×D3×#157—22no—
11C2×D3×#157—22no—
12C2×D3×#157C414yescomplemented
13C2×#6294C2212yes—
14C2×C3×#624C2212yes—
15#88940—22no—
16C3×#3773—22no—
17#88804—22no—
18C3×(C2×D3)⋊Dic3—22no—
19C23×#157—31no—
20C2×#3773—31no—
21C2×#3773D311yescharacteristic, complemented
22#63974—32no—
23D3×(C2×D3)⋊C4—62no—
24#63737—32no—
25C22×C3⋊(C3⋊D3)D411yescharacteristic
26C22×C3×C3⋊D3C2311yescharacteristic
27C22×C32×D3D411yescharacteristic
28C2×D3×C3⋊D3—21no—
29C2×D3×C3⋊D3—21no—
30C2×D3×C3⋊D3—21no—
31C2×D3×C3⋊D3—21no—
32C2×D3×C3⋊D3C2×C412yes—
33C2×D3×C3⋊D3Q812yes—
34C2×C3⋊#157—21no—
35C2×C3⋊#157—21no—
36C2×C3⋊#157C2×C412yes—
37C2×C3×#157C2×C412yes—
38C2×C3×#157D412yes—
39C2×C3×#157D412yes—
40D3×#157—28no—
41C2×C3×C3⋊Dic3—22no—
42C2×C32×Dic3—22no—
43#6294—22no—
44#6294—22no—
45C3×#624—22no—
46C3×#624—22no—
47#4987—91nomaximal
48C23×C3⋊D3—31no—
49C22×D32—62no—
50C22×#157—31no—
51C22×#157—34no—
52C22×#157—62no—
53C22×#157—62no—
54C22×#157C2×D311yescharacteristic
55C2×C3×C4×D3—62no—
56C2×#624—32no—
57C2×#624C2×D312yes—
58C2×D3×Dic3—62no—
59#3773—22no—
60#3773—62no—
61#3752—62no—
62C3×#217—62no—
63C3×(C2×D3)⋊C4—62no—
64#3722—62no—
65#3722—62no—
66#3722—122no—
67(C2×D3)⋊Dic3—22no—
68C22×C33C2×D411yescharacteristic
69C2×C3⋊(C3⋊D3)—21no—
70C2×C3⋊(C3⋊D3)—21no—
71C2×C3⋊(C3⋊D3)C4⋊C412yescomplemented
72C2×C3×C3⋊D3C22×C411yescharacteristic
73C2×C3×C3⋊D3C2×D411yescharacteristic
74C2×C3×C3⋊D3C2×D411yescharacteristic
75C2×C3×C3⋊D3C2×Q811yescharacteristic
76C2×C3×C3⋊D3#4111yescharacteristic
77C2×C3×C3⋊D3#4111yescharacteristic
78C2×C32×D3—21no—
79C2×C32×D3—21no—
80C2×C32×D3C4⋊C412yescomplemented
81D3×C3⋊D3—22no—
82D3×C3⋊D3—22no—
83D3×C3⋊D3—22no—
84D3×C3⋊D3—22no—
85C3⋊#157—22no—
86C3×#157C22⋊C414yescomplemented
87C3×C3⋊Dic3—42no—
88C32×Dic3—42no—
89C22×C4×D3—91no—
90C2×C3×C4⋊C4—91no—
91C2×(C2×D3)⋊C4—92no—
92C2×C4⋊Dic3—91no—
93D3×C22⋊C4—182no—
94C3⋊#95—92no—
95C23×C32—31no—
96C22×C3⋊D3—31no—
97C22×C3⋊D3—32no—
98C22×C3⋊D3—32no—
99C22×C3⋊D3—62no—
100C22×C3⋊D3—61no—
101C22×C3⋊D3—61no—
102C22×C3⋊D3—61no—
103C22×C3⋊D3—61no—
104C22×C3⋊D3—122no—
105C22×C3⋊D3C22×D311yescharacteristic
106C22×C3×D3—22no—
107C22×C3×D3—62no—
108C2×D32—64no—
109C2×D32—64no—
110C2×D32—122no—
111C2×D32—122no—
112C2×D32—122no—
113C2×D32—122no—
114C2×#157—32no—
115C2×#157—61no—
116C2×#157—61no—
117C2×#157—68no—
118C2×#157C4×D312yes—
119C2×#157D1212yes—
120C2×#157#6712yes—
121C2×C32×C4—62no—
122C2×C3⋊Dic3—22no—
123C2×C3×Dic3—22no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—42no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—122no—
129C3×C4×D3—122no—
130C3×C4×D3—122no—
131#624—22no—
132#624—22no—
133#624—62no—
134#624—62no—
135D3×Dic3—122no—
136D3×Dic3—122no—
137#380—271noSylow
138C2×C33C2×C4⋊C411yescharacteristic
139C2×C33#12111yescharacteristic
140C2×C33#12111yescharacteristic
141C3⋊(C3⋊D3)—22no—
142C3×C3⋊D3C2×C22⋊C411yescharacteristic
143C3×C3⋊D3#11711yescharacteristic
144C3×C3⋊D3C22⋊Q811yescharacteristic
