Finite Group Databasebeta

← #11713920

Subgroups — #11713920

Order 768.

447 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C4⋊C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#73065C211yesmaximal, characteristic
3#73065C211yesmaximal, characteristic
4#73065C211yesmaximal, characteristic
5C4⋊#859C211yesmaximal, characteristic, complemented
6#73059C211yesmaximal, characteristic, complemented
7C4×#859C211yesmaximal, characteristic, complemented
8#27554—31nomaximal, Sylow
9C22×GL(2,𝔽3)C2211yescharacteristic
10C2×#898—41nomaximal
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×C4×SL(2,𝔽3)C2211yescharacteristic
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16C2×#859C2211yescharacteristic
17#2512—31no—
18Q8×C4⋊C4D311yescharacteristic, complemented
19#2100—31no—
20#2099—31no—
21#2071—31no—
22#2065—31no—
23#1934—31no—
24#1920—31no—
25#1916—31no—
26#1880—31no—
27#1880—31no—
28#1880—31no—
29#1865—31no—
30#1803—31no—
31C4×#103—31no—
32C22×SL(2,𝔽3)C2311yescharacteristic
33C2×GL(2,𝔽3)—21no—
34C2×GL(2,𝔽3)—21no—
35C2×GL(2,𝔽3)—21no—
36C2×C3×C4⋊C4—41no—
37C2×(C2×D3)⋊C4—41no—
38C2×(C2×D3)⋊C4—41no—
39C2×(C2×D3)⋊C4—41no—
40C2×C4⋊Dic3—41no—
41C2×Dic3⋊C4—41no—
42C2×C4×Dic3—41no—
43#898—48no—
44C4×SL(2,𝔽3)—21no—
45C4×SL(2,𝔽3)—21no—
46C4×SL(2,𝔽3)—21no—
47#859—21no—
48#859—21no—
49#859—21no—
50C22×SD16—31no—
51C2×#126—61no—
52C2×C4×Q8C2×D311yescharacteristic
53C2×C4×Q8C2×D311yescharacteristic
54C2×C4×Q8C2×D311yescharacteristic
55C2×#106—31no—
56C2×C4⋊C8—31no—
57C2×#103—31no—
58C2×#103—31no—
59C2×#103—31no—
60C2×#102—31no—
61C2×#102—31no—
62C2×#102—31no—
63C2×C4×C8—31no—
64#397—61no—
65#396—61no—
66#390—31no—
67#390—31no—
68#390—31no—
69#389—31no—
70#389—31no—
71#389—31no—
72#388—61no—
73#384—31no—
74#384—31no—
75C4×C4⋊C4—31no—
76C4×C4⋊C4—31no—
77#340—31no—
78#340—31no—
79#340—31no—
80#340—31no—
81#340—31no—
82#340—31no—
83#340—31no—
84#340—31no—
85#336—31no—
86#336—31no—
87#336—31no—
88#336—31no—
89C23×D3—41no—
90C22×C3×C4—41no—
91C22×C3×C4—41no—
92C22×C3×C4—41no—
93C22×Dic3—41no—
94C22×Dic3—41no—
95C22×Dic3—41no—
96C2×SL(2,𝔽3)#4111yescharacteristic
97C2×SL(2,𝔽3)#4111yescharacteristic
98C2×SL(2,𝔽3)#4111yescharacteristic
99GL(2,𝔽3)—41no—
100C3×C4⋊C4—44no—
101(C2×D3)⋊C4—44no—
102(C2×D3)⋊C4—44no—
103(C2×D3)⋊C4—44no—
104C4⋊Dic3—44no—
105Dic3⋊C4—44no—
106C4×Dic3—44no—
107C22×Q8C22×D311yescharacteristic
108C22×D4—31no—
109C2×SD16—61no—
110C2×SD16—61no—
111C2×SD16—61no—
112C2×SD16—61no—
113C2×SD16—61no—
114C2×SD16—61no—
115C22×C8—31no—
116C22×C8—31no—
117C22×C8—31no—
118C22×C8—31no—
119#126—124no—
120C4×Q8—21no—
121C4×Q8—21no—
122C4×Q8—21no—
123C4×Q8—61no—
124C4×Q8—61no—
125C4×Q8—61no—
126C2×C4⋊C4—31no—
127C2×C4⋊C4—31no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4—61no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4—61no—
137C2×C4⋊C4S411yescharacteristic
138C2×C22⋊C4—61no—
139C2×C22⋊C4—61no—
140C2×C22⋊C4—61no—
141C2×C22⋊C4—61no—
142C2×C22⋊C4—61no—
143C2×C22⋊C4—61no—
144C2×C42—31no—
145C2×C42—31no—
146C2×C42—31no—
147C2×C42—61no—
148#106—62no—
149C4⋊C8—121no—
150#103—61no—
151#103—61no—
152#103—61no—
153#103—61no—
154#103—61no—
155#103—61no—
156#102—62no—
157#102—121no—
158#102—121no—
159C4×C8—121no—
160#95—31no—
161#95—31no—
162#95—61no—
163#95—61no—
164C23×C3—41no—
165C22×D3—42no—
166C22×D3—81no—
167C22×D3—81no—
168C22×D3—81no—
169C22×D3—81no—
170C22×D3—81no—
171C22×D3—81no—
172C2×C3×C4—42no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—81no—
