Finite Group Databasebeta

← #11722973

Subgroups — #11722973

Order 864.

450 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 80 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×#6289C211yesmaximal, characteristic
2C2×C3×#627C211yesmaximal, characteristic, complemented
3#88815C211yesmaximal, characteristic, complemented
4#88804C211yesmaximal, characteristic
5#88804C211yesmaximal, characteristic
6C3×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
7C3×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
8#63895—31nomaximal
9#63888—31nomaximal
10#63871—31nomaximal
11C22×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C3⋊Dic3C2211yescharacteristic
13C2×C3×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊Dic3C2211yescharacteristic
15C2×C32×Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18#6289—22no—
19C3×#627—22no—
20C2×C4×C3⋊D3—31no—
21C2×C3×Dic6—31no—
22C2×C3×Dic6—31no—
23C2×#627D311yescharacteristic, complemented
24C2×#624—31no—
25C2×#624—31no—
26C2×#624—61no—
27#3752—31no—
28#3752—31no—
29C3×C4⋊Dic3—31no—
30C3×C4⋊Dic3—31no—
31C3×Dic3⋊C4—31no—
32C3×Dic3⋊C4—31no—
33#3724—31no—
34Dic3⋊Dic3—31no—
35Dic3⋊Dic3—31no—
36Dic3⋊Dic3D311yescharacteristic, complemented
37Dic3⋊Dic3D311yescharacteristic, complemented
38#3722—31no—
39#3722—31no—
40#3722—31no—
41#3722—31no—
42#3722—61no—
43#3722—61no—
44C22×C33C2311yescharacteristic
45C2×C3⋊(C3⋊D3)—21no—
46C2×C3⋊(C3⋊D3)—21no—
47C2×C3⋊(C3⋊D3)Q812yescomplemented
48C3×C3⋊Dic3—21no—
49C3×C3⋊Dic3—21no—
50C3×C3⋊Dic3—21no—
51C32×Dic3—21no—
52C32×Dic3—21no—
53C32×Dic3D412yes—
54#946—91no—
55#897—91no—
56#896—91no—
57C22×C3⋊D3—31no—
58C22×C3⋊D3—31no—
59C22×C3⋊D3—31no—
60C22×C3⋊D3—61no—
61C22×C3⋊D3—61no—
62C22×C3⋊D3—61no—
63C22×C3⋊D3—121no—
64C2×C32×C4—31no—
65C2×C32×C4—31no—
66C2×C32×C4—31no—
67C2×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊Dic3C2×D311yescharacteristic
69C2×C3⋊Dic3C2×D311yescharacteristic
70C4×C3⋊D3—62no—
71C2×C3×Dic3—21no—
72C2×C3×Dic3—21no—
73C2×C3×Dic3—21no—
74C2×C3×Dic3—31no—
75C2×C3×Dic3—31no—
76C2×C3×Dic3—31no—
77C2×C3×Dic3—31no—
78C2×C3×Dic3—31no—
79C2×C3×Dic3—31no—
80C2×C3×Dic3—61no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—61no—
83C2×C3×Dic3C2×D311yescharacteristic
84C2×C3×Dic3C2×D311yescharacteristic
85C2×C3×Dic3C2×D311yescharacteristic
86C2×C3×Dic3C2×D311yescharacteristic
87C2×C3×Dic3C2×D311yescharacteristic
88C2×C3×Dic3C2×D311yescharacteristic
89C3×Dic6—62no—
90C3×Dic6—62no—
91#627—22no—
92#624—64no—
93#624—62no—
94#624—62no—
95C2×C33C2×D411yescharacteristic
96C2×C33C2×Q811yescharacteristic
97C2×C33#4111yescharacteristic
98C3⋊(C3⋊D3)—22no—
99C2×C3×Q8—91no—
100C2×C4×D3—91no—
101C2×C4×D3—91no—
102C2×C4×D3—91no—
103C2×C4×D3—181no—
104C2×Dic6—31no—
