Finite Group Databasebeta

← #11723000

Subgroups — #11723000

Order 864.

486 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 72 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×#6292C211yesmaximal, characteristic
2C2×#6289C211yesmaximal, characteristic, complemented
3#88814C211yesmaximal, characteristic, complemented
4#88804C211yesmaximal, characteristic, complemented
5C3×#3724C211yesmaximal, characteristic, complemented
6C3×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
7C3×#3722C211yesmaximal, characteristic, complemented
8#63897—31nomaximal
9#63854—31nomaximal
10#63853—31nomaximal
11C22×C3⋊(C3⋊D3)C2211yescharacteristic
12C22×C3×C3⋊D3C2211yescharacteristic
13C2×C3×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊Dic3C2211yescharacteristic
15C2×C32×Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18#6292—22no—
19#6289—22no—
20C2×C4⋊(C3⋊D3)—31no—
21C2×C4×C3⋊D3—31no—
22C2×#626—31no—
23C2×#626—31no—
24C2×#626—61no—
25C2×#624—31no—
26C2×#624—31no—
27C2×#624—61no—
28#3752—31no—
29C3×(C2×D3)⋊C4—31no—
30C3×(C2×D3)⋊C4—31no—
31C3×C4⋊Dic3—31no—
32C3×Dic3⋊C4—31no—
33C3×Dic3⋊C4—31no—
34C3×Dic3⋊C4—31no—
35#3724D311yescharacteristic, complemented
36Dic3⋊Dic3D311yescharacteristic, complemented
37#3722—31no—
38#3722—31no—
39#3722—31no—
40#3722—61no—
41#3722D311yescharacteristic, complemented
42(C2×D3)⋊Dic3—31no—
43(C2×D3)⋊Dic3—31no—
44C22×C33C2311yescharacteristic
45C2×C3⋊(C3⋊D3)—21no—
46C2×C3⋊(C3⋊D3)—21no—
47C2×C3⋊(C3⋊D3)D412yes—
48C2×C3×C3⋊D3—21no—
49C2×C3×C3⋊D3—21no—
50C2×C3×C3⋊D3—21no—
51C3×C3⋊Dic3—21no—
52C3×C3⋊Dic3—21no—
53C32×Dic3—21no—
54C32×Dic3—21no—
55C32×Dic3—21no—
56#894—91no—
57#894—91no—
58#883—91no—
59C22×C3⋊D3—31no—
60C22×C3⋊D3—31no—
61C22×C3⋊D3—31no—
62C22×C3⋊D3—61no—
63C22×C3⋊D3—61no—
64C22×C3⋊D3—61no—
65C22×C3⋊D3—121no—
66C22×C3⋊D3C2×D311yescharacteristic
67C22×C3×D3—31no—
68C22×C3×D3—31no—
69C22×C3×D3—61no—
70C2×C32×C4—31no—
71C2×C32×C4—31no—
72C2×C32×C4—31no—
73C2×C3⋊Dic3C2×D311yescharacteristic
74C2×C3⋊Dic3C2×D311yescharacteristic
75C4⋊(C3⋊D3)—62no—
76C4×C3⋊D3—62no—
77C2×C3×Dic3—21no—
78C2×C3×Dic3—21no—
79C2×C3×Dic3—21no—
80C2×C3×Dic3—31no—
81C2×C3×Dic3—31no—
82C2×C3×Dic3—31no—
83C2×C3×Dic3—31no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3C2×D311yescharacteristic
87C2×C3×Dic3C2×D311yescharacteristic
88C2×C3×Dic3C2×D311yescharacteristic
89C2×C3×Dic3C2×D311yescharacteristic
90C2×C3×Dic3C2×D311yescharacteristic
91C2×C3×Dic3C2×D311yescharacteristic
92#626—62no—
93#626—62no—
94#626—64no—
95#624—62no—
96#624—62no—
97#624—64no—
98C2×C33C2×D411yescharacteristic
99C2×C33#4111yescharacteristic
100C2×C33#4111yescharacteristic
101C3⋊(C3⋊D3)—22no—
102C3×C3⋊D3—41no—
103C2×#67—91no—
104C2×D12—91no—
105C2×D12—91no—
106C2×D12—181no—
107C2×C4×D3—91no—
108C2×C4×D3—91no—
109C2×C4×D3—91no—
110C2×C4×D3—181no—
111C3×C4⋊C4—91no—
112C3×C4⋊C4—91no—
113C3×C22⋊C4—91no—
114#217—91no—
115(C2×D3)⋊C4—31no—
116(C2×D3)⋊C4—31no—
