Finite Group Databasebeta

← #10591712

Subgroups — #10591712

Order 576.

594 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 104 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×#638C211yesmaximal, characteristic
2C2×C3×#217C211yesmaximal, characteristic, complemented
3C2×C3×#217C211yesmaximal, characteristic, complemented
4C2×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
5C2×#3722C211yesmaximal, characteristic, complemented
6C2×#3722C211yesmaximal, characteristic, complemented
7#63579C211yesmaximal, characteristic, complemented
8#4961—31nomaximal
9#4959—31nomaximal
10C24×C32C2211yescharacteristic
11C23×C3⋊D3C2211yescharacteristic
12C2×#638—22no—
13C2×#638—22no—
14C2×#638—22no—
15C22×C3⋊Dic3C2211yescharacteristic
16C22×C3×Dic3C2211yescharacteristic
17C22×C3×Dic3C2211yescharacteristic
18C22×C3×Dic3C2211yescharacteristic
19C22×C3×Dic3C2211yescharacteristic
20C3×#217—22no—
21C3×#217—22no—
22Dic3⋊Dic3—22no—
23#3722—22no—
24#3722—22no—
25C22×#67—31no—
26C22×#67—31no—
27C22×#67—61no—
28C2×C3×C22⋊C4—31no—
29C2×C3×C22⋊C4—31no—
30C2×#217D311yescharacteristic, complemented
31C2×#217D311yescharacteristic, complemented
32C2×(C2×D3)⋊C4—31no—
33C2×(C2×D3)⋊C4—31no—
34C2×(C2×D3)⋊C4—31no—
35C2×(C2×D3)⋊C4—31no—
36C2×C4⋊Dic3—31no—
37C2×Dic3⋊C4—31no—
38C3⋊#95—31no—
39C3⋊#95—31no—
40C23×C32—21no—
41C23×C32—21no—
42C23×C32—21no—
43C23×C32—21no—
44C23×C32—21no—
45C23×C32—21no—
46C23×C32D412yes—
47C23×C32C2311yescharacteristic
48C22×C3⋊D3—21no—
49C22×C3⋊D3—21no—
50C22×C3⋊D3—21no—
51C22×C3⋊D3—21no—
52C22×C3⋊D3—21no—
53C22×C3⋊D3—21no—
54C22×C3⋊D3D412yes—
55#638—28no—
56C2×C3⋊Dic3—21no—
57C2×C3⋊Dic3—21no—
58C2×C3⋊Dic3D412yes—
59C2×C3×Dic3—21no—
60C2×C3×Dic3—21no—
61C2×C3×Dic3—21no—
62C2×C3×Dic3—21no—
63C2×C3×Dic3—21no—
64C2×C3×Dic3—21no—
65C2×C3×Dic3—21no—
66C2×C3×Dic3—21no—
67C2×C3×Dic3—21no—
68C2×C3×Dic3—21no—
69C2×C3×Dic3—21no—
70C2×C3×Dic3D412yescomplemented
71#394—91noSylow
72C24×C3—21no—
73C24×C3C2×D311yescharacteristic
74C24×C3C2×D311yescharacteristic
75C23×D3—31no—
76C23×D3—31no—
77C23×D3—61no—
78C22×C3×C4—31no—
79C22×C3×C4—31no—
80C22×C3×C4—31no—
81C22×C3×C4—31no—
82C2×#67—62no—
83C2×#67—62no—
84C2×#67—62no—
85C2×#67—62no—
86C2×#67—64no—
87C2×#67—64no—
88C2×#67—64no—
89C2×#67—62no—
90C2×#67—62no—
91C22×Dic3—31no—
92C22×Dic3—31no—
93C22×Dic3—61no—
94C22×Dic3C2×D311yescharacteristic
95C22×Dic3C2×D311yescharacteristic
96C22×Dic3C2×D311yescharacteristic
97C22×Dic3C2×D311yescharacteristic
98C3×C22⋊C4—62no—
99C3×C22⋊C4—62no—
100#217—22no—
101#217—22no—
102(C2×D3)⋊C4—62no—
103(C2×D3)⋊C4—62no—
104(C2×D3)⋊C4—62no—
105(C2×D3)⋊C4—62no—
106C4⋊Dic3—62no—
107Dic3⋊C4—62no—
108C22×C32—22no—
109C22×C32—22no—
110C22×C32—22no—
111C22×C32—41no—
112C22×C32—41no—
113C22×C32—41no—
114C22×C32—41no—
115C22×C32C2×D411yescharacteristic
116C22×C32C2×D411yescharacteristic
117C22×C32C2×D411yescharacteristic
118C22×C32C2×D411yescharacteristic
119C22×C32#4111yescharacteristic