176C2×C3×C4—81no—
177C2×C3×C4—81no—
178C2×C3×C4—81no—
179C2×C3×C4—81no—
180C2×C3×C4—81no—
181C2×Dic3—42no—
182C2×Dic3—42no—
183C2×Dic3—42no—
184C2×Dic3—81no—
185C2×Dic3—81no—
186C2×Dic3—81no—
187C2×Dic3—81no—
188C2×Dic3—81no—
189C2×Dic3—81no—
190SL(2,𝔽3)#12611yescharacteristic, complemented
191C24—61no—
192C2×Q8—31no—
193C2×Q8—31no—
194C2×Q8—31no—
195C2×Q8#23511yescharacteristic
196C2×Q8#23711yescharacteristic
197C2×Q8#23911yescharacteristic
198C2×D4—61no—
199C2×D4—61no—
200C2×D4—61no—
201C2×D4—61no—
202C2×D4—61no—
203C2×D4—61no—
204C22×C4—31no—
205C22×C4—31no—
206C22×C4—31no—
207C22×C4—31no—
208C22×C4—61no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4C2×S411yescharacteristic
212C22×C4C2×S411yescharacteristic
213C22×C4C2×S411yescharacteristic
214SD16—121no—
215SD16—121no—
216SD16—121no—
217SD16—121no—
218C2×C8—61no—
219C2×C8—61no—
220C2×C8—61no—
221C2×C8—61no—
222C2×C8—61no—
223C2×C8—61no—
224C2×C8—61no—
225C2×C8—61no—
226C2×C8—61no—
227C2×C8—61no—
228C2×C8—61no—
229C2×C8—61no—
230C4⋊C4—32no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—121no—
243C4⋊C4—122no—
244C4⋊C4—122no—
245C4⋊C4—241no—
246C4⋊C4—241no—
247C4⋊C4GL(2,𝔽3)14yescomplemented
248C22⋊C4—122no—
249C22⋊C4—122no—
250C22⋊C4—122no—
251C22⋊C4—122no—
252C22⋊C4—241no—
253C22⋊C4—241no—
254C42—61no—
255C42—61no—
256C42—61no—
257C42—61no—
258C42—61no—
259C42—61no—
260C42—122no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—41no—
268C2×D3—82no—
269C2×D3—82no—
270C2×D3—82no—
271C2×D3—161no—
272C2×D3—161no—
273C2×D3—161no—
274C2×D3—161no—
275C3×C4—82no—
276C3×C4—82no—
277C3×C4—82no—
278Dic3—82no—
279Dic3—82no—
280Dic3—82no—
281C23—121no—
282C23—121no—
283C23—121no—
284C23—121no—
285C23—121no—
286C23—121no—
287C23—121no—
288C23C22×S411yescharacteristic
289Q8—31no—
290Q8—31no—
291Q8—31no—
292Q8—61no—
293Q8#89811yescharacteristic, complemented
294D4—121no—
295D4—121no—
296D4—121no—
297D4—121no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4C2×GL(2,𝔽3)12yes—
329C2×C4C2×GL(2,𝔽3)12yes—
330C2×C4C2×GL(2,𝔽3)12yes—
331C8—121no—
332C8—121no—
333C8—121no—
334C8—121no—
335C2×C3—41no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—41no—
339C2×C3—41no—
340C2×C3—41no—
341C2×C3—41no—
342D3—162no—
343C22—241no—
344C22—241no—
345C22—241no—
346C22—241no—
347C22—241no—
348C22—241no—
349C22—241no—
350C22#591211yescharacteristic
351C22#591411yescharacteristic
352C22C22×GL(2,𝔽3)11yescharacteristic
353C22#591911yescharacteristic
354C22#592311yescharacteristic
355C22#592511yescharacteristic
356C22#592711yescharacteristic
357C4—22no—
358C4—22no—
359C4—22no—
360C4—31no—
361C4—31no—
362C4—31no—
363C4—31no—
364C4—121no—
365C4—121no—
366C4—121no—
367C4—241no—
368C4—241no—
369C4—241no—
370C3—41noSylow
371C2—481no—
372C2#8531311yescharacteristic
373C2#8532511yescharacteristic
374C2#8537111yescharacteristic
375C2#8539711yescharacteristic
376C2#8541311yescharacteristic
377C2#8542511yescharacteristic
378C2#8543011yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingQ8×C4⋊C4D3
prime radicalC22×GL(2,𝔽3)C22
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#126
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C4⋊C4×SL(2,𝔽3)C2semidirect
5. C4⋊#859C2semidirect
6. #73059C2semidirect
7. C4×#859C2semidirect
18. Q8×C4⋊C4D3semidirect
190. SL(2,𝔽3)#126semidirect
247. C4⋊C4GL(2,𝔽3)semidirect
293. Q8#898semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×SL(2,𝔽3)2
#730652
C4⋊#8592
#730592
C4×#8592
#275543
C2×#8984