Finite Group Databasebeta

← #11713578

Subgroups — #11713578

Order 768.

544 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 57 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×#5425C211yesmaximal, characteristic
2C2×#5425C211yesmaximal, characteristic
3C2×C4⋊#226C211yesmaximal, characteristic, complemented
4C2×C4×#226C211yesmaximal, characteristic, complemented
5#2341⋊C3C211yesmaximal, characteristic
6#73061C211yesmaximal, characteristic, complemented
7#73059C211yesmaximal, characteristic, complemented
8#25151—31nomaximal, Sylow
9C22×#41⋊C3C2211yescharacteristic
10C22×#226C2211yescharacteristic
11C22×#226C2211yescharacteristic
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×C4×SL(2,𝔽3)C2211yescharacteristic
14#5425—22no—
15#5425—22no—
16C2×#859C2211yescharacteristic
17C2×#859C2211yescharacteristic
18C4⋊#226—22no—
19C4×#226—22no—
20#4647—41nomaximal
21C2×#462—31no—
22C2×#452—31no—
23C2×#452—31no—
24C2×C4×Q16—31no—
25#2429—31no—
26#2341D311yescharacteristic
27#2067—31no—
28#2036—31no—
29#2035—31no—
30#2013—31no—
31#1968—31no—
32#1923—31no—
33#1880—31no—
34#1878—31no—
35#1865—31no—
36C2×#41⋊C3—21no—
37C2×#41⋊C3—21no—
38C2×#41⋊C3—21no—
39C22×SL(2,𝔽3)C2311yescharacteristic
40C2×#226—21no—
41C2×#226—21no—
42C2×#226—21no—
43C2×#226—21no—
44C2×#226—21no—
45C2×#226D412yes—
46C2×C4⋊Dic3—41no—
47C2×Dic3⋊C4—41no—
48C2×Dic3⋊C4—41no—
49C2×Dic3⋊C4—41no—
50C2×C4×Dic3—41no—
51C4×SL(2,𝔽3)—21no—
52C4×SL(2,𝔽3)—21no—
53#859—21no—
54#859D412yes—
55C3×#95—41no—
56C3⋊#95—41no—
57C22×#41C2×D311yescharacteristic
58C22×Q16—31no—
59C22×Q16—31no—
60C22×SD16—31no—
61C22×SD16—31no—
62C2×C4⋊Q8—31no—
63C2×C22⋊Q8—31no—
64C2×C22⋊Q8—31no—
65C2×C4×Q8—31no—
66C2×C4×Q8C2×D311yescharacteristic
67C2×C4×Q8C2×D311yescharacteristic
68#462—62no—
69#462—62no—
70#452—62no—
71#452—62no—
72#452—62no—
73#452—62no—
74C4×Q16—62no—
75C4×Q16—62no—
76C2×#107—31no—
77C2×#106—31no—
78C2×C4⋊C8—31no—
79C2×#103—31no—
80C2×#103—31no—
81C2×#103—31no—
82C2×#103—31no—
83C2×#103—31no—
84C2×#103—31no—
85C2×#102—31no—
86C2×#102—31no—
87C2×C22⋊C8—31no—
88C2×C22⋊C8—31no—
89C2×C4×C8—31no—
90#398—61no—
91#396—61no—
92#391—31no—
93#390—31no—
94#390—31no—
95#389—31no—
96#389—61no—
97#388—61no—
98#380—31no—
99C4×C22⋊C4—31no—
100#340—31no—
101#340—31no—
102#336—31no—
103#336—31no—
104C22×C3×C4—41no—
105C22×C3×C4—41no—
106C22×C3×C4—41no—
107C22×Dic3—41no—
108C22×Dic3—41no—
109C22×Dic3—41no—
110C22×Dic3—41no—
111#41⋊C3—41no—
112C2×SL(2,𝔽3)C2×D411yescharacteristic
113C2×SL(2,𝔽3)C2×D411yescharacteristic
114C2×SL(2,𝔽3)#4111yescharacteristic
115#226—22no—
116#226—41no—
117C4⋊Dic3—82no—
118Dic3⋊C4—44no—
119Dic3⋊C4—82no—
120Dic3⋊C4—82no—
121C4×Dic3—82no—
122C2×#41—21no—
123C2×#41—21no—
124C2×#41—21no—
125C2×#41—61no—
126C2×#41—61no—
127C2×#41—61no—
128C22×Q8—31no—
129C22×Q8—31no—
130C22×Q8C22×D311yescharacteristic
131C22×D4—31no—
132C23×C4—31no—
133C2×Q16—32no—
134C2×Q16—32no—
135C2×Q16—61no—
136C2×Q16—61no—
137C2×Q16—61no—
138C2×Q16—61no—
139C2×Q16—61no—
140C2×Q16—61no—
141C2×Q16—61no—
142C2×Q16—61no—
143C2×Q16—61no—
144C2×Q16—61no—
145C2×SD16—62no—
146C2×SD16—62no—
147C2×SD16—62no—
148C2×SD16—62no—
149C2×SD16—121no—
150C2×SD16—121no—
151C22×C8—31no—
152C22×C8—31no—
153C22×C8—31no—
154C22×C8—31no—
155C4⋊Q8—62no—
156C4⋊Q8—121no—
157C22⋊Q8—62no—
158C22⋊Q8—62no—
159C22⋊Q8—121no—
160C22⋊Q8—121no—
161C4×Q8—21no—
162C4×Q8—21no—
163C4×Q8—61no—
164C4×Q8—61no—
165C4×Q8—62no—
166C4×Q8—121no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C22⋊C4—31no—
178C2×C22⋊C4—31no—
179C2×C22⋊C4—61no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—61no—
182C2×C42—31no—
183C2×C42—31no—
184C2×C42—61no—
185#107—62no—
186#106—121no—
187C4⋊C8—62no—
188#103—32no—
189#103—32no—
190#103—62no—
191#103—62no—
192#103—61no—
193#103—62no—
194#103—61no—
195#103—121no—
196#102—62no—
197#102—62no—
198C22⋊C8—62no—
199C22⋊C8—62no—
200C4×C8—62no—
201#95—31no—
202#95—61no—
203#95—61no—
204#95—61no—
