Finite Group Databasebeta

← #11923086

Subgroups — #11923086

Order 1152.

728 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 81 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×C3×2−1+4⋊C3C211yesmaximal, characteristic
2C22×C3⋊#226C211yesmaximal, characteristic, complemented
3#10593659C212yesmaximal, complemented
4#10593654C211yesmaximal, characteristic, complemented
5#10593653C212yesmaximal, complemented
6#85535—33nomaximal
7#84117—31nomaximal
8C3×2−1+4⋊C3—22no—
9C2×C3×#41⋊C3C2211yescharacteristic
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊#226—22no—
13C2×C3⋊#226C2212yes—
14#64144—41nomaximal
15C3⋊#859C2212yes—
16C2×C3×2−1+4D311yescharacteristic
17C2×2−1+4⋊C3D313yes—
18C22×#226—33no—
19#5825—31no—
20C2×#951—31no—
21C22×#216—31no—
22#5425—36no—
23#5420—33no—
24#5419—36no—
25#5239—32no—
26#5239—32no—
27#5234—31no—
28#5233—31no—
29#5228—32no—
30#5228—32no—
31#5225—31no—
32C2×C32×D4—41no—
33C22×C3⋊Dic3—41no—
34C3×#41⋊C3—21no—
35C2×C3×SL(2,𝔽3)—21no—
36C2×C3×SL(2,𝔽3)D412yes—
37C2×C3×SL(2,𝔽3)C2311yescharacteristic
38C3⋊#226—22no—
39#3757—42no—
40#3757—41no—
41#3750—42no—
42#3011—91noSylow
43C3×2−1+4—22no—
44C3×2−1+4—62no—
45C2×C3×#41—31no—
46C2×C3×#41—31no—
47C2×C3×#41—31no—
48C2×C3×#41C2×D311yescharacteristic
49C22×C3×Q8—31no—
50C22×C3×Q8C2×D311yescharacteristic
51C22×C3×Q8C2×D311yescharacteristic
52C22×Dic6—31no—
532−1+4⋊C3—26no—
54C2×#41⋊C3C2×D313yes—
55C22×SL(2,𝔽3)C2×D313yes—
56C22×SL(2,𝔽3)C2×D313yes—
57C2×#226—36no—
58C2×#226—66no—
59#951—64no—
60#944—61no—
61C2×#216—32no—
62C2×#216—32no—
63C2×#216—32no—
64C2×#216—62no—
65C2×#216—62no—
66#936—61no—
67#935—61no—
68#935—61no—
69#935—123no—
70C2×#214—32no—
71#926—31no—
72#924—32no—
73#924—32no—
74C2×#208—31no—
75C22×C3⋊C8—31no—
76#859—36no—
77#835—32no—
78#835—32no—
79#835—32no—
80#832—32no—
81#830—32no—
82#830—32no—
83C23×C32—41no—
84C23×C32—41no—
85C32×D4—82no—
86C2×C32×C4—41no—
87C2×C3⋊Dic3—42no—
88C2×C3⋊Dic3—42no—
89C2×C3⋊Dic3—82no—
90C3×SL(2,𝔽3)C2×D411yescharacteristic
91C2×2−1+4C3⋊D311yescharacteristic
92C2×#137—91no—
93C22×Q16—91no—
94#536—91no—
95#452—92no—
96#452—92no—
97#451—92no—
98#451—92no—
99#419—91no—
100#416—91no—
101#408—91no—
102C3×#41—21no—
103C3×#41—34no—
104C3×#41—62no—
105C3×#41—62no—
106C3×#41—62no—
107C3×#41—62no—
108C3×#41—62no—
109C3×#41—61no—
110C2×C3×Q8—21no—
111C2×C3×Q8—31no—
112C2×C3×Q8—31no—
113C2×C3×Q8—31no—
114C2×C3×Q8—32no—
115C2×C3×Q8—62no—
116C2×C3×Q8—62no—
117C2×C3×Q8—61no—
