Finite Group Databasebeta

← C23×#10374 #19158000

Subgroups — C23×#10374 #19158000

Order 5888.

508 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2549C211yesmaximal, characteristic, complemented
2C23×#1964C211yesmaximal, characteristic
3C23×#1519C211yesmaximal, characteristic
4C2×C23×#327C211yesmaximal, characteristic
5C23×#1294C211yesmaximal, characteristic, complemented
6C23×#1289C211yesmaximal, characteristic, complemented
7C23×#1282C211yesmaximal, characteristic, complemented
8C2×C23×C22⋊Q8—21no—
9C2×C23×#121C2211yescharacteristic
10C2×C23×#99—21no—
11C2×C23×#99—21no—
12C2×C23×C22⋊C8C2211yescharacteristic
13C2×C23×C22⋊C8C2211yescharacteristic
14C23×#398—21no—
15C23×#395—21no—
16C23×#391—21no—
17C23×#389—21no—
18C23×#385—21no—
19C23×#380C2211yescharacteristic
20C23×#380C2211yescharacteristic
21C4×C22⋊C4×C23C2211yescharacteristic
22C2×C23×#95C2211yescharacteristic
23C23×#327—21no—
24C23×#327—21no—
25C23×#327—21no—
26C23×#327—21no—
27C22×Q8×C23—21no—
28C22×D4×C23—21no—
29C22×D4×C23Q811yescharacteristic, complemented
30C23×C4×C23—21no—
31C23×C4×C23C2311yescharacteristic
32C22×C8×C23—21no—
33C22×C8×C23—21no—
34C23×C22⋊Q8—42no—
35C23×C22⋊Q8—42no—
36C23×#121—21no—
37C23×#121—21no—
38C23×#121—21no—
39C23×#121—21no—
40C2×C4⋊C4×C23—21no—
41C2×C4⋊C4×C23—21no—
42C2×C4⋊C4×C23—21no—
43C2×C4⋊C4×C23—21no—
44C2×C4⋊C4×C23—21no—
45C2×C4⋊C4×C23—21no—
46C2×C4⋊C4×C23—21no—
47C2×C4⋊C4×C23Q811yescharacteristic, complemented
48C2×C22⋊C4×C23—21no—
49C2×C22⋊C4×C23—21no—
50C2×C22⋊C4×C23—21no—
51C2×C22⋊C4×C23D411yescharacteristic
52C2×C22⋊C4×C23D411yescharacteristic
53C2×C42×C23—21no—
54C23×#99—42no—
55C23×#99—42no—
56C23×C22⋊C8—21no—
57C23×C22⋊C8—21no—
58C23×C22⋊C8—21no—
59C23×C22⋊C8—21no—
60C23×#95—21no—
61C23×#95—21no—
62C23×#95—21no—
63C23×#95—21no—
64C23×#95—21no—
65C23×#95D411yescharacteristic
66C23×#95D411yescharacteristic
67C23×#95D411yescharacteristic
68C23×#95D411yescharacteristic
69C24×C23—21no—
70C24×C23C2×D411yescharacteristic
71C2×Q8×C23—42no—
72C2×Q8×C23—42no—
73C2×Q8×C23—81no—
74C2×D4×C23—21no—
75C2×D4×C23—21no—
76C2×D4×C23—41no—
77C2×D4×C23—41no—
78C2×D4×C23—41no—
79C2×D4×C23—41no—
80C2×D4×C23—41no—
81C2×D4×C23—41no—
82C2×D4×C23—81no—
83C22×C4×C23—21no—
84C22×C4×C23—21no—
85C22×C4×C23—21no—
86C22×C4×C23—21no—
87C22×C4×C23—21no—
88C22×C4×C23—21no—
89C22×C4×C23—21no—
90C22×C4×C23—42no—
91C22×C4×C23—41no—
92C22×C4×C23—41no—
93C22×C4×C23—41no—
94C22×C4×C23—81no—
95C22×C4×C23—81no—
96C22×C4×C23C2×D411yescharacteristic
97C22×C4×C23C2×D411yescharacteristic
98C22×C4×C23C2×Q811yescharacteristic
99C22×C4×C23#4111yescharacteristic
100C22×C4×C23#4111yescharacteristic
101C22×C4×C23#4111yescharacteristic
102C2×C8×C23—41no—
103C2×C8×C23—41no—
104C2×C8×C23—81no—
105C2×C8×C23—81no—
106C4⋊C4×C23—21no—
107C4⋊C4×C23—21no—
108C4⋊C4×C23—42no—
109C4⋊C4×C23—42no—
110C4⋊C4×C23—42no—
111C4⋊C4×C23—42no—
112C4⋊C4×C23—81no—
113C4⋊C4×C23—81no—
114C4⋊C4×C23—81no—
115C22⋊C4×C23—21no—
116C22⋊C4×C23—21no—
117C22⋊C4×C23—41no—
118C22⋊C4×C23—41no—
119C22⋊C4×C23—41no—
120C22⋊C4×C23—41no—
121C22⋊C4×C23—41no—
122C22⋊C4×C23—41no—
123C22⋊C4×C23D812yes—
124C22⋊C4×C23SD1612yes—
125C42×C23—81no—