105C2×Dic6—31no—
106C3×C4⋊C4—91no—
107C3×C4⋊C4—91no—
108(C2×D3)⋊C4—91no—
109(C2×D3)⋊C4—91no—
110(C2×D3)⋊C4—91no—
111(C2×D3)⋊C4—91no—
112(C2×D3)⋊C4—91no—
113(C2×D3)⋊C4—91no—
114(C2×D3)⋊C4—181no—
115(C2×D3)⋊C4—181no—
116C4⋊Dic3—31no—
117C4⋊Dic3—31no—
118C4⋊Dic3—91no—
119Dic3⋊C4—31no—
120Dic3⋊C4—31no—
121Dic3⋊C4—91no—
122Dic3⋊C4—91no—
123C22×C32—21no—
124C22×C32—21no—
125C22×C32—21no—
126C22×C32—41no—
127C22×C32C22×D311yescharacteristic
128C22×C32C22×D311yescharacteristic
129C22×C32C22×D311yescharacteristic
130C2×C3⋊D3—32no—
131C2×C3⋊D3—32no—
132C2×C3⋊D3—32no—
133C2×C3⋊D3—62no—
134C2×C3⋊D3—62no—
135C2×C3⋊D3—62no—
136C2×C3⋊D3—61no—
137C2×C3⋊D3—61no—
138C2×C3⋊D3—62no—
139C2×C3⋊D3—62no—
140C2×C3⋊D3—61no—
141C2×C3⋊D3—61no—
142C2×C3⋊D3—61no—
143C2×C3⋊D3—61no—
144C2×C3⋊D3—122no—
145C2×C3⋊D3—121no—
146C2×C3⋊D3—121no—
147C2×C3⋊D3—121no—
148C2×C3⋊D3—121no—
149C2×C3⋊D3—122no—
150C2×C3⋊D3—122no—
151C32×C4—32no—
152C32×C4—61no—
153C32×C4—61no—
154C3⋊Dic3—21no—
155C3⋊Dic3—21no—
156C3⋊Dic3—21no—
157C3×Dic3—21no—
158C3×Dic3—21no—
159C3×Dic3—21no—
160C3×Dic3—21no—
161C3×Dic3—22no—
162C3×Dic3—41no—
163C3×Dic3—41no—
164C3×Dic3—62no—
165C3×Dic3—61no—
166C3×Dic3—61no—
167C3×Dic3—61no—
168C3×Dic3—61no—
169C3×Dic3—61no—
170C3×Dic3—61no—
171C3×Dic3—121no—
172C3×Dic3—121no—
173C3×Dic3D1212yes—
174C3×Dic3#6712yes—
175C22⋊Q8—271noSylow
176C33C22⋊Q811yescharacteristic, complemented, Sylow
177C22×D3—91no—
178C22×D3—91no—
179C22×D3—91no—
180C22×D3—181no—
181C22×D3—181no—
182C22×D3—181no—
183C22×D3—361no—
184C3×Q8—182no—
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—61no—
192C2×C3×C4—61no—
193C2×C3×C4—61no—
194C2×C3×C4—91no—
195C2×C3×C4—91no—
196C2×C3×C4—91no—
197C2×Dic3—31no—
198C2×Dic3—31no—
199C2×Dic3—31no—
200C2×Dic3—31no—
201C2×Dic3—31no—
202C2×Dic3—31no—
203C2×Dic3—61no—
204C2×Dic3—61no—
205C2×Dic3—61no—
206C2×Dic3D3211yescharacteristic
207C2×Dic3D3211yescharacteristic
208C2×Dic3D3211yescharacteristic
209C4×D3—182no—
210C4×D3—182no—
211C4×D3—184no—
212C4×D3—182no—
213Dic6—62no—
214Dic6—62no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32—21no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32C2×D1211yescharacteristic
228C2×C32#23511yescharacteristic
229C2×C32D3×D411yescharacteristic
230C2×C32#23711yescharacteristic
231C2×C32D3×Q811yescharacteristic
232C2×C32D3×Q811yescharacteristic
233C2×C32D3×Q811yescharacteristic
234C2×C32#23911yescharacteristic
235C2×C32C2×#6711yescharacteristic
236C3⋊D3—62no—
237C3⋊D3—64no—
238C3⋊D3—62no—
239C3⋊D3—62no—
240C3⋊D3—122no—
241C3⋊D3—122no—
242C3⋊D3—124no—
243C2×Q8—91no—
244C22×C4—271no—
245C4⋊C4—91no—
246C4⋊C4—91no—
247C4⋊C4—271no—
248C22⋊C4—271no—
249C22⋊C4—271no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—41no—