117(C2×D3)⋊C4—91no—
118(C2×D3)⋊C4—91no—
119(C2×D3)⋊C4—91no—
120(C2×D3)⋊C4—91no—
121(C2×D3)⋊C4—91no—
122(C2×D3)⋊C4—181no—
123C4⋊Dic3—31no—
124Dic3⋊C4—31no—
125Dic3⋊C4—31no—
126Dic3⋊C4—31no—
127C22×C32—21no—
128C22×C32—21no—
129C22×C32—21no—
130C22×C32—41no—
131C22×C32C22×D311yescharacteristic
132C22×C32C22×D311yescharacteristic
133C22×C32C22×D311yescharacteristic
134C2×C3⋊D3—21no—
135C2×C3⋊D3—21no—
136C2×C3⋊D3—21no—
137C2×C3⋊D3—32no—
138C2×C3⋊D3—32no—
139C2×C3⋊D3—32no—
140C2×C3⋊D3—62no—
141C2×C3⋊D3—62no—
142C2×C3⋊D3—62no—
143C2×C3⋊D3—61no—
144C2×C3⋊D3—61no—
145C2×C3⋊D3—62no—
146C2×C3⋊D3—62no—
147C2×C3⋊D3—61no—
148C2×C3⋊D3—61no—
149C2×C3⋊D3—61no—
150C2×C3⋊D3—61no—
151C2×C3⋊D3—122no—
152C2×C3⋊D3—121no—
153C2×C3⋊D3—121no—
154C2×C3⋊D3—121no—
155C2×C3⋊D3—121no—
156C2×C3⋊D3—122no—
157C2×C3⋊D3—122no—
158C2×C3×D3—62no—
159C2×C3×D3—61no—
160C2×C3×D3—61no—
161C2×C3×D3—61no—
162C2×C3×D3—61no—
163C2×C3×D3—61no—
164C2×C3×D3—61no—
165C2×C3×D3—121no—
166C2×C3×D3—121no—
167C32×C4—61no—
168C32×C4—61no—
169C32×C4—61no—
170C3⋊Dic3—21no—
171C3⋊Dic3—21no—
172C3×Dic3—22no—
173C3×Dic3—22no—
174C3×Dic3—21no—
175C3×Dic3—21no—
176C3×Dic3—21no—
177C3×Dic3—21no—
178C3×Dic3—21no—
179C3×Dic3—21no—
180C3×Dic3—41no—
181C3×Dic3—62no—
182C3×Dic3—61no—
183C3×Dic3—61no—
184C3×Dic3—61no—
185C3×Dic3—61no—
186C3×Dic3—121no—
187#123—271noSylow
188C33#12311yescharacteristic, complemented, Sylow
189C23×C3—91no—
190C22×D3—31no—
191C22×D3—31no—
192C22×D3—61no—
193C22×D3—91no—
194C22×D3—91no—
195C22×D3—91no—
196C22×D3—181no—
197C22×D3—181no—
198C22×D3—181no—
199C22×D3—361no—
200C2×C3×C4—31no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—61no—
207C2×C3×C4—61no—
208C2×C3×C4—61no—
209C2×C3×C4—91no—
210C2×C3×C4—91no—
211#67—182no—
212C2×Dic3—31no—
213C2×Dic3—31no—
214C2×Dic3—31no—
215C2×Dic3—31no—
216C2×Dic3—61no—
217C2×Dic3—61no—
218C2×Dic3D3211yescharacteristic
219C2×Dic3D3211yescharacteristic
220C2×Dic3D3211yescharacteristic
221D12—182no—
222D12—182no—
223D12—184no—
224C4×D3—182no—
225C4×D3—182no—
226C4×D3—182no—
227C4×D3—184no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—21no—
232C2×C32—21no—
233C2×C32—21no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32—41no—
238C2×C32—41no—
239C2×C32—41no—
240C2×C32#23511yescharacteristic
241C2×C32#23511yescharacteristic
242C2×C32#23511yescharacteristic
243C2×C32D3×D411yescharacteristic
244C2×C32D3×D411yescharacteristic
245C2×C32D3×D411yescharacteristic
246C2×C32#23711yescharacteristic
247C2×C32#23911yescharacteristic
248C2×C32#23911yescharacteristic
249C3⋊D3—41no—
250C3⋊D3—62no—
251C3⋊D3—64no—
252C3⋊D3—62no—
253C3⋊D3—62no—
254C3⋊D3—122no—
255C3⋊D3—122no—
256C3⋊D3—124no—
257C3×D3—122no—
258C3×D3—121no—
259C3×D3—121no—
260C2×D4—271no—
261C22×C4—271no—
262C4⋊C4—91no—
263C4⋊C4—91no—
264C22⋊C4—91no—
265C22⋊C4—271no—
266C22⋊C4—271no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—41no—
271C22×C3—181no—
272C22×C3—181no—
273C22×C3—181no—
