Finite Group Databasebeta

← #11714025

Subgroups — #11714025

Order 768.

500 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 61 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×#520⋊C3C211yesmaximal, characteristic
2C2×#5389C211yesmaximal, characteristic, complemented
3#73151C211yesmaximal, characteristic
4#73151C211yesmaximal, characteristic
5#73062C211yesmaximal, characteristic, complemented
6C4⋊#859C211yesmaximal, characteristic, complemented
7#73059C211yesmaximal, characteristic, complemented
8#32968—31nomaximal, Sylow
9C22×#41⋊C3C2211yescharacteristic
10#520⋊C3—22no—
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×#859C2211yescharacteristic
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16C2×#859C2211yescharacteristic
17#5389—22no—
18#4982—41nomaximal
19C2×#520D311yescharacteristic
20C2×#479—31no—
21C2×#478—31no—
22#2629—31no—
23#1865—31no—
24#1864—31no—
25#1863—31no—
26#1862—31no—
27#1853—31no—
28#1852—31no—
29#1803—31no—
30#1802—31no—
31#1798—31no—
32#1788—31no—
33#1788—31no—
34C2×#41⋊C3—21no—
35C2×#41⋊C3—21no—
36C2×#41⋊C3Q812yes—
37C22×SL(2,𝔽3)C2311yescharacteristic
38C2×C3×C4⋊C4—41no—
39C2×C4⋊Dic3—41no—
40C2×Dic3⋊C4—41no—
41C2×Dic3⋊C4—41no—
42C4×SL(2,𝔽3)—21no—
43C4×SL(2,𝔽3)D412yes—
44#859—21no—
45#859—21no—
46#859—21no—
47#859—21no—
48C3⋊#95—41no—
49C3⋊#95—41no—
50C3⋊#95—41no—
51C22×#41C2×D311yescharacteristic
52C2×#125—31no—
53#520—22no—
54#520—62no—
55C2×C4×Q8C2×D311yescharacteristic
56C2×C4×Q8C2×D311yescharacteristic
57C2×C4×D4—31no—
58C2×C4×D4—31no—
59C2×#117—31no—
60C22×C4⋊C4—31no—
61#479—62no—
62#479—62no—
63#478—64no—
64C2×#107—31no—
65C2×#107—31no—
66C2×#106—31no—
67C2×#106—31no—
68C2×C4⋊C8—31no—
69C2×C4⋊C8—31no—
70C2×#103—31no—
71C2×#103—31no—
72C2×#103—31no—
73C2×#103—31no—
74C2×#102—31no—
75C2×#102—31no—
76C2×#102—31no—
77C2×#102—31no—
78C2×C22⋊C8—31no—
79C2×C22⋊C8—31no—
80#398—61no—
81#398—61no—
82#396—61no—
83#396—61no—
84#389—31no—
85#389—31no—
86#384—31no—
87#380—31no—
88#380—31no—
89#340—31no—
90#340—31no—
91#340—31no—
92#340—31no—
93#336—31no—
94#336—31no—
95C22×C3×C4—41no—
96C22×C3×C4—41no—
97C22×C3×C4—41no—
98C22×Dic3—41no—
99C22×Dic3—41no—
100C22×Dic3—41no—
101C22×Dic3—41no—
102#41⋊C3—22no—
103C2×SL(2,𝔽3)C2×D411yescharacteristic
104C2×SL(2,𝔽3)C2×Q811yescharacteristic
105C2×SL(2,𝔽3)#4111yescharacteristic
106C3×C4⋊C4—82no—
107C4⋊Dic3—82no—
108Dic3⋊C4—82no—
109Dic3⋊C4—82no—
110C2×#41—21no—
111C2×#41—21no—
112C2×#41—32no—
113C2×#41—61no—
114C2×#41—61no—
115C2×#41Dic612yes—
116C22×Q8C22×D311yescharacteristic
117C22×D4—31no—
118C23×C4—31no—
119C23×C4—61no—
120C22×C8—31no—
121C22×C8—31no—
122C22×C8—31no—
123C22×C8—31no—
124#125—62no—
125#125—62no—
126C4×Q8—21no—
127C4×Q8—32no—
128C4×Q8—61no—
129C4×Q8#6712yes—
130C4×D4—34no—
131C4×D4—62no—
132C4×D4—62no—
133C4×D4—62no—
134#117—62no—
135#117—62no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—31no—
143C2×C4⋊C4—62no—
144C2×C4⋊C4—62no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4S411yescharacteristic
152C2×C22⋊C4—31no—
153C2×C22⋊C4—31no—
154C2×C22⋊C4—61no—
155C2×C22⋊C4—61no—
156C2×C42—31no—
157C2×C42—31no—
158#107—62no—
159#107—62no—
160#106—62no—
161#106—62no—
162C4⋊C8—62no—
163C4⋊C8—62no—
164#103—61no—
165#103—61no—
166#103—61no—
167#103—61no—
168#103—61no—
169#103—61no—
170#103—61no—
171#103—61no—
172#102—62no—
173#102—62no—
174#102—62no—
175#102—62no—
176C22⋊C8—62no—
177C22⋊C8—62no—
178#95—61no—
179#95—61no—
180#95—61no—
181#95—61no—
182C23×C3—41no—
183C2×C3×C4—42no—
184C2×C3×C4—42no—
185C2×C3×C4—81no—