254C22×C3C2×D3211yescharacteristic
255C22×C3C2×D3211yescharacteristic
256C22×C3C2×D3211yescharacteristic
257C2×D3—92no—
258C2×D3—92no—
259C2×D3—92no—
260C2×D3—182no—
261C2×D3—182no—
262C2×D3—182no—
263C2×D3—182no—
264C2×D3—182no—
265C2×D3—181no—
266C2×D3—181no—
267C2×D3—181no—
268C2×D3—181no—
269C2×D3—181no—
270C2×D3—181no—
271C2×D3—362no—
272C2×D3—361no—
273C2×D3—361no—
274C2×D3—361no—
275C2×D3—361no—
276C2×D3—362no—
277C2×D3—362no—
278C3×C4—32no—
279C3×C4—32no—
280C3×C4—62no—
281C3×C4—61no—
282C3×C4—61no—
283C3×C4—61no—
284C3×C4—61no—
285C3×C4—121no—
286C3×C4—121no—
287C3×C4—181no—
288C3×C4—181no—
289C3×C4—181no—
290Dic3—21no—
291Dic3—21no—
292Dic3—62no—
293Dic3—61no—
294Dic3—61no—
295Dic3—61no—
296Dic3—61no—
297Dic3—61no—
298Dic3—61no—
299Dic3—121no—
300Dic3—121no—
301Dic3#62612yes—
302C32—21no—
303C32—21no—
304C32—21no—
305C32—41no—
306C32#89611yescharacteristic, complemented
307C32#89711yescharacteristic, complemented
308C32#94611yescharacteristic, complemented
309C23—271no—
310Q8—182no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—91no—
315C2×C4—91no—
316C2×C4—91no—
317C2×C4—542no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—41no—
328C2×C3—41no—
329C2×C3—41no—
330C2×C3#379611yescharacteristic
331C2×C3#379711yescharacteristic
332C2×C3#379711yescharacteristic
333C2×C3#379711yescharacteristic
334C2×C3#379911yescharacteristic
335C2×C3D3×D1211yescharacteristic
336C2×C3#380411yescharacteristic
337C2×C3C2×#62611yescharacteristic
338C2×C3D3×#6711yescharacteristic
339D3—184no—
340D3—182no—
341D3—182no—
342D3—182no—
343D3—362no—
344D3—362no—
345D3—364no—
346C22—272no—
347C22—541no—
348C22—541no—
349C22D3311yescharacteristic
350C4—32no—
351C4—61no—
352C4—61no—
353C4—181no—
354C4—181no—
355C4—181no—
356C3—21no—
357C3—21no—
358C3—21no—
359C3—41no—
360C3#6387111yescharacteristic, complemented
361C3#6388811yescharacteristic, complemented
362C3#6389511yescharacteristic, complemented
363C2—542no—
364C2D3×#62611yescharacteristic
365C2#8895811yescharacteristic
366C2#8896211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, cosocleC22×C33C23
prime radicalC22×C3⋊(C3⋊D3)C22
hypocenterC33C22⋊Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3×#627C2semidirect
3. #88815C2semidirect
6. C3×Dic3⋊Dic3C2semidirect
7. C3×Dic3⋊Dic3C2semidirect
23. C2×#627D3semidirect
36. Dic3⋊Dic3D3semidirect
37. Dic3⋊Dic3D3semidirect
47. C2×C3⋊(C3⋊D3)Q8semidirect
176. C33C22⋊Q8semidirect
306. C32#896semidirect
307. C32#897semidirect
308. C32#946semidirect
360. C3#63871semidirect
361. C3#63888semidirect
362. C3#63895semidirect

Maximal subgroups

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

subgroupindex
C2×#62892
C2×C3×#6272
#888152
#888042
C3×Dic3⋊Dic32
#638953
#638883
#638713