118C2×C3×Q8#6712yes—
119C2×C3×Q8C22×D311yescharacteristic
120C2×C3×D4—31no—
121C2×C3×D4—31no—
122C2×C3×D4—31no—
123C2×C3×D4—43no—
124C2×C3×D4S411yescharacteristic
125C22×C3×C4—31no—
126C22×C3×C4—31no—
127C22×C3×C4—31no—
128C22×C3×C4—61no—
129C22×Dic3—61no—
130C22×Dic3—123no—
131C2×Dic6—32no—
132C2×Dic6—62no—
133C2×Dic6—62no—
134C2×Dic6—61no—
135#41⋊C3—23no—
136C2×SL(2,𝔽3)—23no—
137C2×SL(2,𝔽3)#6716yes—
138C2×SL(2,𝔽3)C22×D313yes—
139#226—66no—
140#217—62no—
141#217—62no—
142#217—61no—
143#217—126no—
144#217—123no—
145#216—64no—
146#216—62no—
147#216—64no—
148#216—62no—
149#214—64no—
150C4⋊Dic3—32no—
151Dic3⋊C4—62no—
152Dic3⋊C4—62no—
153Dic3⋊C4—126no—
154C4×Dic3—61no—
155#208—62no—
156C2×C3⋊C8—32no—
157C2×C3⋊C8—32no—
158C2×C3⋊C8—62no—
159C22×C32—42no—
160C22×C32—41no—
161C22×C32—82no—
162C22×C32—81no—
163C22×C32—82no—
164C32×C4—81no—
165C3⋊Dic3—82no—
166C3⋊Dic3—82no—
1672−1+4—22no—
1682−1+4—62no—
169C2×#41—31no—
170C2×#41—31no—
171C2×#41—31no—
172C2×#41C2×C3⋊D311yescharacteristic
173C22×Q8—31no—
174C22×Q8—91no—
175C22×Q8C2×C3⋊D311yescharacteristic
176C22×Q8C2×C3⋊D311yescharacteristic
177#137—184no—
178C2×Q16—92no—
179C2×Q16—92no—
180C2×Q16—92no—
181C2×Q16—182no—
182C2×Q16—182no—
183C2×SD16—92no—
184C2×M16—91no—
185C22×C8—91no—
186C4⋊Q8—181no—
187#124—181no—
188#123—181no—
189#123—181no—
190C22⋊Q8—92no—
191C22⋊Q8—92no—
192#117—91no—
193#103—92no—
194#103—92no—
195#103—92no—
196#102—92no—
197C22⋊C8—92no—
198C22⋊C8—92no—
199C23×C3—31no—
200C23×C3—43no—
201C23×C3—43no—
202C23×C3C2×S411yescharacteristic
203C23×C3C2×S411yescharacteristic
204C3×Q8—32no—
205C3×Q8—31no—
206C3×Q8—32no—
207C3×Q8—62no—
208C3×Q8—61no—
209C3×Q8—61no—
210C3×Q8C2×#6711yescharacteristic
211C3×D4—22no—
212C3×D4—32no—
213C3×D4—62no—
214C3×D4—61no—
215C3×D4—62no—
216C3×D4—86no—
217C2×C3×C4—32no—
218C2×C3×C4—32no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—43no—
223C2×C3×C4—61no—
224C2×C3×C4—61no—
225C2×C3×C4—61no—
226C2×C3×C4—61no—
227C2×C3×C4—64no—
228C2×C3×C4C2×S411yescharacteristic
229C2×Dic3—62no—
230C2×Dic3—62no—
231C2×Dic3—126no—
232C2×Dic3—126no—
233C2×Dic3—122no—
234C2×Dic3—246no—
235Dic6—62no—
236Dic6—62no—
237Dic6—122no—
238SL(2,𝔽3)C2×#6713yes—
239C3⋊C8—62no—
240C3⋊C8—62no—
241C2×C32—42no—
242C2×C32—41no—
243C2×C32—82no—
244C2×C32—161no—
245#41—21no—
246#41—34no—
247#41—62no—
248#41—62no—
249#41—62no—
250#41—62no—
251#41—62no—
252#41—61no—
253C2×Q8—21no—
254C2×Q8—31no—
255C2×Q8—31no—
256C2×Q8—31no—