186C2×C3×C4—81no—
187C2×C3×C4—81no—
188C2×C3×C4—81no—
189C2×C3×C4—81no—
190C2×C3×C4—81no—
191C2×C3×C4—81no—
192C2×Dic3—42no—
193C2×Dic3—42no—
194C2×Dic3—81no—
195C2×Dic3—81no—
196C2×Dic3—81no—
197C2×Dic3—81no—
198C2×Dic3—81no—
199C2×Dic3—81no—
200C2×Dic3—81no—
201C2×Dic3—81no—
202C2×Dic3—81no—
203C2×Dic3—81no—
204SL(2,𝔽3)C22⋊Q811yescharacteristic
205C24—31no—
206#41—22no—
207#41—64no—
208#41—62no—
209#41—62no—
210#41—62no—
211C2×Q8—31no—
212C2×Q8—31no—
213C2×Q8—31no—
214C2×Q8C2×Dic611yescharacteristic
215C2×Q8#23511yescharacteristic
216C2×Q8C2×#6711yescharacteristic
217C2×D4—32no—
218C2×D4—32no—
219C2×D4—32no—
220C2×D4—61no—
221C2×D4—61no—
222C2×D4—61no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—32no—
226C22×C4—32no—
227C22×C4—31no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—64no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—241no—
238C22×C4—241no—
239C22×C4C2×S411yescharacteristic
240C22×C4C2×S411yescharacteristic
241C22×C4C2×S411yescharacteristic
242C2×C8—61no—
243C2×C8—61no—
244C2×C8—61no—
245C2×C8—61no—
246C2×C8—61no—
247C2×C8—61no—
248C2×C8—61no—
249C2×C8—61no—
250C2×C8—61no—
251C2×C8—61no—
252C2×C8—61no—
253C2×C8—61no—
254C4⋊C4—22no—
255C4⋊C4—32no—
256C4⋊C4—32no—
257C4⋊C4—62no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—122no—
269C4⋊C4—122no—
270C4⋊C4—241no—
271C4⋊C4—241no—
272C4⋊C4—241no—
273C4⋊C4—241no—
274C22⋊C4—62no—
275C22⋊C4—62no—
276C22⋊C4—241no—
277C22⋊C4—241no—
278C42—32no—
279C42—32no—
280C42—61no—
281C42—61no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—41no—
289C3×C4—82no—
290C3×C4—82no—
291C3×C4—161no—
292Dic3—82no—
293Dic3—82no—
294Dic3—161no—
295Dic3—161no—
296C23—32no—
297C23—61no—
298C23—61no—
299C23—61no—
300C23—61no—
301C23—61no—
302C23—61no—
303C23C22×S411yescharacteristic
304Q8—31no—
305Q8—31no—
306Q8—31no—
307Q8—61no—
308Q8#92411yescharacteristic
309D4—34no—
310D4—62no—
311D4—62no—
312D4—62no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—31no—
321C2×C4—31no—
322C2×C4—31no—
323C2×C4—31no—
324C2×C4—31no—
325C2×C4—31no—
326C2×C4—32no—
327C2×C4—62no—
328C2×C4—62no—
329C2×C4—62no—
330C2×C4—62no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—244no—
349C2×C4A4⋊Q812yes—
350C2×C4GL(2,ℤ/4)12yes—
351C8—121no—
352C8—121no—
353C8—121no—
354C8—121no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—41no—
362C22—62no—
363C22—62no—
364C22—62no—
365C22—121no—
366C22—121no—
367C22—121no—
368C22—121no—
369C22C2×A4⋊Q811yescharacteristic
370C22#591211yescharacteristic
371C22#592411yescharacteristic
372C22#592511yescharacteristic
373C22#592611yescharacteristic
374C22#592711yescharacteristic
375C22C2×GL(2,ℤ/4)11yescharacteristic
376C4—22no—
377C4—22no—
378C4—31no—
379C4—31no—
380C4—31no—
381C4—31no—
382C4—41no—
383C4—62no—
384C4—121no—
385C4—241no—
386C4—241no—
387C4—241no—
388C4—241no—
389C3—41noSylow
390C2—122no—
391C2#8536111yescharacteristic
392C2#8536211yescharacteristic
393C2#8536311yescharacteristic
394C2#8536711yescharacteristic
395C2#8548011yescharacteristic
396C2#8552911yescharacteristic
397C2#8553911yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#520D3
prime radicalC22×#41⋊C3C22
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)C22⋊Q8
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
2. C2×#5389C2semidirect
5. #73062C2semidirect
6. C4⋊#859C2semidirect
7. #73059C2semidirect

Maximal subgroups

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

subgroupindex
C2×#520⋊C32
C2×#53892
#731512
#730622
C4⋊#8592
#730592
#329683
#49824