145C3×C3⋊D3C22⋊Q811yescharacteristic
146C32×D3—22no—
147C23×D3—91no—
148C23×D3—182no—
149C22×C3×C4—92no—
150C22×C3×C4—91no—
151C22×Dic3—92no—
152C22×Dic3—91no—
153C2×C4×D3—62no—
154C2×C4×D3—94no—
155C2×C4×D3—182no—
156C2×C4×D3—182no—
157C2×C4×D3—182no—
158C3×C4⋊C4—182no—
159C3×C22⋊C4—182no—
160#217—62no—
161(C2×D3)⋊C4—62no—
162(C2×D3)⋊C4—182no—
163(C2×D3)⋊C4—182no—
164(C2×D3)⋊C4—182no—
165(C2×D3)⋊C4—362no—
166C4⋊Dic3—182no—
167C22×C32—22no—
168C22×C32—32no—
169C22×C32—42no—
170C22×C32—61no—
171C22×C32—61no—
172C22×C32D3×D411yescharacteristic
173C2×C3⋊D3—32no—
174C2×C3⋊D3—64no—
175C2×C3⋊D3—61no—
176C2×C3⋊D3—61no—
177C2×C3⋊D3—62no—
178C2×C3⋊D3—62no—
179C2×C3⋊D3—62no—
180C2×C3⋊D3—62no—
181C2×C3⋊D3—124no—
182C2×C3⋊D3—122no—
183C2×C3⋊D3—122no—
184C2×C3⋊D3—242no—
185C2×C3⋊D3—242no—
186C2×C3⋊D3C2×C4×D311yescharacteristic
187C2×C3⋊D3C2×D1211yescharacteristic
188C2×C3⋊D3#23711yescharacteristic
189C2×C3⋊D3D3×Q811yescharacteristic
190C2×C3⋊D3#23911yescharacteristic
191C2×C3⋊D3C2×#6711yescharacteristic
192C2×C3×D3—24no—
193C2×C3×D3—42no—
194C2×C3×D3—42no—
195C2×C3×D3—62no—
196C2×C3×D3—62no—
197C2×C3×D3—62no—
198C2×C3×D3—62no—
199C2×C3×D3—62no—
200C2×C3×D3—62no—
201D32—124no—
202D32—124no—
203D32—124no—
204D32—124no—
205#157—62no—
206#157(C2×D3)⋊C414yescomplemented
207C32×C4—122no—
208C3⋊Dic3—42no—
209C3×Dic3—42no—
210C3×Dic3—42no—
211C3×Dic3—82no—
212C3×Dic3—122no—
213C3×Dic3—122no—
214C3×Dic3—242no—
215C23×C4—271no—
216C2×C4⋊C4—91no—
217C2×C4⋊C4—271no—
218C2×C22⋊C4—272no—
219#95—272no—
220C33#38011yescharacteristic, complemented, Sylow
221C23×C3—62no—
222C23×C3—91no—
223C22×D3—62no—
224C22×D3—91no—
225C22×D3—92no—
226C22×D3—92no—
227C22×D3—184no—
228C22×D3—184no—
229C22×D3—182no—
230C22×D3—181no—
231C22×D3—181no—
232C22×D3—181no—
233C22×D3—181no—
234C22×D3—362no—
235C22×D3—362no—
236C22×D3—362no—
237C22×D3—362no—
238C22×D3—362no—
239C22×D3GO+(4,𝔽2)11yescharacteristic
240C2×C3×C4—62no—
241C2×C3×C4—62no—
242C2×C3×C4—92no—
243C2×C3×C4—92no—
244C2×C3×C4—92no—
245C2×C3×C4—122no—
246C2×C3×C4—182no—
247C2×C3×C4—182no—
248C2×C3×C4—182no—
249C2×Dic3—22no—
250C2×Dic3—62no—
251C2×Dic3—62no—
252C2×Dic3—92no—
253C2×Dic3—122no—
254C2×Dic3—182no—
255C2×Dic3—182no—
256C2×Dic3—181no—
257C2×Dic3—181no—
258C4×D3—122no—
259C4×D3—122no—
260C4×D3—188no—
261C4×D3—362no—
262C4×D3—362no—
263C2×C32—22no—
264C2×C32—22no—
265C2×C32—22no—
266C2×C32—42no—
267C2×C32—42no—
268C2×C32—42no—
269C2×C32—62no—
270C2×C32D3×C4⋊C411yescharacteristic
271C2×C32#89511yescharacteristic
272C2×C32#93811yescharacteristic
273C3⋊D3—62no—
274C3⋊D3—124no—
275C3⋊D3—244no—
276C3⋊D3#89211yescharacteristic
277C3⋊D3#89711yescharacteristic
278C3⋊D3C2×(C2×D3)⋊C411yescharacteristic
279C3⋊D3#94611yescharacteristic
280C3×D3—44no—
281C3×D3—62no—
282C3×D3—62no—
283C3×D3—62no—
284C3×D3—62no—
285C24—271no—
286C22×C4—92no—
287C22×C4—91no—
288C22×C4—274no—
289C22×C4—272no—
290C22×C4—271no—
291C22×C4—542no—
292C22×C4—542no—