126#10374C2311yesmaximal, characteristic, direct product factor, Sylow
127C23×C23—41no—
128C23×C23—41no—
129C23×C23—41no—
130C23×C23—41no—
131C23×C23—41no—
132C23×C23—81no—
133C23×C23—81no—
134C23×C23#12011yescharacteristic
135C23×C23C22⋊Q811yescharacteristic
136C23×C23#12411yescharacteristic
137C23×C23C2×D811yescharacteristic
138C23×C23C2×SD1611yescharacteristic
139C23×C23AGL(1,ℤ/8)11yescharacteristic
140C23×C23#13711yescharacteristic
141Q8×C23—84no—
142D4×C23—81no—
143D4×C23—81no—
144D4×C23—81no—
145D4×C23—81no—
146D4×C23—81no—
147D4×C23—81no—
148C2×C4×C23—22no—
149C2×C4×C23—22no—
150C2×C4×C23—22no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—41no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—41no—
158C2×C4×C23—41no—
159C2×C4×C23—41no—
160C2×C4×C23—41no—
161C2×C4×C23—41no—
162C2×C4×C23—41no—
163C2×C4×C23—41no—
164C2×C4×C23—41no—
165C2×C4×C23—41no—
166C2×C4×C23—41no—
167C2×C4×C23—41no—
168C2×C4×C23—41no—
169C2×C4×C23—41no—
170C2×C4×C23—41no—
171C2×C4×C23—41no—
172C2×C4×C23—41no—
173C2×C4×C23—41no—
174C2×C4×C23—41no—
175C2×C4×C23—82no—
176C2×C4×C23—82no—
177C2×C4×C23—81no—
178C2×C4×C23—81no—
179C2×C4×C23—81no—
180C2×C4×C23C22⋊Q811yescharacteristic
181C2×C4×C23C22⋊Q811yescharacteristic
182C2×C4×C23#12411yescharacteristic
183C2×C4×C23#12411yescharacteristic
184C8×C23—81no—
185C8×C23—81no—
186#2549C2×C2311yescharacteristic, complemented
187#1964C2×C2311yescharacteristic
188#1519C2×C2311yescharacteristic
189C2×#327C2×C2311yescharacteristic
190#1294C2×C2311yescharacteristic, complemented
191#1289C2×C2311yescharacteristic, complemented
192#1282C2×C2311yescharacteristic, complemented
193C22×C23—21no—
194C22×C23—21no—
195C22×C23—21no—
196C22×C23—21no—
197C22×C23—21no—
198C22×C23—21no—
199C22×C23—41no—
200C22×C23—41no—
201C22×C23—41no—
202C22×C23—41no—
203C22×C23—81no—
204C22×C23—81no—
205C22×C23—81no—
206C22×C23—161no—
207C22×C23—161no—
208C22×C23#39311yescharacteristic
209C22×C23#44711yescharacteristic
210C22×C23#44811yescharacteristic
211C22×C23#47411yescharacteristic
212C22×C23#47611yescharacteristic
213C22×C23#48611yescharacteristic
214C22×C23#48911yescharacteristic
215C4×C23—21no—
216C4×C23—21no—
217C4×C23—21no—
218C4×C23—21no—
219C4×C23—42no—
220C4×C23—42no—
221C4×C23—42no—
222C4×C23—81no—
223C4×C23—81no—
224C4×C23—81no—
225C4×C23—81no—
226C4×C23—161no—
227C2×C22⋊Q8—21no—
228C2×#121C22×C2311yescharacteristic
229C2×#99—21no—
230C2×#99—21no—
231C2×C22⋊C8C22×C2311yescharacteristic
232C2×C22⋊C8C22×C2311yescharacteristic
233#398—21no—
234#395—21no—
235#391—21no—
236#389—21no—
237#385—21no—
238#380C22×C2311yescharacteristic
239#380C22×C2311yescharacteristic
240C4×C22⋊C4C22×C2311yescharacteristic
241C2×#95C22×C2311yescharacteristic
242#327—21no—
243#327—21no—
244#327—21no—
245#327—21no—
246C2×C23—21no—
247C2×C23—21no—
248C2×C23—21no—
249C2×C23—21no—
250C2×C23—21no—
251C2×C23—21no—
252C2×C23—161no—
253C2×C23#202311yescharacteristic
254C2×C23#203111yescharacteristic
255C2×C23#203211yescharacteristic
256C22×Q8—21no—
257C22×D4—21no—
258C22×D4Q8×C2311yescharacteristic, complemented
259C23×C4—21no—
260C23×C4C23×C2311yescharacteristic
261C22×C8—21no—
262C22×C8—21no—
263C22⋊Q8—42no—