205#95S411yescharacteristic
206C23×C3—41no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4—81no—
210C2×C3×C4—81no—
211C2×C3×C4—81no—
212C2×C3×C4—81no—
213C2×C3×C4—81no—
214C2×C3×C4—81no—
215C2×C3×C4—81no—
216C2×Dic3—42no—
217C2×Dic3—42no—
218C2×Dic3—42no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—81no—
222C2×Dic3—81no—
223C2×Dic3—81no—
224C2×Dic3—81no—
225C2×Dic3—81no—
226C2×Dic3—81no—
227C2×Dic3—81no—
228SL(2,𝔽3)#12111yescharacteristic
229C24—31no—
230#41—41no—
231#41—122no—
232#41—121no—
233#41—121no—
234#41—121no—
235C2×Q8—31no—
236C2×Q8—31no—
237C2×Q8—31no—
238C2×Q8—32no—
239C2×Q8—61no—
240C2×Q8—61no—
241C2×Q8—61no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×Q8—61no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×Q8—61no—
248C2×Q8—61no—
249C2×Q8—61no—
250C2×Q8#23511yescharacteristic
251C2×Q8D3×D411yescharacteristic
252C2×Q8D3×D411yescharacteristic
253C2×D4—32no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—61no—
259C22×C4—31no—
260C22×C4—31no—
261C22×C4—31no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4C2×S411yescharacteristic
273C22×C4C2×S411yescharacteristic
274C22×C4C2×S411yescharacteristic
275Q16—62no—
276Q16—62no—
277Q16—62no—
278Q16—62no—
279Q16—121no—
280Q16—121no—
281Q16—121no—
282Q16—121no—
283SD16—124no—
284SD16—124no—
285C2×C8—32no—
286C2×C8—32no—
287C2×C8—61no—
288C2×C8—61no—
289C2×C8—61no—
290C2×C8—61no—
291C2×C8—61no—
292C2×C8—61no—
293C2×C8—61no—
294C2×C8—61no—
295C2×C8—61no—
296C2×C8—61no—
297C4⋊C4—32no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—122no—
306C4⋊C4—122no—
307C4⋊C4—122no—
308C4⋊C4—122no—
309C4⋊C4—122no—
310C4⋊C4—122no—
311C22⋊C4—62no—
312C22⋊C4—62no—
313C22⋊C4—122no—
314C22⋊C4—122no—
315C22⋊C4—241no—
316C42—61no—
317C42—61no—
318C42—61no—
319C42—61no—
320C42—122no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C3×C4—82no—
329C3×C4—82no—
330C3×C4—161no—
331Dic3—44no—
332Dic3—82no—
333Dic3—82no—
334Dic3—161no—
335C23—61no—
336C23—61no—
337C23—61no—
338C23—61no—
339C23—61no—
340C23—61no—
341C23—61no—
342C23C22×S411yescharacteristic
343Q8—31no—
344Q8—31no—
345Q8—31no—
346Q8—61no—
347Q8—62no—
348Q8—62no—
349Q8—121no—
350Q8—121no—
351Q8—121no—
352Q8—121no—
353Q8—121no—
354Q8—121no—
355Q8#88411yescharacteristic
356D4—62no—
357D4—62no—
358D4—121no—
359D4—121no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—31no—
368C2×C4—31no—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—31no—
372C2×C4—31no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—62no—
376C2×C4—62no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4#98412yes—
396C2×C4#98512yes—
397C8—62no—
398C8—62no—
399C8—121no—
400C8—121no—
401C2×C3—41no—
402C2×C3—41no—
403C2×C3—41no—
404C2×C3—41no—
405C2×C3—41no—
406C2×C3—41no—
407C2×C3—41no—
408C22—121no—
409C22—121no—
410C22—121no—
411C22—121no—
412C22—121no—
413C22—121no—
414C22—121no—
415C22#591211yescharacteristic
416C22D4×S411yescharacteristic
417C22D4×S411yescharacteristic
418C22C2×#98411yescharacteristic
419C22C2×#98511yescharacteristic
420C22#592411yescharacteristic
421C22#592711yescharacteristic
422C4—22no—
423C4—22no—
424C4—31no—
425C4—31no—
426C4—31no—
427C4—31no—
428C4—41no—
429C4—122no—
430C4—121no—
431C4—121no—
432C4—122no—
433C4—241no—
434C4—241no—
435C3—41noSylow
436C2—241no—
437C2#8529911yescharacteristic
438C2#8533511yescharacteristic
439C2#8536211yescharacteristic
440C2#8537411yescharacteristic
441C2#8537611yescharacteristic
442C2#8540411yescharacteristic
443C2#8540711yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
Fitting#2341D3
prime radicalC22×#41⋊C3C22
hypercenter#95S4
hypocenterSL(2,𝔽3)#121
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊#226C2semidirect
4. C2×C4×#226C2semidirect
6. #73061C2semidirect
7. #73059C2semidirect

Maximal subgroups

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

subgroupindex
C2×#54252
C2×C4⋊#2262
C2×C4×#2262
#2341⋊C32
#730612
#730592
#251513
#46474