120C22×C32#4111yescharacteristic
121C22×C32#4111yescharacteristic
122C2×C3⋊D3—22no—
123C2×C3⋊D3—22no—
124C2×C3⋊D3—22no—
125C2×C3⋊D3—41no—
126C2×C3⋊D3—41no—
127C2×C3⋊D3—41no—
128C2×C3⋊D3—41no—
129C3⋊Dic3—22no—
130C3×Dic3—22no—
131C3×Dic3—41no—
132C3×Dic3—41no—
133C3×Dic3—41no—
134C22×D4—91no—
135C2×C4⋊C4—91no—
136C2×C22⋊C4—31no—
137C2×C22⋊C4—31no—
138C2×C22⋊C4—91no—
139C2×C22⋊C4—91no—
140#95—91no—
141C23×C3—22no—
142C23×C3—22no—
143C23×C3—21no—
144C23×C3—21no—
145C23×C3—21no—
146C23×C3—21no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3—21no—
150C23×C3—21no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—21no—
155C23×C3—22no—
156C23×C3—21no—
157C23×C3—41no—
158C23×C3—41no—
159C23×C3—41no—
160C23×C3—41no—
161C23×C3D1212yes—
162C23×C3#6712yes—
163C23×C3C22×D311yescharacteristic
164C23×C3C22×D311yescharacteristic
165C22×D3—32no—
166C22×D3—32no—
167C22×D3—62no—
168C22×D3—62no—
169C22×D3—62no—
170C22×D3—61no—
171C22×D3—61no—
172C22×D3—61no—
173C22×D3—61no—
174C22×D3—61no—
175C22×D3—61no—
176C22×D3—61no—
177C22×D3—61no—
178C22×D3—61no—
179C22×D3—61no—
180C22×D3—61no—
181C22×D3—61no—
182C22×D3—121no—
183C22×D3—121no—
184C22×D3—121no—
185C22×D3—121no—
186C2×C3×C4—32no—
187C2×C3×C4—61no—
188C2×C3×C4—61no—
189C2×C3×C4—61no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—61no—
193C2×C3×C4—61no—
194C2×C3×C4—61no—
195C2×C3×C4—61no—
196C2×C3×C4—61no—
197C2×C3×C4—61no—
198#67—616no—
199#67—68no—
200#67—68no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—32no—
213C2×Dic3—32no—
214C2×Dic3—62no—
215C2×Dic3—62no—
216C2×Dic3—62no—
217C2×Dic3—61no—
218C2×Dic3—61no—
219C2×Dic3—61no—
220C2×Dic3—61no—
221C2×Dic3#6712yescomplemented
222C2×C32—42no—
223C2×C32#12011yescharacteristic
224C2×C32#12111yescharacteristic
225C2×C32#12111yescharacteristic
226C2×C32#12111yescharacteristic
227C2×C32#12311yescharacteristic
228C2×C32#12311yescharacteristic
229C2×C32#12411yescharacteristic
230C3⋊D3—42no—
231C24—91no—
232C24D3211yescharacteristic
233C2×D4—182no—
234C2×D4—182no—
235C2×D4—182no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—91no—
241C4⋊C4—182no—
242C22⋊C4—62no—
243C22⋊C4—62no—
244C22⋊C4—182no—
245C22⋊C4—182no—
246C22×C3—24no—
247C22×C3—22no—
248C22×C3—22no—
249C22×C3—22no—
250C22×C3—22no—
251C22×C3—22no—
252C22×C3—22no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—42no—
261C22×C3—42no—
262C22×C3—42no—
263C22×C3—42no—
264C22×C3—42no—
265C22×C3—42no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3C2×D1211yescharacteristic
275C22×C3#23511yescharacteristic
276C22×C3#23511yescharacteristic
277C22×C3#23511yescharacteristic
278C22×C3D3×D411yescharacteristic
279C22×C3D3×D411yescharacteristic
280C22×C3D3×D411yescharacteristic
281C22×C3D3×D411yescharacteristic
282C22×C3D3×D411yescharacteristic
283C22×C3#23711yescharacteristic
284C22×C3#23711yescharacteristic
285C22×C3#23711yescharacteristic
286C22×C3C2×#6711yescharacteristic