264C22⋊Q8—42no—
265#121—21no—
266#121—21no—
267#121—21no—
268#121—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4Q8×C2311yescharacteristic, complemented
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4D4×C2311yescharacteristic
281C2×C22⋊C4D4×C2311yescharacteristic
282C2×C42—21no—
283#99—42no—
284#99—42no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95D4×C2311yescharacteristic
295#95D4×C2311yescharacteristic
296#95D4×C2311yescharacteristic
297#95D4×C2311yescharacteristic
298C23#1037411yescharacteristic, direct product factor, Sylow
299C24—21no—
300C24C2×D4×C2311yescharacteristic
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—81no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—81no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—81no—
325C22×C4—81no—
326C22×C4C2×D4×C2311yescharacteristic
327C22×C4C2×D4×C2311yescharacteristic
328C22×C4C2×Q8×C2311yescharacteristic
329C22×C4#41×C2311yescharacteristic
330C22×C4#41×C2311yescharacteristic
331C22×C4#41×C2311yescharacteristic
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—81no—
335C2×C8—81no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—81no—
343C4⋊C4—81no—
344C4⋊C4—81no—
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4D8×C2312yes—
354C22⋊C4SD16×C2312yes—
355C42—81no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—81no—
362C23—81no—
363C23C23×#12011yescharacteristic
364C23C23×C22⋊Q811yescharacteristic
365C23C23×#12411yescharacteristic
366C23C2×D8×C2311yescharacteristic
367C23C2×SD16×C2311yescharacteristic
368C23C23×AGL(1,ℤ/8)11yescharacteristic
369C23C23×#13711yescharacteristic
370Q8—84no—
371D4—81no—
372D4—81no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4C23×C22⋊Q811yescharacteristic
410C2×C4C23×C22⋊Q811yescharacteristic
411C2×C4C23×#12411yescharacteristic
412C2×C4C23×#12411yescharacteristic
413C8—81no—
414C8—81no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—161no—
429C22—161no—
430C22C23×#39311yescharacteristic
431C22C23×#44711yescharacteristic
432C22C23×#44811yescharacteristic
433C22C23×#47411yescharacteristic
434C22C23×#47611yescharacteristic
435C22C23×#48611yescharacteristic
436C22C23×#48911yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—42no—
443C4—42no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—161no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—21no—
454C2—21no—
455C2—161no—
456C2C23×#202311yescharacteristic
457C2C23×#203111yescharacteristic
458C2C23×#203211yescharacteristic
subgroupG/H
center, socleC22×C23#393
commutator, Frattini, cosocleC23×C4C23×C23
prime radicalC2×C23×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2549C2semidirect
5. C23×#1294C2semidirect
6. C23×#1289C2semidirect
7. C23×#1282C2semidirect
29. C22×D4×C23Q8semidirect
47. C2×C4⋊C4×C23Q8semidirect
126. #10374C23direct (N × H)
186. #2549C2×C23semidirect
190. #1294C2×C23semidirect
191. #1289C2×C23semidirect
192. #1282C2×C23semidirect
258. C22×D4Q8×C23semidirect
276. C2×C4⋊C4Q8×C23semidirect
298. C23#10374direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25492
C23×#19642
C23×#15192
C2×C23×#3272
C23×#12942
C23×#12892
C23×#12822
#1037423