293C4⋊C4—182no—
294C4⋊C4—542no—
295C22⋊C4—182no—
296C22⋊C4—542no—
297C22×C3—22no—
298C22×C3—42no—
299C22×C3—64no—
300C22×C3—91no—
301C22×C3—91no—
302C22×C3—91no—
303C22×C3—91no—
304C22×C3—91no—
305C22×C3—91no—
306C22×C3—122no—
307C22×C3—122no—
308C22×C3C2×GO+(4,𝔽2)11yescharacteristic
309C2×D3—21no—
310C2×D3—21no—
311C2×D3—62no—
312C2×D3—62no—
313C2×D3—62no—
314C2×D3—62no—
315C2×D3—62no—
316C2×D3—62no—
317C2×D3—92no—
318C2×D3—184no—
319C2×D3—182no—
320C2×D3—182no—
321C2×D3—182no—
322C2×D3—182no—
323C2×D3—181no—
324C2×D3—181no—
325C2×D3—364no—
326C2×D3—364no—
327C2×D3—364no—
328C2×D3—364no—
329C2×D3—364no—
330C2×D3—362no—
331C2×D3—362no—
332C2×D3—722no—
333C2×D3—722no—
334C2×D3#377312yescomplemented
335C3×C4—94no—
336C3×C4—122no—
337C3×C4—122no—
338C3×C4—242no—
339C3×C4—362no—
340Dic3—42no—
341Dic3—122no—
342Dic3—122no—
343Dic3—182no—
344Dic3—242no—
345C32—22no—
346C32—42no—
347C32#498711yescharacteristic, complemented
348C23—31no—
349C23—91no—
350C23—272no—
351C23—272no—
352C23—271no—
353C23—541no—
354C23—541no—
355C23—541no—
356C23—541no—
357C2×C4—62no—
358C2×C4—92no—
359C2×C4—92no—
360C2×C4—92no—
361C2×C4—182no—
362C2×C4—182no—
363C2×C4—182no—
364C2×C4—272no—
365C2×C4—548no—
366C2×C4—542no—
367C2×C4—542no—
368C2×C4—541no—
369C2×C4—541no—
370C2×C3—22no—
371C2×C3—22no—
372C2×C3—22no—
373C2×C3—42no—
374C2×C3—42no—
375C2×C3—42no—
376C2×C3—91no—
377C2×C3—91no—
378C2×C3—91no—
379C2×C3—91no—
380C2×C3—124no—
381C2×C3C2×#377311yescharacteristic
382C2×C3#6424211yescharacteristic
383C2×C3#6424211yescharacteristic
384D3—22no—
385D3—62no—
386D3—62no—
387D3—62no—
388D3—62no—
389D3—182no—
390D3—364no—
391D3—724no—
392C22—32no—
393C22—61no—
394C22—61no—
395C22—91no—
396C22—91no—
397C22—91no—
398C22—91no—
399C22—91no—
400C22—91no—
401C22—272no—
402C22—542no—
403C22—542no—
404C22—542no—
405C22—542no—
406C22—541no—
407C22—541no—
408C22D3×GO+(4,𝔽2)11yescharacteristic
409C4—94no—
410C4—122no—
411C4—362no—
412C4—542no—
413C3—22no—
414C3—42no—
415C3#1059351611yescharacteristic, complemented
416C2—62no—
417C2—91no—
418C2—91no—
419C2—91no—
420C2—91no—
421C2—542no—
422C2D3×#377311yescharacteristic
423C2#1172411411yescharacteristic
424C2#1172411611yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutatorC2×C3×C3⋊D3C22×C4
Fitting, socleC22×C33C2×D4
prime radicalC22×D3×C3⋊D3C22
cosocleC22×C3×C3⋊D3C23
hypocenterC33#380
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×#88940C2semidirect
3. C2×C3×#3773C2semidirect
5. #11722769C2semidirect
12. C2×D3×#157C4semidirect
21. C2×#3773D3semidirect
71. C2×C3⋊(C3⋊D3)C4⋊C4semidirect
80. C2×C32×D3C4⋊C4semidirect
86. C3×#157C22⋊C4semidirect
206. #157(C2×D3)⋊C4semidirect
220. C33#380semidirect
334. C2×D3#3773semidirect
347. C32#4987semidirect
415. C3#10593516semidirect

Maximal subgroups

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

subgroupindex
C22×D3×#1572
C2×#889402
C2×C3×#37732
#117229042
#117227692
#105935163
#49879