287C22×C3C2×#6711yescharacteristic
288C2×D3—64no—
289C2×D3—62no—
290C2×D3—62no—
291C2×D3—62no—
292C2×D3—62no—
293C2×D3—62no—
294C2×D3—62no—
295C2×D3—122no—
296C2×D3—122no—
297C2×D3—122no—
298C2×D3—122no—
299C2×D3—122no—
300C2×D3—122no—
301C2×D3—121no—
302C2×D3—121no—
303C2×D3—121no—
304C2×D3—121no—
305C2×D3—121no—
306C2×D3—121no—
307C2×D3—121no—
308C2×D3—121no—
309C3×C4—62no—
310C3×C4—121no—
311C3×C4—121no—
312C3×C4—121no—
313Dic3—22no—
314Dic3—41no—
315Dic3—41no—
316Dic3—41no—
317Dic3—64no—
318Dic3—62no—
319Dic3—62no—
320C32#39411yescharacteristic, complemented, Sylow
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—92no—
328C23—181no—
329C23—181no—
330C23—181no—
331C23—181no—
332C23—181no—
333C23—181no—
334C23#62612yes—
335C23C2×D3211yescharacteristic
336D4—188no—
337C2×C4—32no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—92no—
350C2×C4—181no—
351C2×C4—181no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—44no—
360C2×C3—42no—
361C2×C3—42no—
362C2×C3#88211yescharacteristic
363C2×C3#88311yescharacteristic
364C2×C3#88311yescharacteristic
365C2×C3#88411yescharacteristic
366C2×C3#88411yescharacteristic
367C2×C3#88511yescharacteristic
368C2×C3#88511yescharacteristic
369C2×C3#88611yescharacteristic
370C2×C3#92911yescharacteristic
371C2×C3#93011yescharacteristic
372C2×C3#93711yescharacteristic
373C2×C3#93811yescharacteristic
374C2×C3#93911yescharacteristic
375C2×C3#93911yescharacteristic
376D3—124no—
377D3—122no—
378D3—122no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—182no—
387C22—182no—
388C22—182no—
389C22—361no—
390C22—361no—
391C22—361no—
392C22—361no—
393C22#380411yescharacteristic
394C22#380411yescharacteristic
395C22#380411yescharacteristic
396C22C2×#62611yescharacteristic
397C22D3×#6711yescharacteristic
398C22#381111yescharacteristic
399C22#381111yescharacteristic
400C4—62no—
401C4—121no—
402C4—121no—
403C4—121no—
404C4—182no—
405C3—21no—
406C3#495911yescharacteristic, complemented
407C3#496111yescharacteristic, complemented
408C2—42no—
409C2—362no—
410C2#6395311yescharacteristic
411C2#6395811yescharacteristic
412C2#6396611yescharacteristic
413C2#6397111yescharacteristic
414C2#6397511yescharacteristic
415C2#6397811yescharacteristic
416C2#6398111yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, cosocleC23×C32C23
FittingC24×C32C22
prime radicalC22×#638C2
hypercenterC24D32
hypocenterC32#394
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×C3×#217C2semidirect
3. C2×C3×#217C2semidirect
4. C2×Dic3⋊Dic3C2semidirect
5. C2×#3722C2semidirect
6. C2×#3722C2semidirect
7. #63579C2semidirect
30. C2×#217D3semidirect
31. C2×#217D3semidirect
70. C2×C3×Dic3D4semidirect
221. C2×Dic3#67semidirect
320. C32#394semidirect
406. C3#4959semidirect
407. C3#4961semidirect

Maximal subgroups

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

subgroupindex
C22×#6382
C2×C3×#2172
C2×Dic3⋊Dic32
C2×#37222
#635792
#49613
#49593