274C22×C3C2×D3211yescharacteristic
275C22×C3C2×D3211yescharacteristic
276C22×C3C2×D3211yescharacteristic
277C2×D3—62no—
278C2×D3—61no—
279C2×D3—61no—
280C2×D3—61no—
281C2×D3—61no—
282C2×D3—61no—
283C2×D3—61no—
284C2×D3—92no—
285C2×D3—92no—
286C2×D3—92no—
287C2×D3—121no—
288C2×D3—121no—
289C2×D3—182no—
290C2×D3—182no—
291C2×D3—182no—
292C2×D3—182no—
293C2×D3—182no—
294C2×D3—181no—
295C2×D3—181no—
296C2×D3—181no—
297C2×D3—181no—
298C2×D3—181no—
299C2×D3—181no—
300C2×D3—362no—
301C2×D3—361no—
302C2×D3—361no—
303C2×D3—361no—
304C2×D3—361no—
305C2×D3—362no—
306C2×D3—362no—
307C3×C4—62no—
308C3×C4—62no—
309C3×C4—61no—
310C3×C4—61no—
311C3×C4—61no—
312C3×C4—61no—
313C3×C4—61no—
314C3×C4—61no—
315C3×C4—121no—
316C3×C4—181no—
317C3×C4—181no—
318Dic3—21no—
319Dic3—21no—
320Dic3—21no—
321Dic3—62no—
322Dic3—61no—
323Dic3—61no—
324Dic3—61no—
325Dic3—61no—
326Dic3—121no—
327C32—21no—
328C32—21no—
329C32—21no—
330C32—41no—
331C32#88311yescharacteristic, complemented
332C32#89411yescharacteristic, complemented
333C32#89411yescharacteristic, complemented
334C23—91no—
335C23—271no—
336D4—542no—
337C2×C4—31no—
338C2×C4—31no—
339C2×C4—31no—
340C2×C4—91no—
341C2×C4—91no—
342C2×C4—542no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3—21no—
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—361no—
356C2×C3#379611yescharacteristic
357C2×C3#379811yescharacteristic
358C2×C3#379911yescharacteristic
359C2×C3#379911yescharacteristic
360C2×C3#379911yescharacteristic
361C2×C3#380211yescharacteristic
362C2×C3#380411yescharacteristic
363C2×C3#381111yescharacteristic
364C2×C3#381111yescharacteristic
365D3—122no—
366D3—121no—
367D3—121no—
368D3—184no—
369D3—182no—
370D3—182no—
371D3—182no—
372D3—362no—
373D3—362no—
374D3—364no—
375C22—181no—
376C22—181no—
377C22—181no—
378C22—272no—
379C22—541no—
380C22—541no—
381C22D3311yescharacteristic
382C4—61no—
383C4—61no—
384C4—61no—
385C4—181no—
386C4—181no—
387C3—21no—
388C3—21no—
389C3—21no—
390C3—41no—
391C3#6385311yescharacteristic, complemented
392C3#6385411yescharacteristic, complemented
393C3#6389711yescharacteristic, complemented
394C2—361no—
395C2—542no—
396C2#8895411yescharacteristic
397C2#8896211yescharacteristic
398C2#8896511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, cosocleC22×C33C23
prime radicalC2×#6292C2
hypocenterC33#123
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×#6289C2semidirect
3. #88814C2semidirect
4. #88804C2semidirect
5. C3×#3724C2semidirect
6. C3×Dic3⋊Dic3C2semidirect
7. C3×#3722C2semidirect
35. #3724D3semidirect
36. Dic3⋊Dic3D3semidirect
41. #3722D3semidirect
188. C33#123semidirect
331. C32#883semidirect
332. C32#894semidirect
333. C32#894semidirect
391. C3#63853semidirect
392. C3#63854semidirect
393. C3#63897semidirect

Maximal subgroups

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

subgroupindex
C2×#62922
C2×#62892
#888142
#888042
C3×#37242
C3×Dic3⋊Dic32
C3×#37222
#638973
#638543
#638533