257C2×Q8—32no—
258C2×Q8—62no—
259C2×Q8—62no—
260C2×Q8—61no—
261C2×Q8—92no—
262C2×Q8—181no—
263C2×Q8—182no—
264C2×Q8—182no—
265C2×Q8#63812yes—
266C2×Q8C22×C3⋊D311yescharacteristic
267C2×D4—31no—
268C2×D4—31no—
269C2×D4—31no—
270C2×D4C3⋊S411yescharacteristic
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—61no—
275C22×C4—181no—
276Q16—184no—
277Q16—182no—
278Q16—184no—
279Q16—182no—
280SD16—184no—
281M16—182no—
282C2×C8—92no—
283C2×C8—92no—
284C2×C8—182no—
285C4⋊C4—92no—
286C4⋊C4—182no—
287C4⋊C4—182no—
288C22⋊C4—182no—
289C22⋊C4—181no—
290C22⋊C4—182no—
291C42—181no—
292C22×C3—22no—
293C22×C3—22no—
294C22×C3—21no—
295C22×C3—46no—
296C22×C3—43no—
297C22×C3—62no—
298C22×C3—61no—
299C22×C3—86no—
300C22×C3—83no—
301C22×C3—86no—
302C22×C3GL(2,ℤ/4)12yes—
303C22×C3C22×S411yescharacteristic
304C3×C4—21no—
305C3×C4—31no—
306C3×C4—31no—
307C3×C4—32no—
308C3×C4—61no—
309C3×C4—83no—
310Dic3—122no—
311Dic3—122no—
312Dic3—246no—
313Dic3—246no—
314C32—41noSylow
315C23—31no—
316C23C2×C3⋊S411yescharacteristic
317C23C2×C3⋊S411yescharacteristic
318Q8—32no—
319Q8—31no—
320Q8—32no—
321Q8—62no—
322Q8—61no—
323Q8—61no—
324Q8—182no—
325Q8—182no—
326Q8—362no—
327Q8C2×#63811yescharacteristic
328D4—22no—
329D4—32no—
330D4—62no—
331D4—61no—
332D4—62no—
333C2×C4—32no—
334C2×C4—32no—
335C2×C4—31no—
336C2×C4—31no—
337C2×C4—31no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—64no—
343C2×C4—182no—
344C2×C4—182no—
345C2×C4—362no—
346C2×C4C2×C3⋊S411yescharacteristic
347C8—182no—
348C8—182no—
349C2×C3—22no—
350C2×C3—41no—
351C2×C3—46no—
352C2×C3—43no—
353C2×C3—86no—
354C2×C3—121no—
355C2×C3—163no—
356C2×C3#592712yes—
357C2×C3C2×GL(2,ℤ/4)11yescharacteristic
358C22—22no—
359C22—22no—
360C22—21no—
361C22—62no—
362C22—61no—
363C22#6426912yes—
364C22C22×C3⋊S411yescharacteristic
365C4—21no—
366C4—31no—
367C4—31no—
368C4—32no—
369C4—61no—
370C4—362no—
371C4—362no—
372C3—43no—
373C3#8553511yescharacteristic, complemented
374C2—22no—
375C2—41no—
376C2—121no—
377C2#1059659012yes—
378C2C2×#6426911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC2×C3×2−1+4D3
socleC22×C3C22×S4
prime radicalC2×C3×2−1+4⋊C3C2
hypercenterC2×D4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊#226C2semidirect
3. #10593659C2semidirect
4. #10593654C2semidirect
5. #10593653C2semidirect
373. C3#85535semidirect

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C22×C3⋊#2262
#105936592
#105936542
#105936532
#855353
#841173
#641444